proceedings of the conference on analysis, modeling and

Proceedings of theConference on Analysis, Modeling and

Computation of PDE and Multiphase FlowAugust 3-5, 2004

Stony Brook UniversityStony Brook, NY

Xiaolin LiYan YuEditors

c© Draft date September 27, 2005

Contents

Contents i

Preface 1

1 Three-phase Flow in Petroleum Reservoirs 3

2 Stable and Unstable Approximations to Hyperbolic Systems ofConservation laws 15

3 The Glimm Scheme and Compressible Fluid Flows 31

4 Numerical Simulation of Nonisothermal Multiphase Multicompo-nent Flow in Porous Media 71

5 Numerical Simulation of High Mach Number Astrophysical Jets 89

6 A residual-based error estimator for Lagrangian hydrocodes 99

7 Central SchemesCentral Schemes and Central Discontinuous GalerkinMethods on Overlapping Cells 111

8 Dual-Family Viscous Shock Waves in Systems of Conservation Laws:A Surprising Example 125

9 Inviscid and Viscous CFD Modeling of Plume Dynamics in LaserAblation 137

10 Linearized Richtmyer-Meshkov Flow for Elastic Materials 147

i

ii CONTENTS

11 Modeling of cavitating and bubbly flows and applications 159

12 Component-based Adaptive Mesh Control Procedures 169

13 The Hubble Length as a Critical Length Scale in Shock Wave Cos-mology 181

Preface

This proceedings is a collection of original research papers from the conference onAnalysis, Modeling and Computation of PDE and Multiphase Flow, held duringAugust 3-5, 2004. This conference also celebrated the 70th birthday of Dr. JamesGlimm for his important contribution to the field.

Complex and multiphase fluid flows play important role in scientific and en-gineering research. Diverse analytical, numerical and statistical modeling methodshave been developed by researchers in this interdisciplinary field. Emphasis has beenput on the resolution of discontinuous wave structures inside the flow, especially atthe interface between two different phases. Mathematically, such discontinuitiesform the weak solution of the hyperbolic PDE. Numerically, it is essential that suchweak solution structures are respected and well treated to ensure good quality ofcomputational results. The difference between under-solved and well-solved solu-tions could have dramatic impact on the scientific validity of the simulated physicalproblems.

The conference took a retrospective and prospective look at analysis, modelingand computations of multiphase flow, and allow for a solid assessment of the stateof the art. Contributions were invited covering a broad set of applications in areasof analytical and numerical methods for hyperbolic and elliptic PDEs, the trackingand capture methods for moving interface, the mesh generation and adaption forcomputation and the analytical and statistical modeling for the multiphase flow.

The editors express their gratitude to all of the participants in the conference,to the authors of the papers appearing in this proceedings, to the National ScienceFoundation for their generous support through Grant 0422640.

1

2 CONTENTS

Chapter 1

Three-phase Flow in PetroleumReservoirsTransitional shock waves in heterogeneous rock

E. AbreuDepartamento de Modelagem Computacional, Instituto Politecnico/UERJ,Postal Box 97.282, Zip Code 28.601-970, Nova Friburgo, RJ, [email protected]

F. FurtadoDepartment of Mathematics, University of Wyoming,Laramie, WY 82071-3036, [email protected]

D. MarchesinInstituto Nacional de Matematica Pura e Aplicada,Estrada Castorina, 110, Rio de Janeiro, Zip Code 22.460-320, RJ, [email protected]

F. PereiraDepartamento de Modelagem Computacional, Instituto Politecnico/UERJ,Postal Box 97.282, Zip Code 28.601-970, Nova Friburgo, RJ, [email protected]

Abstract: We are concerned with the accurate numerical approximation of thesystem governing three-phase, immiscible displacement in heterogeneous porous me-dia. We describe a two-level operator-splitting procedure for three-phase flows thattakes into account capillary pressure and apply it to indicate the existence of non-

3

4 CHAPTER 1. THREE-PHASE FLOW IN PETROLEUM RESERVOIRS

classical transitional waves in multidimensional flows, thereby extending previousresults for one-dimensional problems. Our numerical procedure combines a secondorder, conservative central difference scheme for a pertinent system of conservationlaws modeling the convective transport of the fluid phases with locally conservativemixed finite elements for the associated parabolic and elliptic problems.

Keywords: Three-phase flow, transitional waves, heterogeneity, conservationlaws, mixed finite elements, central difference scheme

1.1 INTRODUCTION

Prof. James Glimm has introduced a number of innovative ideas and methods oflasting importance for the analysis of the scale-up problem for multiphase flow inmultiscale heterogeneous porous media, Glimm and Sharp, 1991; An et al., 1995;Glimm et al., 1998; Glimm and Sharp, 1998; Glimm and Sharp, 1999; Glimm et al.,2001. He also proposed the study of three-phase flow generalizing Buckley-Leveret’ssolution for two-phase flow. Following his ideas, we are currently investigating theinfluence of rock heterogeneity on three-phase flows, as a first step in a numericalstudy of the scale-up problem for these flows.

Distinct models have been proposed for phase relative permeabilities for three-phase immiscible displacement problems, Corey et al., 1956; Stone, 1970; Dria et al.,1993; Juanes and Patzek, 2004-a; Juanes and Patzek, 2004-b; Juanes and Patzek,2004-c. In this work we adopt the Corey-Pope model (Corey et al., 1956; Dria et al.,1993), which has been used extensively in petroleum engineering. One peculiarityof this model is that the 2× 2 system of first-order PDEs (the saturation equations)which results when capillarity (diffusive) effects are neglected fails to be strictlyhyperbolic somewhere in the interior of the saturation triangle: for a particularstate in the interior of the saturation triangle the characteristic speeds coincide, orresonate. Such a state, whose location is determined by the fluid viscosities, is calledan umbilic point, Isaacson et al., 1992. It plays an important role in three-phaseflow; in particular, its existence leads to the occurrence of nonclassical transitionalshock waves in the solutions of the three-phase flow model. Crucial to calculatingtransitional shock waves is the correct modeling of capillarity effects, Isaacson et al.,1990.

Our new numerical procedure is used to indicate, through a set of high resolu-tion numerical simulations, the existence of transitional waves in multidimensionalflows for some reasonable capillary pressure functions.

1.2. THE MODEL PROBLEM 5

1.2 THE MODEL PROBLEM

We consider two-dimensional, horizontal flow of three immiscible fluid phases in aporous medium. For concreteness, the phases will be referred to as gas, oil andwater, and indicated by the subscripts g, o, and w, respectively. We assume thatthere are no sources or sinks. Compressibility, thermal and gravitational effects areneglected. The governing equations (in dimensionless form) become:

Saturation equations:

∂Sw

∂t+ ∇ · (vfw(Sw, Sg)) = ∇ · ww, (1.1)

∂Sg

∂t+ ∇ · (vfg(Sw, Sg)) = ∇ · wg. (1.2)

Pressure-Velocity equations:

∇ · v = 0, (1.3)

v = −K(x)λ(Sw, Sg)∇po + vwo + vgo. (1.4)

The flux terms ww and wg are given by

[ww,wg]T = K(x)B(Sw, Sg) [∇Sw,∇Sg]

T . (1.5)

Here, [a,b] denotes the 2-by-2 matrix with column vectors a and b, and B(Sw, Sg) =Q(Sw, Sg)P

′(Sw, Sg), where

Q =

⎡⎣ λw(1 − fw) −λwfg

−λgfw λg(1 − fg)

⎤⎦ , P ′ =

⎡⎢⎢⎢⎢⎣∂pwo

∂Sw

∂pwo

∂Sg

∂pgo

∂Sw

∂pgo

∂Sg

⎤⎥⎥⎥⎥⎦ . (1.6)

K is the absolute permeability of the porous medium; Si, ki and µi are, respectively,the saturation, relative permeability, and viscosity of phase i; po is the oil pressure;the correction velocities vwo, vgo, are defined by vij = −K(x)λi(Sw, Sg)∇pij wherepij = pi − pj denote the capillary pressure between phases i and j, i = j, which areexperimentally measured as functions of the saturations. Also, λi = ki/µi denotethe phase mobilities, given in terms of the phase relative permeabilities and phaseviscosities; λ(Sw, Sg) =

∑i λi is the total mobility and fi(Sw, Sg) = λi/λ is the

fractional flow function of phase i. An additional constraint is given by∑

i Si = 1;as a consequence, any pair of saturations inside the saturation triangle, defined by


:= (Si, Sj) : Si, Sj ≥ 0, Si + Sj ≤ 1, i = j, can be chosen to describe thestate of the fluid phases (So, Sw, Sg) in the porous medium. In our model we choose,for convenience, to work with the saturations of water and gas. The diffusive termis represented by the right-hand side of the system (1.1)-(1.2) and it incorporatescapillary pressure effects. For our model one can verify that this term is strictlyparabolic in the interior of the saturation triangle, Azevedo et al., 2002.

We adopt the model by Corey-Pope (Corey et al., 1956; Dria et al., 1993) forphase relative permeabilities: kw = S2

w, ko = S2o and kg = S2

g . For other models ofthree-phase flow used in petroleum engineering, such as certain models of Stone (Stone, 1970), the umbilic point is generally replaced by an elliptic region, in whichthe characteristic speeds are not real.

We also adopt the Leverett and Lewis, 1941 model for capillary pressures givenby pwo = 5ε(2− Sw)(1− Sw) and pgo = 5ε(2− Sg)(1− Sg), where the dimensionlesscoefficient ε controls the relative importance of capillary/dispersive and advectiveforces. In our numerical experiments (see below) we take ε = 1.0 × 10−3.

In our two-dimensional experiments we consider the Riemann problem RPgiven by:

RP =

⎧⎨⎩

SLw = 0.721, SR

w = 0.05,

SLg = 0.279, SR

g = 0.15.(1.7)

The viscosities of the fluids are µo = 1.0, µw = 0.5, and µg = 0.3. Boundary condi-tions for (1.1)-(1.6) must be imposed to complete the definition of the mathematicalmodel. We remark that for the choice of parameters described above a transitionalshock wave appears in the two-dimensional solution of RP .

1.3 THE NUMERICAL SIMULATOR

We employ a two-level operator-splitting procedure (see Abreu et al., 2004) for thenumerical solution of the three-phase flow system. The splitting allows time stepsfor the pressure-velocity calculation that are longer than those for the diffusivecalculation, which are in turn longer than those for advection. Thus, we introducethree time steps: ∆ta for the solution of the hyperbolic problem for the advection,∆td for the diffusive calculation and ∆tp for the pressure-velocity calculation so that∆tp ≥ ∆td ≥ ∆ta. We remark that in practice, variable time steps are always useful,especially for the advection micro-steps subject dynamically to a CFL condition.

The oil pressure and the Darcy velocity are approximated at times tm = m∆tp,m = 0, 1, 2, . . . . Locally conservative mixed finite elements are used to discretizethe pertinent elliptic equation (see Abreu et al., 2004; Douglas et al., 1997).

1.4. THE EFFECT OF HETEROGENEITY ON THREE-PHASE FLOW 7

The saturations Sw and Sg are approximated at times tn = n∆td, n = 1, 2, . . .in the diffusive calculation; recall that they are specified at t = 0. Locally conserva-tive mixed finite elements are used to discretize the spatial operators in the diffusionsystem; the time discretization of the latter is performed by means of the implicitbackward Euler method (see Abreu et al., 2004).

In addition, there are values for the saturations computed at intermediatetimes tn,κ = tn + k∆ta for tn < tn,κ ≤ tn+1 that take into account the advectivetransport of water and gas but ignore the diffusive effects. In these intermediatetimes the system of conservation laws is approximated by a non-oscillatory, secondorder, conservative central difference scheme (see Abreu et al., 2004; Nessyahu andTadmor, 1990).

We refer the reader to Abreu et al., 2004 for a detailed description of ournumerical simulator.

1.4 THE EFFECT OF HETEROGENEITY ON

THREE-PHASE FLOW

In this section we consider two-dimensional numerical experiments.

1.4.1 A grid refinement study

A numerical convergence study is performed for the permeability field displayed inthe top picture of Figure 1.1; the remaining pictures of this figure show saturationsurface plots for the gas phase.

A mixture (72.1 % of water and 27.9 % of gas) is injected at a constant ratealong the left vertical boundary, x = 0 and y ∈ [0, 1], of the computational region.“No-flow” conditions are imposed on the horizontal boundaries: y = 0, Y and x ∈[0, 1]. The initial conditions for the system are given by SR

w = 0.05 and SRg = 0.15;

these data correspond to (1.7).

Two values are specified for the permeability field in the computational region:0.01 in a rectangle that touches the top boundary and 1.0 elsewhere. The computa-tional grids used were 64×64 (middle left picture), 128×128 (middle right picture),256 × 256 (bottom left picture) and 512 × 512 (bottom right picture). Clearly the64 × 64 grid resolves the main features of the flow and captures the transitional(intermediate) wave; as the grid is refined the fronts in the numerical solution getsharper, indicating numerical convergence of our new procedure. Note that spuriousoscillations do not occur in the numerical solutions.


1.4.2 Transitional Waves

Transitional waves have a strong dependency upon the physical diffusion being mod-eled (see Marchesin and Plohr, 1999 and references therein); note that rock hetero-geneity introduces variability in the diffusion term of Eqs. (1.1-1.2). The goal of thenumerical experiments reported below is the investigation of the occurrence of tran-sitional waves in two-dimensional, multiscale heterogeneous problems; such waveshave been observed only in one-dimensional homogeneous problems (Bertozzi et al.,1990; Isaacson et al., 1990; Isaacson et al., 1992; Falls and Schulte, 1992; Marchesinand Plohr, 1999).

As a model for multi-length scale rock heterogeneity we consider scalar, log-normal permeability fields, so that ξ(x) = logK(x) is Gaussian and its distribution isdetermined by its mean and covariance function. We assume that the distribution isstationary, isotropic and fractal (self-similar). Thus the mean is an absolute constantand the covariance is given by a power law: Cov(x,y) = ‖x − y‖−β, β > 0. Thescaling exponent β controls the nature of multiscale heterogeneity: As it increases,the heterogeneities concentrated in the larger length scales are emphasized and thefield becomes more regular (locally). In our simulation studies we also consider log-normal, statistically independent random fields. We refer to such fields by β = ∞.See Furtado et al., 1990 for a discussion of the numerical construction of fractalfields.

The spatially variable permeability fields are defined on 512 × 128 grids withthree values for the coefficient of variation Cv (standard deviation/mean): 0.5, 1.0and 2.0. Cv is used as a dimensionless measure of the permeability field heterogene-ity. The computed fluid flows are defined in a bounded two-dimensional reservoirΩ = [0, X] × [0, Y ] with aspect ratio X/Y = 4, discretized in a uniform grid of 512× 128 cells. Again, a mixture (72.1 % of water and 27.9 % of gas) is injected atconstant rate along the left horizontal boundary, x = 0 and y ∈ [0, Y ], of the compu-tational region and “no-flow” conditions are imposed on the horizontal boundaries:y = 0, Y and x ∈ [0, 1]. The initial conditions for the system correspond to (1.7).

In Figure 1.2 we show gas saturation surface plots, displayed as a function ofposition. Uncorrelated (β = ∞) permeability fields were considered with Cv = 0.5,1.0 and 2.0, from top to bottom. Note in each of these plots the clear presence of atransitional (viscous) shock wave (the intermediate wave).

Correlated (β = 0.5) permeability fields were used in the simulations reportedin Figure 1.3. Again, the values Cv = 0.5, 1.0 and 2.0 were considered (fromtop to bottom). Gas saturation surface plots are shown in Figure 1.3. For weakheterogeneities (Cv = 0.5), the presence of a transitional (intermediate) wave is stillevident. However, the separate identity of this transitional wave and its precursor

1.5. CONCLUDING DISCUSSION 9

Buckley-Leverett wave is dimmed, as the heterogeneity strength increases.

1.5 CONCLUDING DISCUSSION

An accurate, two-level operator splitting technique is introduced for the numericalsolution of three-phase, immiscible displacement problems in heterogeneous porousmedia. The nonlinear advection, diffusion and pressure-velocity problems resultingfrom the splitting are approximated sequentially by a nonoscillatory, second-order,conservative central difference scheme (for advection) and locally conservative mixedfinite elements (for diffusion and pressure-velocity). The numerical tests performedindicate that the two-dimensional simulator is accurate.

The new simulator has been used to uncover the existence of nonclassicaltransitional shock waves in two-dimensional, immiscible three-phase flow in hetero-geneous porous media. Currently the authors are using this simulator to study thescale-up problem for such flows.

ACKNOWLEDGMENTS

E.A. thanks CAPES/Brazil for a fellowship. F.F. was supported by NSF grant INT-0104529. D.M. was supported by CNPq grant 301532/ 2003-6, CNPq/NSF grant690047/01-0, and FAPERJ grant E-26/152.163 /2002. F.P. wishes to acknowl-edge the support of the Brazilian Council for Development of Science and Tech-nology CNPq through Grants 472199/01-3 CTPetro/CNPq, 470216/2003-4 CNPq,504733/2004-4 CTPetro/CNPq, and 690047/01-0 CNPq/NSF.

References

Abreu E., Douglas, Jr., J., Furtado, F., Marchesin, D. and Pereira, F., Three-PhaseImmiscible Displacement in Heterogeneous Petroleum Reservoirs, 2004. Submit-ted to: Journal of Applied Numerical Mathematics. (Available at http://www.labtran.iprj.uerj.br/P

An, L., Glimm, J., Sharp, D. H., and Zhang, Q., 1995, Scale up of flow in porousmedia. In A. P. Bourgeat, C. Carasso, S. Luckhaus, and A. Mikelic, editors, Math-ematical Modelling of Flow Through Porous Media, pp. 26-44. World Scientific,New Jersey.

Azevedo, A., Marchesin, D., Plohr, B. J., and Zumbrun, K., 2002, Capillary insta-bility in models for three-phase flow, Zeit. Angew. Math. Phys, vol. 53, no. 5, pp.


713-746.

Bertozzi, A., Munch, A. and Shearer, M., 1990, Undercompressive shocks in thinflim, Physica D, vol. 134, pp. 431-464.

Corey, A., Rathjens, C., Henderson, J., and Wyllie, M., 1956, Three-phase relativepermeability, Trans. AIME, 207, pp. 349-351.

Douglas, Jr. J., Furtado, F., and Pereira, F, 1997, On the numerical simulation ofwaterflooding of heterogeneous petroleum reservoirs, Computational Geosciences,1, pp. 155-190.

Dria, D. E., Pope, G.A, and Sepehrnoori, K., 1993, Three-phase gas/oil/ brine rela-tive permeabilities measured under CO2 flooding conditions, Society of PetroleumEngineers, SPE 20184, pp. 143-150.

Falls, A. and Schulte, W., 1992, Theory of three-component, three-phase displacementin porous media, SPE Reservoir Eng., vol. 7, pp. 377-384.

Furtado F, Glimm, J., Lindquist, B., and Pereira F., 1990, Multi-length scale calcu-lations of mixing length growth in tracer floods, In Kovarik, F. (ed), Proceedingsof the Emerging Technologies Conference, Texas.

Glimm, J., Hou, S., Lee, Y., Sharp, D. H., Ye, K., 2001, Prediction of oil produc-tion with confidence intervals, SPE 66350, Society of Petroleum Engineers. SPEReservoir Simulation Symposium held in Houston, TX, 11-14 Feb.

Glimm J., Kim, H., Sharp, D. and Wallstrom, T., 1998, A stochastic analysis of thescale up problem for flow in porous medium, Comput. Appl. Math., 17, pp. 67-79.

Glimm, J., and Sharp, D. H., 1991, A random field model for anomalous diffusionin heterogeneous porous media, J. Stat. Phys., 62, pp. 415-424.

Glimm, J., and Sharp, D. H., 1998, Stochastic methods for the prediction of complexmultiscale phenomena, Quarterly J. Appl. Math., 56, pp. 741-765.

Glimm, J., and Sharp, D. H., 1999, Prediction and the quantification of uncertainty,Physica D, 133, pp. 152-170.

Isaacson, E., Marchesin, D., and Plohr, B., 1990, Transitional waves for conservationlaws, SIAM J. Math. Anal., vol. 21, pp. 837-866.

Isaacson, E., Marchesin, D., Plohr, B. and Temple, J. B., 1992, Multiphase flowmodels with singular Riemann problems, Comput. Appl. Math., vol. 11, pp. 147-166.

Juanes, R. and Patzek, T. W., Relative permeabilities for strictly hyperbolic modelsof three-phase flow in porous media, to appear in Transport in Porous Media,2004.


Juanes, R. and Patzek, T. W., Analytical solution to the Riemann problem of three-phase flow in porous media, Transport in Porous Media, 55 (1):47-70, 2004.

Juanes, R. and Patzek, T. W., Three-phase displacement theory: An improved de-scription of relative permeabilities, to appear in SPE Journal, 2004.

Leverett, M. C., and Lewis, W. B., 1941, Steady flow of gas-oil-water mixturesthrough unconsolidated sands, Trans. AIME 142, pp. 107-116.

Marchesin, D. and Plohr, B. J., SPE 56480, 1999, and 2001, Wave structure in WAGrecovery, SPEJ, vol. 6, no. 2, pp. 209-219.

Nessyahu, N., and Tadmor, E., 1990, Non-oscillatory central differencing for hyper-bolic conservation laws, Journal of Computational Physics, pp. 408-463.

Stone, H. L., 1970, Probability model for estimating three-phase relative permeability.JPT, 23(2), pp. 214-218, February 1970. Petrol. Trans. AIME, 249.


0 8 16 24 32 40 48 56 64

08

1624

3240

4856

64

00.075

0.150.225

0.30.375

Y

X

Sg

Figure 1.1: Mesh refinement study in two spatial dimensions. The perme-ability field, shown in the top plot, is piecewise constant with only twodistinct values: 0.01 in the small rectangular region and 1.0 elsewhere.The remaining plots display the computed gas saturations at a fixed timeusing increasingly refined meshes.


Figure 1.2: Numerical solutions of the three-phase flow problem in 2Dheterogeneous formations with Riemann data RP (1.7). From top tobottom, gas saturation surface plots at a fixed time corresponding toCv = 0.5, 1.0 and 2.0, respectively. Uncorrelated permeability fields (β =∞) are considered. The intermediate wave in each of these three plots isa nonclassical transitional viscous shock.


Figure 1.3: Gas saturation surface plots at a fixed time corresponding tocorrelated permeability fields with β = 0.5 and Cv = 0.5, 1.0 and 2.0, re-spectively. The Riemann data is as in the previous figure. A nonclassicaltransitional viscous shock is clearly visible when Cv = 0.5. However, theseparate identity of this wave and its precursor Buckley-Leverett shockis dimmed, as the heterogeneity strength increases.

Chapter 2

Stable and UnstableApproximations to HyperbolicSystems of Conservation laws

Alberto BressanDepartment of Mathematics, Penn State UniversityUniversity Park, Pa. 16802, [email protected]

Abstract: For strictly hyperbolic systems in one space dimension, the Glimmscheme provided the first rigorous construction of global weak solutions. Since then,various other approximations methods have been analyzed. Recent results on theirstability and convergence properties are reviewed in this paper.

Keywords: Hyperbolic system of conservation laws, entropy weak solution.

2.1 Introduction

In this brief survey paper we review various approximation methods for hyperbolicconservation laws, and discuss their stability and convergence properties.

A system of conservation laws in one space dimension has the form

ut + f(u)x = 0 . (1.1)

The components of the vector u = (u1, . . . , un) ∈ IRn are the conserved quanti-ties, while the components of the function f = (f1, . . . , fn) : IRn → IRn are the

15

16 CHAPTER 2. APPROXIMATIONS TO CONSERVATION LAWS

corresponding fluxes. For smooth solutions, (1.1) is equivalent to the quasilinearsystem

ut + A(u)ux = 0 , (1.2)

where A(u).= Df(u) is the n × n Jacobian matrix of the flux function f . The

system is strictly hyperbolic if this Jacobian matrix has n real distinct eigenvalues,λ1(u) < · · · < λn(u) for every u ∈ IRn. In this case, each matrix A(u) admits a basisof eigenvectors r1(u), . . . , rn(u). Most of the earlier literature has been concernedwith systems having the additional property

(H) For each i = 1, . . . , n, the i-th field is either genuinely nonlinear, so thatDλi(u) · ri(u) > 0 for all u, or linearly degenerate, with Dλi(u) · ri(u) = 0 forall u.

In 1965, J. Glimm introduced his famous approximation scheme [20], and gavea rigorous proof of its convergence. This provided the first result on global existenceof weak solutions to the Cauchy problem for the system (1.1). In addition, severalother approximation methods have now been studied. In particular:

(i) Front tracking.

(ii) Vanishing viscosity.

(iii) Relaxation approximations.

(iv) Semi-discrete schemes.

(v) Fully discrete numerical schemes.

In all cases, the theoretical analysis of these methods still relies on some ofthe ideas introduced in Glimm’s pioneering work [20]. In the following sections, weshall review these various approximation techniques, discussing their stability andtheir convergence to the solution of (1.1).

Throughout this paper, we shall be concerned with the Cauchy problem

ut + f(u)x = 0 , u(0, x) = u(x), (1.3)

assuming that the system is strictly hyperbolic and that the initial data u has smalltotal variation. According to the analysis in [28, 9], the Cauchy problem (1.3) has aunique, global weak solution which satisfies the Liu entropy admissibility condition[26]. Moreover, this solution depends Lipshitz continuously on the initial data, inthe L1 norm [9, 10, 11, 13]. For a general introduction to systems of conservationlaws, see one of the monographs [10, 18, 21, 33, 34]. We remark that, in the casewhere the total variation is large, solutions can actually blow up in finite time [22].For physically relevant systems, the validity of uniform BV bounds and the globalexistence of weak solutions remains a difficult open problem.

2.2. THE GLIMM SCHEME 17

2.2 The Glimm scheme

For simplicity, we discuss here a version of the Glimm scheme, in the case where allcharacteristic speed satisfy λi(u) ∈ [0, 1]. The general case can be easily recoveredby a linear change in the t-x coordinates. To construct an approximate solution uε

of the Cauchy problem (1.3), we consider a grid in the t-x plane with equal timeand space steps ∆t = ∆x = ε. Moreover, we let (θk)k≥0 be a sequence of numberswithin the interval [0, 1].

On the initial strip 0 ≤ t < ε, the function uε is defined as the exact solutionof (1.1) with initial condition

uε(0, x) = u((j + θ0)ε

)if jε < x < (j + 1)ε .

Now assume that uε has been constructed for 0 ≤ t < kε. At time t = kε, thefunction u(kε− , ·) is replaced by a piecewise constant function, having jumps exactlyat the nodes of the grid. This is achieved by a random sampling procedure, defining

uε(kε, x) = uε(kε− , (j + θk)ε

)if jε < x < (j + 1)ε . (2.1)

On the strip kε ≤ t < (k + 1)ε, uε is defined as the exact solution of (1.1) withstarting condition (2.1). By induction, using suitable a-priori bounds on the totalvariation, the approximate solution uε can be constructed for all t ≥ 0.

Repeating this construction with the same values θk but letting the mesh sizeε → 0, one obtains a sequence of approximate solutions uε. By compactness, thereexists a subsequence which converges to some limit function u in L1

loc. If the valuesθk are uniformly distributed, it was proved in [27] that u is a weak solution of (1.3).We recall that the sequence (θk)k≥0 is uniformly distributed on [0, 1] if

limN→∞

#k ; 0 ≤ k < N , θk ∈ [0, λ]

N

= λ for every λ ∈ [0, 1] . (2.2)

The rate of convergence of these approximations heavily depends on the rateat which the sequence θk approximates the uniform distribution. For example, wecould choose an irrational number ω and define

θk.= kω − [[kω]] ,

where [[s]] denotes the integer part of a number s. Very different convergence ratescan be here obtained, depending on the choice of ω.

A nearly optimal rate can be achieved by the sequence

θ0 = 0 , θ1 = 0.1 , . . . , θ362 = 0.263 , . . . , θ40775 = 0.57704 , . . . (2.3)


In other words, to construct the number θk ∈ [0, 1], we write the decimal digitsof the integer k in inverse order, then put a zero in front. In connection with thespecific sequence (2.3), the following estimate was proved in [14].

Theorem 1. Given any initial condition u with small total variation, call uexact bethe unique entropy weak solution of the Cauchy problem (1.3). Let uGlimm be thecorresponding Glimm approximation, obtained using the sequence θk at (2.3) and anequally spaced grid with step size ∆x = ∆t. Then, for every T > 0 one has

lim∆x→0

∥∥uGlimm(T, ·) − uexact(T, ·)∥∥

L1√∆x | ln ∆x|

= 0 . (2.4)

The limit (2.4) is uniform w.r.t. u, as long as Tot.Var.u remains uniformly small.

In other words, the error in the approximation, measured in the L1 norm,converges to zero at a rate slightly slower that

√∆x, where ∆x = ∆t is the mesh of

the grid.

2.3 Front tracking approximations

The first example of front tracking algorithm was introduced by C. Da-fermos [17].The technique has now been extended to fully general n × n systems [1, 2]. Bythis technique, one obtains a piecewise constant approximate solution to (1.3), withjumps along a finite number of straight lines in the t-x plane. The constructionstarts by taking a piecewise constant approximation of the initial data u. At eachpoint of jump, we approximately solve the Riemann problem by a piecewise constantfunction, having jumps along a finite number of straight lines. By piecing togetherthese Riemann solutions, one obtains a piecewise constant approximation, definedon some initial time interval. At the first time t1 where two or more wave frontsinteract, we again solve the Riemann problem within a class of piecewise constantfunctions. The solution is then prolonged up to the next time t2 where two wavefronts meet, and so on. In the case of n×n systems, some technical provisions mustbe added, to prevent the number of fronts becoming infinite in finite time. For adetailed analysis of this method, see [1, 21].

The front tracking method shares many features in common with the Glimmscheme. Solutions to the general Cauchy problem are constructed by piecing to-gether Riemann solutions, and the total variation is estimated by means of thesame functionals as in [20]. The major difference is that now the Riemann prob-lems are not solved on fixed grid, but at points of interaction, determined by the

2.4. VANISHING VISCOSITY 19

particular solution at hand. Moreover, no restarting procedure is needed, and theapproximate solution t → u(t, ·) is continuous as a function of time with values inL1

loc(IR). Thanks to these additional properties, error estimates for front trackingapproximations are much easier to obtain than for the Glimm scheme. We describehere the main result. On a domain D of functions with suitably small total variation,it is well known that the evolution equation (1.1) generates a continuous semigroup.More precisely, calling u(t, ·) = Stu the solution corresponding to the initial data u,the semigroup S : D × [0,∞[ → D satisfies an estimate of the form∥∥Ssu− Stv

∥∥L1 ≤ L ‖u− v‖L1 + L′ |s− t| , (3.2)

for some Lipschitz constants L,L′. At any time τ ≥ 0, the distance between afront tracking approximation w(τ, ·) and the exact solution Sτ u of (1.3) can now beestimated by the formula∥∥w(τ) − Sτ u

∥∥L1

≤ L ·∥∥w(0) − u

∥∥L1

+L ·∫ T

0

(lim suph→0+

∥∥w(t+h)−Shw(t)

∥∥L1

h

)dt.

(3.3)

Here L is the Lipschitz constant of the semigroup w.r.t. the initial data, introducedat (3.2). According to (3.3), the error at time τ is bounded by the sum of the initialerror plus the integral of a running error, all amplified by the factor L. Notice that,if w is a front tracking approximation, the instantaneous error rate

E(t).= lim sup

h→0+

∥∥w(t+ h) − Shw(t)∥∥

L1

h

is computed by a finite sum, ranging over all the fronts of w(t). See [10] for details.

2.4 Vanishing viscosity

Adding a small viscosity to the system (1.2), one obtains the parabolic problem

uεt + A(uε)uε

x = ε uεxx , uε(0, x) = u(x). (4.1)

If the initial data u has small total variation, the analysis in [9] has shown that thecorresponding solutions uε of (1.4) have uniformly bounded variation and convergeto the unique solution of the hyperbolic system as ε → 0 satisfying the generalentropy admissibility conditions introduced by T. P. Liu in [26].

Theorem 2. Assume that the matrices A(u) are strictly hyperbolic, smoothly de-pending on u in a neighborhood of the origin. Then there exist constants C,L, L′


and δ > 0 such that the following holds. If

‖u‖BV.= Tot.Var.u + ‖u‖L∞ < δ, (4.2)

then for each ε > 0 the Cauchy problem (4.1) has a unique solution uε, defined forall t ≥ 0. Adopting a semigroup notation, this will be written as t → uε(t, ·) .

= Sεt u.

In addition, one has

BV bounds : Tot.Var.Sε

t u≤ C Tot.Var.u . (4.3)

L1 stability :∥∥Sε

t u− Sεt v∥∥

L1 ≤ L∥∥u− v

∥∥L1 , (4.4)

∥∥Sεt u− Sε

s u∥∥

L1 ≤ L′(|t− s| +

∣∣√εt−√εs

∣∣) . (4.5)

Convergence: As ε → 0+, the solutions uε converge to the trajectories of asemigroup S such that∥∥Stu− Ssv

∥∥L1 ≤ L ‖u− v‖L1 + L′ |t− s| . (4.6)

In the conservative case where A(u) = Df(u) for some flux function f : IRn →IRn, the vanishing viscosity limit is a weak solution of the corresponding hyperbolicCauchy problem (1.3), satisfying the Liu admissibility conditions.

In the case where all characteristic fields are genuinely nonlinear, the rate ofconvergence of these viscous approximations has been analyzed in [16]. At any timeτ > 0, the L1 distance between the solution u of the hyperbolic Cauchy problem(1.1) and the approximations uε can be estimated by∥∥uε(τ, ·) − u(τ, ·)

∥∥L1 = O(1) · (1 + τ)

√ε| ln ε| Tot.Var.u . (4.7)

As usual, the Landau order symbol O(1) denotes some uniformly bounded quantity.

We remark that the above analysis applies only to parabolic systems with”artificial viscosity”, i.e. in the case where the viscosity matrix is the identity. Ex-tending these results to more general vanishing viscosity approximations remains achallenging task. In particular one would like to understand if uniform BV estimatescan hold for systems with ”physical viscosity”, having the form

uεt + A(uε)uε

x = ε(B(uε)uε

x

)x.

Here the diffusion matrix B = B(u) can be any positive semi-definite matrix. Forsome results in this direction, see [5]. An extension of Theorem 2, valid for solutionsto the initial-boundary value problem on the half line, was recently proved in [8].

2.5. RELAXATION APPROXIMATIONS. 21

2.5 Relaxation approximations.

In connection with the general n × n system of conservation laws (1.1), in [23] Jinand Xin introduced the semilinear system with source

ut + vx = 0vt + Λ2 ux = ε−1

[f(u) − v

] (5.1)

Here u, v ∈ IRn. Letting ε → 0+, one expects that the second equation will forcev = f(u) in the limit. In turn, the solution to the first equation should thus approachthe one in (1.1). Based on physical considerations, one might also replace (5.1) bymore general relaxation systems of the form

ut + vx = 0 ,vt + g(u)x = ε−1

[f(u) − v

],

(5.2)

see [31] for a survey of recent literature. For a system of equations modelling chro-matography, the first proof of a priori BV bounds and convergence for relaxationapproximations appeared in [15]. Very recently, a breakthrough in the understandingof the Jin-Xin relaxation approximations was achieved by S.Bianchini [7]. Assumethat, by a linear transformation of variables, the system (5.1) can be written in theform

ut + vx = 0 ,vt + ux = ε−1

[f(u) − v

].

(5.3)

This is equivalent to the second order equation

ut + A(u)ux = ε(uxx − utt) , (5.4)

where A(u) = Df(u) is the Jacobian matrix of f . A special case of the result in [7]can be stated as follows.

Theorem 3. Assume that the eigenvalues of each matrix A(u) = Df(u) satisfy

−1 < λ1(u) < · · · < λn(u) < 1 . (5.5)

Then there exists δ > 0 such that the following holds. Let u be any smooth functionwith ‖u‖BV ≤ δ . Then the Cauchy problem (5.4) with initial data

u(0, x) = u(x) , ut(0, x) = 0

has a unique, globally defined solution uε. Letting ε→ 0+, one has the convergenceuε → u in L1

loc, where u is the unique entropy weak solution to the hyperbolic problem(1.3).


2.6 Semi-discrete numerical schemes

Starting with the hyperbolic system (1.1), one can define an approximation by dis-cretizing the space variable, and still keeping time as a continuous variable. Moreprecisely, fix a step size ∆x and define

xj.= j∆x , uj(t)

.= u(t, xj) j ∈ ZZ .

Replacing the first order spatial derivative with a finite difference, we obtain aninfinite system of O.D.E’s for the functions uj. For example, assuming that all char-acteristic speeds λi(u) are strictly positive, we can define the upwind approximationscheme

d

dtuj(t) =

f(uj−1(t)

)− f

(uj(t)

)∆x

j ∈ ZZ . (6.1)

In connection with the Cauchy problem (1.3), it is natural to impose the initial data

uj(0) = u(j∆x) . (6.2)

Letting ∆x→ 0+, one expects to recover solutions of the original problem (1.3), inthe limit. This was indeed proved by Bianchini in [6]. In the following, for a fixed∆x, given

(uj(t)

)j∈ZZ

we define the interpolated function

u∆x(t, x) = θuj+1(t) + (1 − θ)uj(t) if x = (j + θ) ∆x , θ ∈ [0, 1] . (6.3)

The main result in [6] is as follows.

Theorem 4. Assume that the system (1.1) is strictly hyperbolic, and that all char-acteristic speeds are positive. Then there exists δ > 0 such that the following holds.If ‖u‖BV ≤ δ, then for all ∆x > 0 the Cauchy problem (6.1)-(6.2) admits a uniqueglobal solution t →

(uj(t)

)j∈ZZ

. The corresponding function u∆x(t, ·) has uniformly

bounded total variation. Letting ∆x → 0, one has the convergence u∆x → u, whereu is the unique entropy weak solution to the Cauchy problem (1.3).

7 - Fully discrete numerical schemes.

For computational purposes, fully discrete numerical schemes are certainly themost important class of approximations. In this case, we discretize both space andtime, and construct a grid in the t-x plane with mesh ∆t,∆x. An approximatesolution Uk,j ≈ u(k∆t , j∆x) is obtained by replacing partial derivatives in (1.1)with finite differences. For example, if for all u the eigenvalues of the Jacobianmatrix Df(u) satisfy ∣∣λi(u)

∣∣ < ∆x/∆t i = 1, . . . , n ,

2.6. SEMI-DISCRETE NUMERICAL SCHEMES 23

one can use the Lax-Friedrichs scheme

Uk+1,j = Uk,j +∆t

2∆x

[f(Uk,j−1) − f(Uk,j+1)

]. (7.1)

In the case where

0 < λi(u) < ∆x/∆t i = 1, . . . , n ,

one can also use the upwind Godunov scheme

Uk+1,j = Uk,j +∆t

∆x

[f(Uk,j−1) − f(Uk,j+1)

]. (7.2)

As for all previous methods, it is natural to expect that, if the initial datahas small total variation, then the approximations constructed by finite differenceschemes will have uniformly small variation, for all positive times. Surprisingly, thisis not true. Indeed, the analysis in [3], has brought to light a subtle mechanismfor instability of fully discrete schemes, due to possible resonances between thespeed of a shock and the ratio ∆x/∆t in the mesh of the grid. As shown bythe counterexample in [4], these resonances can prevent the validity of a priori BVbounds and the L1 stability, for these approximate solutions.

We remark that all previous results about BV stability for viscous, semidis-crete, and relaxation approximations, relied on the local decomposition of a solutionas a superposition of travelling wave profiles. To implement this approach, it isessential to work with a center manifold of travelling profiles smoothly dependingon parameters. For the difference schemes (7.1) or (7.2), a travelling profile withspeed σ is a continuous function U = U(ξ) such that the assignment

Uk,j = U(j∆x− σ k∆t)

provides a solution to the equation (7.1) or (7.2), respectively. The existence ofdiscrete travelling profiles was proved by Majda and Ralston [30] in the case ofrational wave speeds, and by Liu and Yu [29] in the case of irrational, diofantinespeeds. However, as remarked by Serre [32], these discrete profiles cannot dependcontinuously on the wave speed σ. In particular, for general n×n hyperbolic systems,no regular manifold of discrete travelling profiles exists.

A detailed example, showing how continuous dependence fails for Lax-Friedrichswave profiles, was constructed in [3]. The same ideas apply to the Godunov schemeas well. Consider a 2 × 2 strictly hyperbolic system of the form

ut + f(u)x = 0vt + 1

2vx + g(u)x = 0

(7.3)


We assume that f ′(u) ≈ 2/3, so that the system is strictly hyperbolic, with charac-teristic speeds contained inside the interval [0, 1]. To fix the ideas, let ∆t = ∆x = 1.The Godunov approximation then takes the form

uk+1,j = uk,j +[f(uk,j−1) − f(uk,j)

], (7.4)

vk+1,j =vk,j−1 + vk,j

2+ gk,j , (7.5)

where

gk,j = g(uk,j−1) − g(uk,j) .

Notice that the u-component of the solution satisfies a scalar difference equation.Moreover, the v-component satisfies a linear difference equation with source termsgk,j derived from the first equation. The solution of (7.5) can be explicitly computedin terms of binomial coefficients. Namely

vm,i =∑

0≤k<m, i−(m−k)≤j≤i

B(m− k, i− j) gk,j (7.6)

with

B(m, ) =m!

! (m− )!· 2−m . (7.7)

Assume that the u-component is a travelling shock profile connecting the statesu−, u+, with speed σ ∈ ]0, 1[ . We are interested in the oscillations of the mapi → vm,i, for large m. To achieve a better understanding of the solution (7.6), twoapproximations will be performed.

(1) Relying on the central limit theorem, we can replace the binomial coefficients(7.7) with a Gaussian kernel.

(2) Choosing a function g such that g′(u) = 0 except for u in a small neighborhoodof (u+ + u−)/2, we can assume that gk,j is nonzero only at the integer points(k, j) immediately to the left of the line x = σt. More precisely,

gn,j =

1 if j = [[σn]]

0 otherwise.(7.8)

After a linear rescaling of variables, instead of (7.6), (7.8), we are thus led tostudy the heat equation with point sources

vt − vxx = δn,[[σn]] . (7.9)


In turn, the above equation can be compared with

vt − vxx = δn,σn . (7.10)

Notice that in (7.10) the point sources occur at integer times, along the straightline x = σt. On the other hand, in (7.9) these sources are located at the point withinteger coordinates, immediately to the left of the line x = σt.

For the solution of (7.10), by repeated integration by parts one can show thatdownstream oscillations are rapidly decreasing, i.e.∣∣vx(t, σt− y)

∣∣ = O(1) · y−k

as y → ∞, for all k ≥ 1. The estimate of downstream oscillations for (7.9) can nowbe achieved by a comparison with (7.10). In particular, if the speed of the sourceis close to, but not exactly rational, resonances will occur. For example, assumethat σ = 1 + ε, with 0 < ε << 1. Calling Ψε(y) the solution profile at a distance ydownstream from the shock, we find

Ψε(y).=

∑n≥1

G(n, y + [[σn]]

)=

∑n≥1

G(n, y + σn

)−

∑n≥1

[G(n, y + σn

)−G

(n, y + [[σn]]

)]

≈ 1

σ−

∑n≥1

Gx(n, y + σn)(σn− [[σn]]

)≈ 1

σ−

∫ ∞

0

Gx(t, y + σt)(εt− [[εt]]

)dt

(7.11)

A direct analysis of the last integral in (7.11) shows that nontrivial oscillationsoccur when y = O(1) · ε−2. More precisely, for y ∈ Iε

.= [ε−2/2, ε−2], the function

y → Ψε(y) oscillates several times. The amplitude of each oscillation is ≥ c0 ε, andthe distance between a peak and the next one is O(1) · ε−1. Therefore, the totalvariation of Ψε on the interval Iε remains uniformly positive, as ε→ 0. In particular,the family of profiles Ψε cannot converge in the BV norm, as ε → 0. This alreadyexplains why the discrete travelling profiles do not depend smoothly on the wavespeed.

Motivated by the previous analysis, in [4] an example is constructed where thetotal variation of the solution computed by the Godunov scheme becomes arbitrarilylarge, for large times. The hyperbolic system has again the form

ut +(ln(1 + eu)

)x

= 0

vt +1

2vx + g(u)x = 0

u(0, x) = u(x)v(0, x) = v(x)

(7.12)


The special choice of the flux function for the u-component is motivated by anobservation of P. Lax [24]. If zn,j > 0 provide a solution to the linear differenceequation

zn+1,j =zn,j + zn,j−1

2, (7.13)

then the nonlinear transformation

un,j = ln

(zn,j−1

zn,j

).

provides a solution to

un+1,j = un,j +(

ln(1 + eun,j−1

)− ln

(1 + eun,j

))Notice the similarity of this formula with the Hopf-Cole transformation, valid forthe viscous Burgers’ equation.

Explicit solutions for (7.13) are easy to construct. In particular, for any ξ > 0,the function

z(t, x) = e−ξ [x−σ(ξ) t] σ(ξ).=

ln(1 + eξ) − ln 2

ξ,

provides a discrete travelling wave solution. Namely, zm,j = z(m, j) solves (7.13).More generally, since the equation is linear homogeneous, any integral combinationof the form

z(t, x) =

∫a(ξ) e−ξ [x−σ(ξ)t] dξ (7.14)

provides yet another solution. By carefully choosing the coefficient a(ξ) in (7.14),we can construct a discrete approximation to a solution u = u(t, x) having a shocklocated along a prescribed curve x = γ(t). The main result which follows from theanalysis in [4] can be stated as follows.

Let 0 < u+ < u− be the right and left states of a shock for the scalar conservationlaw

ut +[ln(1 + eu)

]x

= 0 ,

having a rational speed σ ∈ ] 1/2 , 1[ . Then there exists a smooth flux function gand a sequence of smooth perturbations φν : IR → IR such that

Tot.Var.φν → 0 , ‖φν‖Ck → 0 as ν → ∞

for all k ≥ 1, and such that the following holds. The Godunov approximations(uν , vν) to the Cauchy problem for (7.12) with initial data

uν(0, x) =

u− + φν(x) if x < 0 ,

u+ if x > 0 ,vν(0, x) = 0 ,


satisfy

Tot.Var.vν(Tν , ·)

→ ∞ as ν → ∞ ,

for some sequence of times Tν → ∞.

Notice that in this case the total variation of the exact solutions would remainuniformly bounded for all times. The small perturbations φν slightly change thespeed of the shock in the u-component of the solution, thus producing a resonancewith the grid. Over a large interval of time, this determines an arbitrarily largeamount of downstream oscillations in the numerically computed solution.

Counterexamples of this type point out a basic limitation of rigorous theoreticalanalysis: For exact solutions, all basic existence-uniqueness theorems are based ona priori bounds on the total variation. On the other hand, any attempt to analyzefinite difference schemes cannot rely on a priori BV bounds or L1 stability estimates,simply because these are not valid.

Our personal understanding is that these oscillations in the numerical approx-imations are “mild”, in the sense that they are spread out over a large number ofgrid points and should not prevent the convergence to the exact solution. Moreover,they are “rare”, in the sense that they occur only for a very small set of initial data.However, no rigorous result is yet known in this direction.

At present, positive results on the stability and convergence of numericalschemes for systems of conservation laws are known in two main cases:

1. For the 2 × 2 system modelling isentropic gas dynamics, convergence of a sub-sequence of finite difference approximations has been proved in [19], by the methodof compensated compactness.

2. For Temple class systems, and more generally for n× n systems where all shockcurves are straight lines, uniform BV bounds, stability and convergence of numericalapproximations were proved in [25] and in [12], respectively.

Establishing the convergence of any finite difference scheme, in connection withgeneral n× n hyperbolic systems, remains an outstanding open problem.

References

[1] F. Ancona and A. Marson, A front tracking algorithm for general nonlinearhyperbolic systems, Preprint, 2004.


[2] P. Baiti and H. K. Jenssen, On the front tracking algorithm, J. Math. Anal. Appl.217 (1998), 395-404.

[3] P. Baiti, A. Bressan and H. K. Jenssen, Instability of travelling wave profiles forthe Lax-Friedrichs scheme Discr. Cont. Dynam. Syst., submitted.

[4] P. Baiti, A. Bressan and H. K. Jenssen, An instability of the Godunov scheme,to appear.

[5] S. Bianchini, Interaction estimates and Glimm functional for general hyperbolicsystems, Discr. Cont. Dynam. Syst. 9 (2003), 133-166.

[6] S. Bianchini, BV solutions of the semidiscrete upwind scheme. Arch. Rat. Mech.Anal. 167 (2003), 1-81.

[7] S. Bianchini, Hyperbolic limit of the Jin-Xin relaxation model, preprint IAC,Rome 2004.

[8] S. Bianchini and F. Ancona, Vanishing viscosity solutions for general hyperbolicsystems with boundary, preprint IAC-CNR, Rome, 2004.

[9] S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolicsystems, Annals of Mathematics (2005), to appear.

[10] A. Bressan, Hyperbolic Systems of Conservation Laws. The One DimensionalCauchy Problem, Oxford University Press, 2000.

[11] A. Bressan, G. Crasta and B. Piccoli, Well posedness of the Cauchy problemfor n× n conservation laws, Amer. Math. Soc. Memoir 694 (2000).

[12] A. Bressan and H. K. Jenssen, On the convergence of Godunov scheme fornonlinear hyperbolic systems, Chinese Ann. Math. B - 21 (2000), 1-16.

[13] A. Bressan, T. P. Liu and T. Yang, L1 stability estimates for n×n conservationlaws, Arch. Rat. Mech. Anal. 149 (1999), 1-22.

[14] A. Bressan and A. Marson, Error bounds for a deterministic version of theGlimm scheme, Arch. Rational Mech. Anal. 142 (1998), 155-176.

[15] A. Bressan and W. Shen, BV estimates for multicomponent chromatographywith relaxation. Discr. Contin. Dynam. Systems, 6 (2000), 21-38.

[16] A. Bressan and T. Yang, On the convergence rate of vanishing viscosity approx-imations, Comm. Pure Appl. Math. 57, 1075-1109.

[17] C. Dafermos, Polygonal approximations of solutions of the initial value problemfor a conservation law, J. Math. Anal. Appl. 38 (1972), 33-41.

[18] C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin 1999.


[19] X. Ding, G. Q. Chen and P. Luo, Convergence of the fractional step Lax-Friedrichs scheme for the isentropic system of gas dynamics, Comm. Math. Phys.121 (1989), 63-84.

[20] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations,Comm. Pure Appl. Math. 18 (1965), 697-715.

[21] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws,Springer Verlag, New York 2002.

[22] H. K. Jenssen, Blowup for systems of conservation laws, SIAM J. Math. Anal.,31 (2000), 894-908.

[23] S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws inarbitrary space dimensions, Comm. Pure Appl. Math. 48 (1955), 235-277.

[24] Lax, P.: Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math.10 (1957), 537-566.

[25] R. LeVeque and B. Temple, Stability of Godunov’s method for a class of 2 × 2systems of conservation laws. Trans. Amer. Math. Soc. 288 (1985), 115-123.

[26] T. P. Liu, The entropy condition and the admissibility of shocks, J. Math. Anal.Appl. 53 (1976), 78-88.

[27] T. P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys.57 (1977), 135-148.

[28] T. P. Liu, Admissible solutions of hyperbolic conservation laws, Amer. Math.Soc. Memoir 240 (1981).

[29] T. P. Liu and S. H. Yu, Continuum shock profiles for discrete conservation lawsI. Construction. Comm. Pure Appl. Math. 52 (1999), 85–127.

[30] A. Majda and J. Ralston, Discrete shock profiles for systems of conservationlaws. Comm. Pure Appl. Math. 32 (1979), 445-482.

[31] R. Natalini, Recent results on hyperbolic relaxation problems, in Analysis ofSystems of Conservation Laws, H. Freisthuler Ed., Chapman & Hall/CRC, 1998,pp.128-198.

[32] D. Serre, Remarks about the discrete profiles of shock waves. Mat. Contemp.11 (1996), 153-170.

[33] D. Serre, Systems of Conservation Laws 1, 2, Cambridge University Press, 2000.

[34] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag,New York, 1983.

Chapter 3

The Glimm Scheme andCompressible Fluid FlowsExistence and Stability of Multidimensional Shock Waves and SupersonicVortex Sheets

Dedicated to James Glimm on the occasion of his 70th birthday

Gui-Qiang ChenDepartment of Mathematics, Northwestern University2033 Sheridan Road, Evanston, IL 60208-2730, [email protected]

Abstract: The Glimm scheme has played an essential role in the developmentof the BV theory for one-dimensional hyperbolic systems of conservation laws. Inthis paper, we analyze some of further successful applications of the Glimm schemeand related methods to nonlinear stability issues for two-dimensional steady super-sonic flows in various physical problems. These applications especially include thenonlinear stability of two-dimensional shocks in steady supersonic Euler flows pastinfinite nonsmooth wedges under a BV perturbation of the obstacle and the nonlin-ear stability of supersonic vortex sheets in steady supersonic Euler flows past infinitenonsmooth walls under the BV perturbation of the boundary. Some related issuesregarding the existence and stability of multidimensional transonic shocks in severalphysical problems are discussed, and an analytical framework for entropy solutionswithout bounded variations for hyperbolic conservation laws is also presented.

Keywords: Glimm scheme, interaction, reflection, Glimm functional, existence,

31

32 CHAPTER 3. THE GLIMM SCHEME

stability, shock, vortex sheets, compressible, Euler equations, conservation laws, su-personic, transonic, free boundary problems, iteration, partial hodograph, frame-work, divergence-measure fields, entropy inequality, entropy-entropy flux, Whitneyparadox, Gauss-Green formula, traces.

Introduction

The Glimm scheme, introduced by James Glimm in 57 in 1965, has played an es-sential role in the development of the BV theory for one-dimensional hyperbolicsystems of conservation laws. In Section 1, we first briefly describe the Glimmscheme and related methods, analyze how these methods have been employed toestablish the BV theory for one-dimensional hyperbolic systems of conservationlaws, and discuss how the Glimm’s ideas have motivated the development of the L∞

theory for the isentropic Euler equations in gas dynamics. Then we analyze someof recent further successful applications of the Glimm scheme and related methodsto nonlinear stability issues for two-dimensional steady supersonic flows in variousphysical problems. These applications especially include the nonlinear stability oftwo-dimensional shocks in steady supersonic Euler flows past infinite nonsmoothwedges under a BV perturbation of the obstacle in Section 2 and the nonlinearstability of supersonic vortex sheets in steady supersonic Euler flows past infinitenonsmooth walls under the BV perturbation of the boundary in Section 3. Somerelated issues regarding the existence and stability of multidimensional transonicshocks in several physical problems are discussed in Section 4. An analytical frame-work for entropy solutions without bounded variations for hyperbolic conservationlaws is also presented in Section 5.

3.1 The Glimm Scheme and One-Dimensional Hy-

perbolic Systems of Conservation Laws

Consider the Cauchy problem for one-dimensional hyperbolic systems of conserva-tion laws:

∂tu + ∂xf(u) = 0, u ∈ IRm, x ∈ IR, (3.1.1)

with Cauchy initial data:

u|t=0 = u0(x), (3.1.2)

where f : IRm → IRm is a nonlinear mapping. The hyperbolicity of system (3.1.1)requires that the Jacobian matrix ∇f(u)m×m have m real eigenvalues λi(u), i =

3.1. HYPERBOLIC CONSERVATION LAWS 33

1, 2, · · · ,m, and be diagonalizable. System (3.1.1) is called strictly hyperbolic in adomain D ⊂ IRm if all the eigenvalues are distinct at any state u ∈ D.

One of the main difficulties in dealing with (3.1.1)–(3.1.2) is that solutionsof the Cauchy problem generally develop singularities in a finite time, because ofthe physical phenomena of breaking of waves and the development of shock waves,among others. For this reason, attention focuses on solutions in the space of dis-continuous functions. Therefore, one can not directly use the classical analytictechniques that predominate in the theory of partial differential equations of othertypes. On the other hand, weak solutions are not unique and an additional physicalcondition, the entropy condition, is required to single out physical relevant solu-tions, entropy solutions. That is, an admissible function u(t, x) is called an entropysolution if it satisfies the Lax entropy inequality:

∂tη(u) + ∂xq(u) ≤ 0 (3.1.3)

in the sense of distributions for any C2 convex entropy-entropy flux pair (η, q) :IRm → IR× IR, that is,

∇2η(u) ≥ 0, ∇q(u) = ∇η(u)∇f(u). (3.1.4)

An archetype of hyperbolic systems of conservation laws is the Euler equationsfor compressible fluids in Eulerian coordinates:⎧⎪⎪⎨

⎪⎪⎩∂tρ+ ∂xm = 0,

∂tm+ ∂x

(m2

ρ+ p

)= 0,

∂tE + ∂x

(mρ(E + p)

)= 0.

(3.1.5)

System (3.1.5) is closed by the constitutive relations

p = p(ρ, e), E =1

2

|m|2ρ

+ ρe. (3.1.6)

In (3.1.5)–(3.1.6), τ = 1/ρ is the deformation gradient (specific volume for fluids,strain for solids), v is the fluid velocity with ρv = m the momentum vector, p isthe scalar pressure, and E is the total energy with e the internal energy which isa given function of (τ, p) or (ρ, p) defined through thermodynamical relations. Theother two thermodynamic variables are temperature θ and entropy S. If (ρ, S) arechosen as the independent variables, then the constitutive relations can be writtenas

(e, p, θ) = (e(ρ, S), p(ρ, S), θ(ρ, S)) (3.1.7)

governed by

θdS = de+ pdτ = de− p

ρ2dρ. (3.1.8)


For a polytropic gas,

p = Rρθ, e = cvθ, γ = 1 +R

cv, (3.1.9)

andp = p(ρ, S) = κργeS/cv , e = e(ρ, S) =

κ

γ − 1ργ−1eS/cv , (3.1.10)

where R > 0 may be taken to be the universal gas constant divided by the effectivemolecular weight of the particular gas, cv > 0 is the specific heat at constant volume,γ > 1 is the adiabatic exponent, and κ > 0 can be any positive constant underscaling. The Lax entropy inequality (3.1.3) becomes the Clausius inequality

∂t(ρa(S)) + ∂x(ma(S)) ≥ 0 (3.1.11)

in the sense of distributions for any a(S) ∈ C2 with a′(S) ≥ 0.

The Euler equations for a compressible fluid that flows isentropically take thefollowing simpler form:

∂tρ+ ∂xm = 0,

∂tm+ ∂x (m2/ρ+ p) = 0,(3.1.12)

where the pressure is regarded as a function of density, p = p(ρ, S0), with constantS0. For a polytropic gas,

p(ρ) = κ0ργ, γ > 1, (3.1.13)

where κ0 > 0 is any positive constant under scaling. The case γ = 1 corresponds tothe isothermal fluid.

System (3.1.5) can be rewritten in Lagrangian coordinates in one-to-one cor-respondence so long as the fluid flow stays away from vacuum ρ = 0:⎧⎪⎨

⎪⎩∂tτ − ∂xv = 0,

∂tv + ∂xp = 0,

∂t(e+ v2/2) + ∂x(pv) = 0,

(3.1.14)

where the coordinates (t, x) are the Lagrangian coordinates, which are different fromthe Eulerian coordinates in (3.1.12); for simplicity of notations, we do not distinguishthem.

The Glimm scheme 57 has been an effective nonlinear method to constructentropy solutions in BV for hyperbolic systems of conservation laws. Its novelingredients especially include:


(i). Randomness into finite difference approximate solutions;

(ii). Interaction estimates among nonlinear waves;

(iii). The Glimm functional for the approximate solutions, which leads to therequired uniform BV estimates;

(iv). Compactness arguments in BV for the Glimm approximate solutions.

The impact of the Glimm scheme on mathematics and other sciences has beentremendous, and it is especially a real milestone in the development of mathematicaltheory of hyperbolic systems of conservation laws (cf. 10; 34; 41; 43; 58; 59; 61;66; 75; 97). In particular, the ideas and techniques introduced in 57 and theirfurther developments have completely changed the ways in which the problems wereapproached.

3.1.1 BV Theory for One-Dimensional Hyperbolic Systems

Assume that system (3.1.1) is strictly hyperbolic and each characteristic field iseither genuinely nonlinear or linearly degenerate in a neighborhood of a constantstate u. Then the solution u(t, x) ∈ BV of the Cauchy problem can be obtainedas the limit of the approximate solutions uh(t, x), when h → 0, constructed by theGlimm scheme, as described below.

The Glimm Scheme

Fix h > 0, a space-step size, and choose the corresponding time-step size ∆t = h/Λsatisfying the Courant-Friedrichs-Lewy condition, where Λ is an upper bound ofthe characteristic speeds |λi|, 1 ≤ i ≤ m. Then we partition the upper half-planeIR2

+ := (t, x) : t ≥ 0, x ∈ IR into the strips

Sk = (t, x) : k∆t ≤ t < (k + 1)∆t, x ∈ IR, k ∈ Z+,

and locate the mesh points (k∆t, jh) with k ∈ Z+, j ∈ Z, and j+k even. Choose anyrandom sequence of numbers a = a0, a1, a2, · · · ⊂ (−1, 1) which is equidistributedin (−1, 1) in the following sense: for any subinterval I ⊂ (−1, 1) of length |I|,

liml→∞

2Nl

l= |I| uniformly with respect to I,

where Nl is the number of indices k ≤ l with ak ∈ I. Set the sampling points asP k

j = (k∆t, (j + ak)h) with j + k odd. Then the approximate solution uh(t, x) isdefined by induction on k = 0, 1, 2, · · · in each strip Sk. Define u0

j = u0((j + a0)h)and

ukj = uh(k∆t− 0, (j + ak)h− 0)


for j + k odd and k ≥ 1. Set

uh(k∆t, x) = ukj for x ∈ ((j − 1)h, (j + 1)h), j + k odd.

Define the solution uh(t, x) for t ∈ [k∆t, (k + 1)∆t), x ∈ [(j − 1)h, (j + 1)h], j + keven, as the solution of the Riemann problem of the system with initial data

u|t=k∆t =

uk

j−1, x < jh,

ukj+1, x > jh.

Then uh(t, x) is well defined: it is the exact entropy solution in each strip Sk, itis continuous at the interfaces x = jh, k∆t ≤ t < (k + 1)∆t with j + k odd, andit experiences jump discontinuities across the lines t = k∆t, k = 0, 1, 2, · · · . Thewaves emanating from the neighboring discontinuing mesh points (k∆t, jh) and(k∆t, (j + 2)h), j + k even, do not intersect.

Existence Theory in BV

If it is proved that uh(t, x) is uniformly bounded in h in IR2+, then Λ can be chosen

and the Glimm approximate solutions are constructed for all t ≥ 0. Then thelimit of the approximate solutions is the entropy solution of the Cauchy problem(3.1.1)–(3.1.2) as in the following theorem.

Theorem 3.1.1 There exist two positive constants ε1 and ε2 such that, for initialdata u0 satisfying

‖u0 − u‖L∞(IR) ≤ ε1, TVIR(u0) ≤ ε2, (3.1.15)

the Cauchy problem (3.1.1)–(3.1.2) has a global entropy solution u(t, x) for (t, x) ∈IR2

+ satisfying the entropy inequality (3.1.3) in the sense of distributions for anyconvex entropy-entropy flux pair and

‖u(t, ·) − u‖L∞(IR) ≤ C ‖u0 − u‖L∞(IR), (3.1.16)

TVIR(u(t, ·)) ≤ C TVIR(u0), (3.1.17)

‖u(t1, ·) − u(t2, ·)‖L1(IR) ≤ C |t1 − t2|TVIR(u0), (3.1.18)

for any t, t1, t2 ∈ IR+ and for some constant C > 0.

In order to show that the approximate solutions uh(t, x) converge to a solution ofthe Cauchy problem (3.1.1) and (3.1.2), it is required to establish

(i) The compactness of the approximate solutions in order to ensure that thereexists a subsequence (still denoted by) uh(t, x) converging to u(t, x) a.e. for (t, x) ∈IR2

+;


(ii) The consistency of the scheme in order to guarantee that the limit u(t, x)is indeed an entropy solution of (3.1.1)–(3.1.2).

For the compactness of the Glimm approximate solutions under assumption(3.1.15), the following estimates in BV can be established:

‖uh(t, ·) − u‖L∞(IR) ≤ C‖u0 − u‖L∞(IR), (3.1.19)

TVIR(uh(t, ·)) ≤ C TVIR(u0), (3.1.20)

‖uh(t1, ·) − uh(t2, ·)‖L1(IR) ≤ C(|t1 − t2| + h) TVIR(u0), (3.1.21)

for any t, t1, t2 ∈ IR+ and for some constant C > 0. Estimate (3.1.19) guarantees thatthe approximate solutions uh(t, x) can be constructed globally for all t ∈ IR+ if ε1 in(3.1.15) is sufficiently small. These compactness estimates imply that the family ofapproximate solutions uh(t, x) has uniformly bounded variation and thus convergesalmost everywhere, by the Helly theorem, to a function u(t, x) in BV . It can beshown that, for any equidistributed random sequence a = a0, a1, a2, · · · ⊂ (−1, 1),the limit function u(t, x) is an entropy solution of the Cauchy problem (3.1.1)–(3.1.2), which also satisfies the entropy inequality (3.1.3).

For the compactness estimates, (3.1.21) is an immediate consequence of (3.1.20)since the waves emanating from each mesh point propagate with speed not exceedingΛ. To establish (3.1.20), note first that, for any t ∈ (k∆t, (k + 1)∆t), TVIR(u(t, ·))is constant and can be measured by the sum of the strengths of waves that emanatefrom the mesh points (k∆t, jh) with j + k even. To estimate how the sum of wavestrengths changes from the strip Sk to Sk+1, consider the family of diamond-shapedregions ♦jk, j + k odd, with vertices P k

j , P k+1j+1 , P k+2

j , and P k+1j−1 . A wave fan of m

waves (δ1, · · · , δm) emanates from the mesh point P k+1j inside ♦jk. Through the

side of ♦jk connecting the two vertices P kj and P k+1

j−1 , there crosses a fan of waves(α1, · · · , αm) which is part (possibly none or all, as some of the components αi

could be zero) of the wave fan emanating from the mesh point P kj−1 and, through

the side of ♦jk connecting the two vertices P kj and P k+1

j+1 , there crosses a fan ofwaves (β1, · · · , βm) which is part (possibly none or all, as some of the componentsβi could be zero) of the wave fan emanating from the mesh point P k

j+1. Indeed, thewave fan (δ1, · · · , δm) approximates the wave pattern that would have resulted ifthe wave fans (α1, · · · , αm) and (β1, · · · , βm) had been allowed to propagate beyondt = (k + 1)∆t and thus interact. It can be shown that the strengths of incomingand outgoing waves are related by

m∑i=1

|δi| =m∑

i=1

(|αi| + |βi|) +O(1)Qjk (3.1.22)

with Qjk =∑

i,j|αi||βj| : αi and βj interacting. If the quadratic term Qjk were

not present, the total variation of uh(t, ·), as measured by the strengths of waves,would not increase from Sk to Sk+1.


The effect of the quadratic term can be controlled through the following Glimmfunctional: For the polygonal curve Jk whose arcs connect nodes P k

j , P k+1j−1 , and P k+1

j+1 ,j + k odd, define the Glimm functional associated with the curve Jk as

F(Jk) = L(Jk) +KQ(Jk), (3.1.23)

whereL(Jk) :=

∑|α| : any wave α crossing Jk

is the linear part measuring the total variation,

Q(Jk) :=∑

|α||β| : α, β interacting waves crossing Jk

is the quadratic part measuring the potential wave interaction, and K is a largepositive constant. The functional F(Jk) is well defined and essentially equivalent toTVIR(uh(t, ·)) for k∆t ≤ t < (k + 1)∆t. It can be shown from (3.1.22) that F(Jk)is nonincreasing in k as long as the total variation remains small, which impliesestimate (3.1.20).

For the details of the proof of Theorem 3.1.1, see Glimm 57 and Liu 74; 81;also see Dafermos 43.

The proof of Theorem 3.1.1 is based on the estimate showing that the effect ofinteractions is of second-order for the general system of m conservation laws, thatis, the change in magnitude of waves due to interaction is of second-order in themagnitude of waves before interaction. For a system of two conservation laws, thereexists a coordinate system of Riemann invariants, and the effect of interaction isof third-order, that is, the system is uncoupled modulo the third-order of the totalvariation of the solution. Therefore, Theorem 1 holds for the initial data of smalloscillation but of larger total variation in the case of two conservation laws. This,in particular, applies to the isentropic Euler equations in (3.1.12) away from thevacuum. For the isothermal case, γ = 1, the condition of small oscillation can alsobe removed since the quadratic term in (3.1.22) do not present (see Nishida 86). Forthe full Euler equations in gas dynamics (3.1.14) with the Cauchy data:

(τ, v, S)|t=0 = (τ0, v0, S0)(x), (3.1.24)

The following existence theorem holds which is due to Liu 76 (also see Temple 99and Chen-Wagner 33).

Theorem 3.1.2 Let K ⊂ (τ, v, S) : τ > 0 be a compact set in IR+ × IR2, andlet N ≥ 1 be any positive constant. Then there exists a constant C0 = C0(K,N),independent of γ ∈ (1, 5/3], such that, for every initial data (τ0, v0, S0)(x) ∈ K withTVIR(τ0, v0, S0) ≤ N , when

(γ − 1)TVIR(τ0, v0, S0) ≤ C0


for any γ ∈ (1, 5/3], the Cauchy problem (3.1.14) and (3.1.24) has a global entropysolution (τ, v, S)(t, x) which is bounded and satisfies

TVIR(τ, v, S)(t, ·) ≤ C TVIR(τ0, v0, S0)

for some constant C > 0 independent of γ.

For the isentropic case, the existence result of Theorem 3.1.2 was proved inNishida-Smoller 87 and DiPerna 49. For extensions to the initial-boundary valueproblems, see 78; 88. In the direction of relaxing the requirement of small totalvariation, see Zhang-Guo 107, Ding-Chang-Hsiao-Li-Wang 46, Temple-Young 100;101, and Schochet 94. For extensions of the Glimm scheme to nonhomogeneousbalance laws, see Dafermos-Hsiao 45, Liu 81, and Chen-Wagner 33.

For the decay of entropy solutions in BV with periodic data or compact sup-port, see Glimm-Lax 58, Liu 79; 80, DiPerna 50; 52, Dafermos 43, and Chen-Frid25. For additional further discussions and references about the Glimm scheme, seeDafermos 43, Serre 95, Smoller 97, and Chen-Wang 34.

L1-Stability for Entropy Solutions in BV

The existence proof in Theorem 1 based on compactness arguments does not provideinformation on the stability of solutions to the Cauchy problem (3.1.1)–(3.1.2). Bymonitoring the time evolution of a certain functional, it can be shown that theGlimm solutions depend continuously on their initial data.

Let u(t, x) and v(t, x) be two approximate solutions of (3.1.1) constructed bythe Glimm scheme with small total variation. The main point is to examine howthe distance ‖u(t, ·) − v(t, ·)‖L1(IR) changes in time. Denote by

s → Ri(s)(u−), s → Si(s)(u−), i = 1, · · · ,m,

the i-rarefaction and i-shock curve of (3.1.1) through the state u−, parametrized byarc-length, and set

Υi(s)(u−) =

Ri(s)(u−), s ≥ 0,

Si(s)(u−), s < 0.

For any fixed point (t, x), consider the scalar function qi(t, x), which can be regardedintuitively as the strength of the i-shock wave in the jump (u(t, x),v(t, x)), definedimplicitly by

v(t, x) = Sm(qm(t, x)) · · · S1(q1(t, x))(u(t, x)).


It is clear that

C−11 |u(t, x) − v(t, x)| ≤

m∑i=1

|qi(t, x)| ≤ C1|u(t, x) − v(t, x)|

for some constant C1 > 0. For each i = 1, · · · ,m, define

Wi(t, x) =(∑

−+∑

++∑

0

)(|α(u(t, x))| + |α(v(t, x))|). (3.1.25)

In (3.1.25),∑

− sums the strengths |α(u(t, x))| (and |α(v(t, x))|) of all kα-wavesxα(t) < x of u(t, x) (and v(t, x)) with i < kα ≤ m, respectively;

∑+ sums the

strengths |α(u(t, x))| (and |α(v(t, x))|) of all kα-waves xα > x of u(t, x) (andv(t, x)) with 1 ≤ kα < i, respectively; and

∑0 sums the strengths |α(u(t, x))|

(and |α(v(t, x))|) of all i-waves, here kα = i, with xα < x (and xα > x) of u(t, x)(and v(t, x)) if qi(t, x) < 0, or with xα > x (and xα < x) of u(t, x) (and v(t, x)) ifqi(t, x) > 0, respectively. Define a functional, equivalent to the L1 distance of u(t, x)and v(t, x), as

Φ(u,v)(t)=

∑mi=1

∫IR|qi(t, x)|(1 +K1(Fu(mN∆t) + Fv(mN∆t)) +K2Wi(t, x))dx

for each t ∈ (mN∆t, (m+ 1)N∆t), where K1 and K2 are sufficiently large positiveconstants, N is the constant in Liu’s wave tracing method, Fu and Fv are theGlimm functionals defined in (3.1.23) for u(t, x) and v(t, x), respectively, valued atthe end time t = mN∆t. The definition of this functional is given by Liu-Yang82, and is similar to those used in Bressan-Liu-Yang 12 and Hu-LeFloch 62 for thesolutions constructed by the wave-front tracking algorithm. The key estimate isthat the functional Φ(u,v)(t) can be controlled by its initial value Φ(u,v)(0), upto a certain error term which approaches zero as the mesh size tends to zero. FromTheorem 1, there exist subsequences of the approximate solutions which converge tothe exact Glimm solutions, locally in the L1 norm. Therefore, one has the followingtheorem on the L1-stability of the exact Glimm solutions.

Theorem 3.1.3 If the initial data functions u0(x) and v0(x) have sufficiently smalltotal variation and u0 − v0 ∈ L1(IR), then, for the corresponding exact Glimmsolutions u(t, x) and v(t, x) of the Cauchy problem (3.1.1)–(3.1.2), there exists aconstant C > 0 such that

‖u(t, ·) − v(t, ·)‖L1(IR) ≤ C ‖u0 − v0‖L1(IR) for all t > 0. (3.1.26)

An immediate consequence of this theorem is that the whole sequence of theapproximate solutions constructed by the Glimm scheme converges to the unique


entropy solution of (3.1.1)–(3.1.2) as the mesh size tends to zero. See also Bressan9 for the uniqueness of limits of Glimm’s random choice method. The details of theproof of Theorem 3.1.3 can be found in Liu-Yang 82; 83.

Other related methods include wave-front tracking method (see Dafermos44, DiPerna 51, Bressan 8, and Risebro 91) and vanishing viscosity method (seeBianchini-Bressan 7). In the direction of relaxing the requirement of small totalvariation for the L1-stability, see Lewicka-Trivisa 68 and Lewicka 67. For more ex-tensive discussions and references about the L1-stability of BV entropy solutionsand related topics, we refer to Bressan 10, Dafermos 43, Holden-Risebro 61, andLeFloch 66.

3.1.2 L∞ Theory for One-Dimensional Hyperbolic Systems

The strong compactness in BV (by the Helly Theorem) can not be used in generalwhen the Cauchy data is allowed to be arbitrarily large, since entropy solutionsmay no longer be in BV . Motivated by the Glimm scheme 57, new compactnessframeworks should be developed to replace the BV compactness framework. Somecompensated compactness frameworks have been successfully developed for hyper-bolic systems of conservation laws to serve this purpose. One of such successfulexamples is a compactness framework for the the Euler equations for isentropicfluids.

Consider the Cauchy problem for the isentropic Euler equations (3.1.12) withinitial data:

(ρ,m)|t=0 = (ρ0,m0)(x), (3.1.27)

where ρ0 and m0 are in the physical region (ρ,m) : ρ ≥ 0, |m| ≤ C0ρ for someC0 > 0. The pressure function p(ρ) is a smooth function in ρ > 0 (nonvacuumstates) satisfying

p′(ρ) > 0, ρp′′(ρ) + 2p′(ρ) > 0 when ρ > 0, (3.1.28)

and

p(0) = p′(0) = 0, limρ→0

ρp(j+1)(ρ)

p(j)(ρ)= cj > 0, j = 0, 1. (3.1.29)

More precisely, we consider a general situation of pressure law that there exist asequence of exponents

1 < γ := γ1 < γ2 < · · · < γJ ≤ 3γ − 1

2< γJ+1


and a function P (ρ) such that

p(ρ) =J∑

j=1

κjργj + ργJ+1P (ρ), (3.1.30)

with lim supρ→0(|P (ρ)| + |ρ3P ′′′(ρ)|) < ∞ for some κj, j = 1, · · · , J, with κ1 > 0.For a polytropic gas obeying the γ-law (3.1.13), or a mixed ideal polytropic fluid,

p(ρ) = κ1ργ1 + κ2ρ

γ2 , γ1, γ2 > 1, κ2 > 0,

the pressure function clearly satisfies (3.1.28) and (3.1.30).

System (3.1.12) is strictly hyperbolic at the states with ρ > 0, and stricthyperbolicity fails at the vacuum states (ρ,m/ρ) : ρ = 0, |m/ρ| <∞. Let (η, q) :IR2

+ → IR2 be an entropy-entropy flux pair of system (3.1.12). An entropy η(ρ,m) iscalled a weak entropy if η = 0 at the vacuum states. A bounded measurable functionu(t, x) = (ρ,m)(t, x) is an entropy solution of (3.1.12) and (3.1.27)–(3.1.29) in IR2

+

if u(t, x) satisfies the following:

(i) There exists C > 0 such that

0 ≤ ρ(t, x) ≤ C, |m(t, x)| ≤ Cρ(t, x); (3.1.31)

(ii) The entropy inequality (3.1.3) holds in the sense of distributions in IR2+ for

any weak entropy pair (η, q)(u) with convex η(u).

Theorem 3.1.4 Consider the Euler equations (3.1.12) satisfying (3.1.28)–(3.1.30).Let (ρh,mh)(t, x), h > 0, be a sequence of functions satisfying

0 ≤ ρh(t, x) ≤ C, |mh(t, x)| ≤ C ρh(t, x) for a.e. (t, x), (3.1.32)

such that, for any weak entropy pair (η, q),

∂tη(ρh,mh) + ∂xq(ρ

h,mh) is compact in H−1loc (IR

2+). (3.1.33)

Then the sequence (ρh,mh)(t, x) is compact in L1loc(IR

2+).

The compactness framework in Theorem 4 was established in Chen-LeFloch27; 29 to replace the BV compactness framework, where the detailed proof can befound. For a gas obeying the γ-law, the case γ = (N + 2)/N,N ≥ 5 odd, was firsttreated by DiPerna 54, while the case 1 < γ ≤ 5/3 for usual gases was first completedby Chen 18 and Ding-Chen-Luo 47. The cases γ ≥ 3 and 5/3 < γ < 3 were treatedby Lions-Perthame-Tadmor 73 and Lions-Perthame-Souganidis 72, respectively. For


the isothermal case γ = 1, a similar compactness theorem was recently establishedby Huang-Wang 63.

In order to establish Theorem 4, it requires to establish the correspondingreduction theorem: a Young measure satisfying the Tartar commutation relations⟨

ν(t,x), η1 q2 − η2 q1⟩

(3.1.34)

=⟨ν(t,x), η1

⟩ ⟨ν(t,x), q2

⟩−

⟨ν(t,x), η2

⟩ ⟨ν(t,x), q1

⟩for a.e. (t, x)

for all weak entropy pairs is a Dirac mass. These conditions are derived by themethod of compensated compactness, especially the div-curl lemma (see Tartar 98and Murat 85). The proof was based on new properties of cancellation of singulari-ties of the entropy kernels χ and entropy flux kernel σ in the following combination

E(ρ, v; s1, s2) := χ(ρ, v; s1)σ(ρ, v; s2) − χ(ρ, v; s2)σ(ρ, v; s1),

a fractional derivative technique first introduced in 18; 47, and an important ob-servation that the following identity is an elementary consequence of the symmetricform of (3.1.34):⟨

ν(t,x), χ(ρ, v; s1)⟩ ⟨ν(t,x), ∂

λ+1s2

∂λ+1s3

E(ρ, v; s2, s3)⟩

+⟨ν(t,x), ∂

λ+1s2

χ(ρ, v; s2)⟩ ⟨ν(t,x), ∂

λ+1s3

E(ρ, v; s3, s1)⟩

+⟨ν(t,x), ∂

λ+1s3

χ(ρ, v; s3)⟩ ⟨ν(t,x), ∂

λ+1s2

E(ρ, v; s1, s2)⟩

= 0 (3.1.35)

for all s1, s2, and s3, where the derivatives are understood in the sense of distribu-tions. It was proved that, when s2, s3 → s1, the second and third terms converge inthe weak-star sense of measures to the same term but with opposite sign. The firstterm is more singular and contains the products of functions of bounded variation bybounded measures, which are known to depend upon regularization. The first termin (3.1.35) converges to a non-trivial limit which is determined explicitly. Finally,the genuine nonlinearity on p(ρ) is required to conclude that the Young measure νeither reduces to a Dirac mass or is supported on the vacuum line.

This compactness framework has successfully been applied to prove the con-vergence of the Lax-Friedrichs scheme, the Godunov scheme, and the vanishingviscosity method, and to establish the compactness of solution operators and thedecay of periodic solutions. See Chen 18, Ding-Chen-Luo 47; 48, Chen-Frid 25,Lions-Perthame-Souganidis 72, and the references cited in Chen-Wang 33. Theother compactness frameworks for various hyperbolic systems of conservation lawscan be found in DiPerna 53, Serre 96, Chen-Li-Li 30, Perthame-Tzavaras 90, and thereferences cited therein. Some effort has been made for analyzing the BV regularityof some nonlinear functions of entropy solutions of (3.1.12) with BV initial data viathe Glimm scheme by Young 106.


3.2 The Glimm Scheme and Shock Waves in Su-

personic Flows past Lipschitz Wedges

We now discuss an application of the Glimm scheme to the existence and stabil-ity of two-dimensional supersonic flows past infinite nonsmooth wedges. The two-dimensional steady Euler flows are generally governed by⎧⎪⎪⎨

⎪⎪⎩∂x(ρu) + ∂y(ρv) = 0,∂x(ρu

2 + p) + ∂y(ρuv) = 0,∂x(ρuv) + ∂y(ρv

2 + p) = 0,∂x(u(E + p)) + ∂y(v(E + p)) = 0,

(3.2.1)

where (u, v) is the velocity and E the total energy, and the constitutive relationsamong the thermodynamical variables ρ, p, e, θ, and S are determined by (3.1.6)–(3.1.10). For the isentropic or isothermal case p = p(ρ) = κ0ρ

γ, γ ≥ 1, the first threeequations in (3.2.1) form the barotropic Euler equations, a self-contained system.The quantity c =

√pρ(ρ, S) is defined as the sonic speed and, for polytropic gases,

c =√γp/ρ. System (3.2.1) governing a supersonic flow, that is, u2 +v2 > c2, has all

real eigenvalues and is hyperbolic, while system (3.2.1) governing a subsonic flow,that is, u2 + v2 < c2, has complex eigenvalues and is elliptic-hyperbolic mixed andcomposite.

The mathematical study of two-dimensional steady supersonic flows past wedgescan date back 1940’s since the stability of such flows is fundamental in applications(cf. Courant-Friedrichs 42 and Whitham 103). When the wedge vertex angle is lessthan the critical angle, local solutions around the wedge vertex were first constructedin Gu 60, Li 69, Schaeffer 93, and the references cited therein. Global potential so-lutions were constructed in 37; 38; 39; 40; 42 when the wedge has some convexityor the wedge is a small perturbation of the straight wedge with fast decay in theflow direction and in 108; 109 for piecewise smooth curved wedges which are a smallperturbation of straight wedge.

As is well-known, the potential flow equation is an excellent model for the flowcontaining only weak shocks since it approximates to the isentropic Euler equationsup to the third order of the shock strengths. For the flows containing shocks oflarge strength, the full Euler equations (3.2.1) are required to govern the physicalflows. For the wedge problem, when the vertex angle is large, the flow containsa large attached shock front emanating from the wedge vertex and, for this case,the Euler equations should take the position to describe the physical flow. Thus itis important to study the two-dimensional steady supersonic flows governed by theEuler equations, rather than the potential flow equation, for the wedge problem witha large vertex angle. When a wedge is straight and the wedge vertex angle is less than

3.2. SHOCK WAVES IN SUPERSONIC FLOWS 45

the critical angle, there exists a supersonic shock front emanating from the wedgevertex so that the constant states on both sides of the shock are supersonic; thecritical angle condition is necessary and sufficient for the existence of the supersonicshock. This can be seen through the shock polar (see Figs. 3.1–3.2; also see 35; 42).

Figure 3.1: Supersonic shock emanating from the wedge vertex

sonic line

Figure 3.2: Shock polar in the (u, v)-plane

Consider the two-dimensional steady supersonic Euler flows past two-dimensionalLipschitz curved wedges whose vertex angles are less than the critical angle ωcrit,along which the total variation of the tangent angle function is suitably small. Morespecifically,

(i).. There exists a Lipschitz function g ∈ Lip(IR+; IR) with g′ ∈ BV (IR+; IR) andg(0) = 0 such that ω0 := arctan(g′(0+)) < ωcrit,

TV (g′(·)) < ε for some constant ε > 0, (3.2.2)

Ω := (x, y) : y > g(x), x ≥ 0, Γ := (x, y) : y = g(x), x ≥ 0,and n(x±) = (−g′(x±),1)√

(g′(x±))2+1are the outer normal vectors to Γ at the points x±

respectively (see Fig. 3.2);


(ii).. The uniform upstream flow U− = (u−, 0, p−, ρ−) satisfies

u− > c− :=√γp−/ρ−

so that a strong supersonic shock emanates from the wedge vertex.

Figure 3.3: Supersonic flow past a curved wedge

With this setup, the wedge problem can be formulated into the following prob-lem of initial-boundary value type for system (3.2.1):

Cauchy Condition:U |x=0 = U−; (3.2.3)

Boundary Condition:

(u, v) · n = 0 on Γ. (3.2.4)

Definition 1 (Entropy Solutions). A function U = U(x, y) ∈ BV (Ω; IR2×IR+×IR+)is called an entropy solution of problem (3.2.1) and (3.2.3)–(3.2.4) provided that

(i). U is a weak solution of (3.2.1), that is, U satisfies the equations in thesense of distributions and the Cauchy and boundary conditions (3.2.3)–(3.2.4) inthe trace sense;

(ii). U satisfies the entropy inequality in the sense of distributions:

∂x(ρua(S)) + ∂y(ρva(S)) ≥ 0 (3.2.5)

for any a ∈ C2 with a′(S) ≥ 0.

Then we have

Theorem 3.2.1 (Existence and Stability) There exists ε0 > 0 and C > 0 suchthat, if (3.2.2) holds for ε ≤ ε0, then there exists a pair of functions

U ∈ BV (IR; IR2 × IR+ × IR+), σ ∈ BV (IR; IR)

3.2. SHOCK WAVES IN SUPERSONIC FLOWS 47

with χ =∫ x

0σ(t)dt ∈ Lip(IR; IR) such that

(i). U is a global entropy solution of problem (3.2.1) and (3.2.3)–(3.2.4) in Ωwith

TV U(x, ·) : [g(x),−∞) ≤ C TV (g′(·)) for every x ∈ IR+,

(u, v) · n|y=g(x) = 0 in the trace sense;

(ii). The curve y = χ(x) is a strong shock front with χ(x) > g(x) for any x > 0and

U |y>χ(x) = U−,√u2 + v2|g(x)<y<χ(x) < u−;

(iii). There exist constants p∞ and σ∞ such that

limx→∞

sup|p(x, y) − p∞| : g(x) < y < χ(x) = 0,

limx→∞

|σ(x) − σ∞| = 0,

andlim

x→∞sup| arctan (v(x, y)/u(x, y)) − ω∞| : g(x) < y < χ(x) = 0,

where ω∞ = limx→∞

arctan(g′(x+)).

This theorem has been established in Chen-Zhang-Zhu 35. It indicates that,under the BV perturbation of the wedge boundary so long as the wedge vertexangle is less than ωcrit, the strong shock front emanating from the wedge vertex isnonlinearly stable in structure globally, although there may be many weak shocksand vortex sheets between the wedge boundary and the strong shock front. Thisasserts that any supersonic shock for the wedge problem is nonlinearly stable.

In order to establish this theorem, we first developed a modified Glimm schemewhose mesh grids are designed to follow the slop of the Lipschitz wedge boundary,which are not standard rectangle mesh grids, so that the lateral Riemann buildingblocks contain only one shock or rarefaction wave emanating from the mesh pointson the boundary. Such a design makes the BV estimates more convenient for theGlimm approximate solutions. Then careful interaction estimates were made. Oneof the essential estimates is that of the strength, δ1, of the reflected 1-waves in theinteraction between the 4-strong shock front and weak waves (α1, β2, β3, β4), that is,

δ1 = α1 +Ks1β4 +O(1)|α1|(|β2| + |β3|) with |Ks1| < 1.

The second essential estimate is the interaction estimate between the wedge bound-ary and weak waves.


Based on the construction of the modified Glimm scheme and interaction es-timates, we successfully identified a Glimm-type functional by incorporating thecurved wedge boundary and the strong shock front naturally and by tracing theinteractions not only between the wedge boundary and weak waves but also the in-teraction between the strong shock front and weak waves. In particular, the Glimm-type functional on the mesh curve Jk in (3.1.23) is replaced by

F(Jk) = C∗|σJk − σ0| + L(Jk) +KQ(Jk),

and the linear part is replaced by

L(Jk) = K0L0(Jk) + L1(Jk) +K2L2(Jk) +K3L3(Jk) +K4L4(Jk)

withL0(Jk) =

∑|ω(Cl)| : Cl ∈ ΩJk

,Lj(Jk) =

∑|αj| : αj crosses Jk, 1 ≤ j ≤ 4,

where ΩJkis the set of the mesh corner points lying in Jk and the boundary, σJk

stands for the speed of the strong shock crossing Jk, the constant K, C∗, K0, K2, K3,and K4 can be appropriate chosen with the aid of the important fact that |Ks1| < 1so that the Glimm functional monotonically decreases in the flow direction. Anotheressential estimate is to trace the approximate strong shocks in order to establish thenonlinear stability and asymptotic behavior of the strong shocks emanating fromthe wedge vertex under the wedge perturbation.

Condition (3.2.2) can be relaxed by combining the analysis in 35 with theargument in 100; 101. We also remark that, in Lien-Liu 71, the nonlinear stabilityof a self-similar three-dimensional gas flow past an infinite cone with small vertexangle was established upon the perturbation of the obstacle. It would be interestingto combine the analysis in 35 with the argument in 71 to study the nonlinear stabilityof a self-similar three-dimensional gas flow past an infinite cone with arbitrary vertexangle.

3.3 The Glimm Scheme and Vortex Sheets in Su-

personic Flows past Lipschitz Walls

In this section, we discuss another application of the Glimm scheme to the stabilityof supersonic vortex sheets over nonsmooth walls along which the total variation ofthe tangent angle functions is suitably small. More precisely,

(i).. There exists a Lipschitz function g ∈ Lip(IR+; IR) with g(0) = 0, g′(0+) = 0,and g′ ∈ BV (IR+; IR) such that

TV (g′(·)) < ε for some constant ε > 0, (3.3.1)

3.3. VORTEX SHEETS IN SUPERSONIC FLOWS 49

Ω = (x, y) : y > g(x), x ≥ 0, Γ = (x, y) : y = g(x), x ≥ 0,and n(x±) = (−g′(x±),1)√

(g′(x±))2+1are the outer normal vectors to Γ at the points x±

respectively (see Fig. 3.4);

(ii).. The upstream flow consists of one supersonic straight vortex sheet y = y0 > 0and two constant vectors U0 = (u0, 0, p0, ρ0) when y > y0 > 0 and U1 =(u1, 0, p0, ρ1) when 0 < y < y0 satisfying

u1 > u0 > 0, ui > ci, i = 0, 1,

where ci = γpi/ρi is the sonic speed of states Ui, i = 0, 1.

With this setup, the vortex sheet problem can be formulated into the followingproblem of initial-boundary value type for system (3.2.1):

Cauchy Condition:

U |x=0 =

U0, 0 < y < y0,U1, y > y0;

(3.3.2)

Boundary Condition:

(u, v) · n = 0 on Γ. (3.3.3)

Figure 3.4: Stability of the supersonic vortex sheet

The stability of supersonic vortex sheets has been studied by classical linearizedstability analysis, large-scale numerical simulations, and asymptotic analysis. Inparticular, it has been predicted that the nonlinear development of instabilities ofsupersonic vortex sheets at high Mach number as time evolves, see Woodward 104,Artola-Majda 5, and the references cited therein. Motivated by the phenomenon ofevolution instabilities, we are interested in whether steady supersonic vortex sheets,as time-asymptotics, are stable under a BV perturbation of the Lipschitz walls. Incontrast with the prediction of the instability in time, it has been proved that steadysupersonic vortex sheets, as time-asymptotics, are stable in structure globally, evenunder the BV perturbation of the Lipschitz walls in Chen-Zhang-Zhu 36.


Theorem 3.3.1 (Existence and Stability) There exists ε0 > 0 and C > 0 suchthat, if (3.3.1) holds for ε ≤ ε0, then there exists a pair of functions

U ∈ BV (IR+; IR), χ ∈ Lip(IR+; IR)

with χ(0) = y0 such that

(i). U is a global entropy solution of problem (3.2.1) and (3.3.2)–(3.3.3) in Ωwith

TV U(x, ·) : [g(x),∞) ≤ C TV (g′(·)) for every x ∈ [0,∞),

(u, v) · n|y=g(x) = 0 in the trace sense;

(ii). The curve y = χ(x) is a strong supersonic vortex sheet with χ(x) > g(x)for any x > 0 and

|U|g(x)<y<χ(x) − U0| ≤ Cε, |U|y>χ(x) − U1| ≤ Cε;

(iii). There exist constants p∞ and χ∞ such that

limx→∞

sup|p(x, y) − p∞| : g(x) < y < χ(x) = 0,

limx→∞

|χ(x) − χ∞| = 0,

andlim

x→∞sup| arctan (v(x, y)/u(x, y)) − ω∞| : y > g(x) = 0,

where ω∞ = limx→∞

arctan(g′(x+)).

This theorem indicates that the strong supersonic vortex sheets are nonlinearlystable in structure, although there may be many weak shocks and supersonic vortexsheets away from the strong vortex sheet, under theBV perturbation of the Lipschitzwall.

In order to establish this problem, as in Section 2, we first developed a modi-fied Glimm scheme whose mesh grids are designed to follow the slop of the Lipschitzboundary, which are not standard rectangle mesh grids, so that the lateral Rie-mann building blocks contain only one wave emanating from the mesh points on theboundary. For this case, one of the essential estimates is that of the strength, δ1, ofthe reflected 1-wave in the interaction between the 4-weak wave, α4, and the strongvortex sheet from below is less than one, that is,

δ1 = K01α4, |K01| < 1.

3.4. MULTIDIMENSIONAL TRANSONIC SHOCKS 51

Another essential estimate is that of the strength, δ1, of the reflected 1-wave in theinteraction between the 1-weak wave, β1, and the strong vortex sheet from above isalso less than one, that is,

δ1 = K11β1, |K11| < 1.

The third essential estimate is the interaction estimate between the boundary andweak waves.

Based on the construction of the modified Glimm scheme and the new interac-tion estimates, we successfully identified a Glimm-type functional by incorporatingthe Lipschitz wall and the strong shock front naturally and by tracing the inter-actions not only between the boundary and weak waves but also the interactionbetween the strong vortex sheet and weak waves so that the Glimm-type functionalmonotonically decreases in the flow direction. Another essential estimate is to tracethe approximate supersonic vortex sheets in order to establish the nonlinear sta-bility and asymptotic behavior of the strong vortex sheet under the BV boundaryperturbation. For more details, see Chen-Zhang-Zhu 36.

3.4 Existence and Stability of Multidimensional

Transonic Shocks

Motivated by the stability of two-dimensional supersonic shocks via the Glimmscheme, we have initiated a program with M. Feldman on the existence and stabilityof multidimensional transonic shocks and developed three new analytical approachessince 1999. To explain two of the three approaches more clearly, we focus here onthe potential flow equation for the velocity potential ϕ : Ω ⊂ R

n → R, which is asecond-order nonlinear equation of mixed elliptic-hyperbolic type:

div (ρ(|Dϕ|2)Dϕ) = 0, x ∈ Ω ⊂ Rn, (3.4.1)

where the density ρ(q2) is ρ(q2) = (1 − θq2)1/(γ−1)

with adiabatic exponent γ >1. Equation (3.4.1) is elliptic at Dϕ with |Dϕ| = q if ρ(q2) + 2q2ρ′(q2) > 0 andhyperbolic if ρ(q2) + 2q2ρ′(q2) < 0. We remark that, for the isothermal case γ = 1,the same results as below can be obtained by following similar arguments. We focushere on the case γ > 1.

We are interested in compressible potential flows with shocks. Let Ω+ and Ω−

be open subsets of Ω such that Ω+ ∩ Ω− = ∅,Ω+ ∪ Ω− = Ω, and S = ∂Ω+ ∩ Ω. Letϕ ∈ Lip(Ω) be a weak solution of (3.4.1) and be in C1(Ω±) so that Dϕ experiencesa jump across S that is an (n− 1)-dimensional smooth surface. Then ϕ satisfies the


following Rankine-Hugoniot conditions on S:

[ϕ]S = 0, [ρ(|Dϕ|2)Dϕ · n]S = 0, (3.4.2)

where n is the unit normal to S from Ω− to Ω+, and the bracket denotes thedifference between the values of the function along S on the Ω± sides. Moreover,a function ϕ ∈ C1(Ω±), which satisfies |Dϕ| ≤

√2/(γ − 1), (3.4.2), and equation

(3.4.1) in Ω± respectively, is a weak solution of (3.4.1) in the whole domain Ω. Setϕ± = ϕ|Ω± . Then we can also write (3.4.2) as

ϕ+ = ϕ− on S (3.4.3)

andρ(|Dϕ+|2)Dϕ+ · n = ρ(|Dϕ−|2)Dϕ− · n on S. (3.4.4)

Note that the function

Φ(p) :=(1 − θp2

)1/(γ−1)p (3.4.5)

is continuous on[0,√

2/(γ − 1)]

and satisfies

Φ(p) > 0 for p ∈(0,√

2/(γ − 1)), Φ(0) = Φ

(√2/(γ − 1)

)= 0, (3.4.6)

0 < Φ′(p) < 1 on (0, c∗), Φ′(p) < 0 on(c∗,

√2/(γ − 1)

), (3.4.7)

Φ′′(p) < 0 on (0, c∗], (3.4.8)

where c∗ =√

2/(γ + 1) is the sonic speed, for which a flow is called supersonic if|Dϕ| > c∗ and subsonic if |Dϕ| < c∗.

Suppose that ϕ ∈ C1(Ω±) is a weak solution satisfying

|Dϕ| < c∗ in Ω+, |Dϕ| > c∗ in Ω−, Dϕ± · n|S > 0. (3.4.9)

Then ϕ is a transonic shock solution with transonic shock S dividing Ω into thesubsonic region Ω+ and the supersonic region Ω− and satisfying the physical entropycondition (see Courant-Friedrichs 42):

ρ(|Dϕ−|2) < ρ(|Dϕ+|2) along S. (3.4.10)

Note that equation (3.4.1) is elliptic in the subsonic region and hyperbolic in thesupersonic region.

Let (x′, xn) be the coordinates in Rn, where x′ = (x1, . . . , xn−1) ∈ R

n−1 andxn ∈ R. Fix V0 ∈ R

n, and let

ϕ0(x) := V0 · x, x ∈ Rn.


If |V0| ∈ (0, c∗) (resp. |V0| ∈ (c∗,√

2/(γ − 1))), then ϕ0(x) is a subsonic (resp.supersonic) solution in R

n, and V0 = Dϕ0 is its velocity.

Let V ′0 ∈ R

n−1 and q+0 > 0 be such that the vector V +

0 := (V ′0 , q

+0 ) satisfies

|V +0 | < c∗. Then, using the properties of function (3.4.5), we conclude from (3.4.6)–

(3.4.8) that there exists a unique q−0 > 0 such that(1 − γ − 1

2(|V ′

0 |2 + |q+0 |2)

) 1γ−1

q+0 =

(1 − γ − 1

2(|V ′

0 |2 + |q−0 |2)) 1

γ−1

q−0 . (3.4.11)

The entropy condition (3.4.10) implies q−0 > q+0 . By denoting V −

0 := (V ′0 , q

−0 ) and

defining functions ϕ±0 (x) := V ±

0 · x on Rn, then ϕ+

0 (resp. ϕ−0 ) is a subsonic (resp.

supersonic) solution. Furthermore, from (3.4.4) and (3.4.11), the function

ϕ0(x) := min(ϕ+0 (x), ϕ−

0 (x)) =

V −

0 · x, x ∈ Ω−0 := x ∈ R

n : xn < 0,V +

0 · x, x ∈ Ω+0 := x ∈ R

n : xn > 0(3.4.12)

is a plane transonic shock solution in Rn, Ω+

0 and Ω−0 are respectively its subsonic

and supersonic regions, and S := xn = 0 is a transonic shock. Note that, ifV ′

0 = 0, then the velocities V ±0 are orthogonal to the shock S and, if V ′

0 = 0, thenthe velocities are not orthogonal to S.

In order to deal with multidimensional transonic shocks in an unbounded do-main Ω, we define the following weighted Holder semi-norms and norms in a domainD ⊂ R

n: Let x → δx be a given nonnegative function defined on D, which will bespecified in each case we consider below. Let δx,y := min(δx, δy) for x,y ∈ D. Fork ∈ R, α ∈ (0, 1), and m ∈ Z+, we define

[u](k)m,0,D =

∑|β|=m

supx∈D

(δm+kx |Dβu(x)|

),

[u](k)m,α,D =

∑|β|=m

supx,y∈D,x =y

(δm+α+kx,y

|Dβu(x) −Dβu(y)||x − y|α

),

(3.4.13)

‖u‖(k)m,0,D =

m∑j=0

[u](k)j;0;D, ‖u‖(k)

m,α,D = ‖u‖(k)m,0,D + [u]

(k)m,α,D,

where Dβ = ∂β1x1

· · · ∂βnxn

, β = (β1, . . . , βn) is a multi-index with βj ≥ 0, βj ∈ Z+, and|β| = β1 + · · · + βn. We denote by ‖u‖m,α,D the (non-weighted) Holder norms in adomain D, i.e., the norms defined as above with δx = δx,y = 1.

3.4.1 Transonic Shock Problems

We first discuss some physical transonic shock problems.


Transonic Shocks in Infinite Nozzles of Arbitrarily Cross-Sections

Consider the following infinite nozzle Ω with arbitrary smooth cross-sections:

Ω = Ψ(Λ × IR) ∩ xn ≥ −1, (3.4.14)

where Λ ⊂ Rn−1 is an open bounded connected set with a smooth boundary, and

Ψ : Rn → R

n is a smooth map, which is close to the identity map. For simplicity,we assume that

∂Λ is in C [n2]+3,α, ‖Ψ − Id‖[n

2]+3,α,Rn ≤ σ (3.4.15)

for some α ∈ (0, 1) and small σ > 0, where [s] is the integer part of s, Id : Rn → R

n

is the identity map, ∂lΩ := Ψ(∂Λ×IR))∩xn > −1. Such nozzles especially includethe slowly varying de Laval nozzles 42; 103. For concreteness, we also assume thatthere exists L > 1 such that

Ψ(x) = x for any x = (x′, xn) with xn > L, (3.4.16)

that is, the nozzle slowly varies in a bounded domain as the de Laval nozzles.

In the two-dimension case, the domain Ω defined above has the following simpleform: Ω = (x1, x2) : x1 ≥ −1, b−(x2) < x1 < b+(x2), where ‖b± − d±‖4,α,R ≤σ and b± ≡ d± on [L,∞) for some constants d± satisfying d+ > d−. For themultidimensional case, the geometry of the nozzles is much richer.

Note that our setup implies that ∂Ω = ∂oΩ ∪ ∂lΩ with

∂lΩ := Ψ(∂Λ × (−∞,∞)) ∩ (x′, xn) : xn > −1,∂oΩ := Ψ(Λ × (−∞,∞)) ∩ (x′, xn) : xn = −1.

Then the transonic nozzle problem can be formulated into the following form:

Problem 4.1: Transonic Nozzle Problem. Given the supersonic upstream flowat the entrance ∂oΩ:

ϕ = ϕ−e , ϕxn = ψ−

e on ∂oΩ, (3.4.17)

the slip boundary condition on the nozzle boundary ∂lΩ:

Dϕ · n = 0 on ∂lΩ, (3.4.18)

and the uniform subsonic flow condition at the infinite exit xn = ∞:

‖ϕ(·) − q∞xn‖C1(Ω∩xn>R) → 0 as R → ∞ for some q∞ ∈ (0, c∗), (3.4.19)

find a multidimensional transonic flow ϕ of the problem (3.4.1) and (3.4.17)–(3.4.19)in Ω.


The standard local existence theory of smooth solutions for the initial-boundaryvalue problem (3.4.17)–(3.4.18) for second-order quasilinear hyperbolic equationsimplies that, as σ is sufficiently small in (3.4.15) and (3.4.22) below, there existsa supersonic solution ϕ− of (3.4.1) in Ω1 := −1 ≤ xn ≤ 1, which is a C l+1

perturbation of ϕ−0 = q−0 xn: For any α ∈ (0, 1],

‖ϕ− − ϕ−0 ‖l,α,Ω1 ≤ C0 σ, l = 1, 2, (3.4.20)

for some constant C0 > 0, and satisfies

Dϕ− · n = 0 on ∂lΩ1, (3.4.21)

provided that (ϕ−e , ψ

−e ) on ∂oΩ satisfies

‖ϕ−e − q−0 xn‖Hs+l + ‖ψ−

e − q−0 ‖Hs+l−1 ≤ σ, l = 1, 2, (3.4.22)

for some integer s > n/2 + 1 and the compatibility conditions up to order s + l,where the norm ‖ · ‖Hs is the Sobolev norm with Hs = W s,2.

Theorem 3.4.1 Let q+0 ∈ (0, c∗) and q−0 ∈

(c∗,

√2/(γ − 1)

)satisfy (3.4.11), and

let ϕ0 be the transonic shock solution (3.4.12) with V ′0 = 0. Then there exist σ0 > 0,

C1, and C2, depending only on n, α, γ, q+0 , Λ, and L such that, for every σ ∈

(0, σ0), any map Ψ satisfying (3.4.15) and (3.4.16), and any supersonic upstream flow(ϕ−

e , ψ−e ) on ∂oΩ satisfying (3.4.22) with l = 1, there exists a solution ϕ ∈ Lip(Ω)

of Problem 4.1 satisfying

Ω+(ϕ) = xn > f(x′), Ω−(ϕ) = xn < f(x′),

‖ϕ− ϕ−0 ‖1,α,Ω− ≤ C1σ, ‖Dϕ− q+

0 en‖0,0,Ω+ ≤ C2σ. (3.4.23)

Moreover, this solution satisfies ϕ ∈ Lip(Ω)∩C∞(Ω+) and the following properties:

(i). The constant q∞ in (3.4.19) must be q+:

q∞ = q+, (3.4.24)

where q+ is the unique solution in the interval (0, c∗) of the equation

ρ((q+)2)q+ = Q+ (3.4.25)

with Q+ = 1|Λ|

∫∂oΩ

ρ(|Dx′ϕ−e |2 + (ψ−

e )2)ψ−e dHn−1. Thus ϕ satisfies

‖ϕ− q+xn‖C1(Ω∩xn>R) → 0 as R → ∞ (3.4.26)

and q+ satisfies|q+ − q+

0 | ≤ C2σ; (3.4.27)


(ii). The function f(x′) in (3.4.23) satisfies

‖f‖1,α,Rn−1 ≤ C1σ, (3.4.28)

and the surface S = (x′, f(x′)) : x′ ∈ Rn−1 ∩ Ω is orthogonal to ∂lΩ at

every intersection point;

(iii). Furthermore, ϕ ∈ C1,α(Ω+) with

‖ϕ− q+xn‖1,α,Ω+ ≤ C2σ. (3.4.29)

In addition, if the upstream flow (ϕ−e , ψ

−e ) on ∂oΩ satisfies (3.4.22) with l = 2, then

the solution ϕ ∈ C2,α(Ω+) with ‖ϕ − q+xn‖2,α,Ω+ ≤ C2σ, and the solution with atransonic shock is unique and stable with respect to the nozzle boundaries and smoothsupersonic upstream flows at the entrance.

When the upstream flow (ϕ−e , ψ

−e ) ≡ (−ψ−

e , ψ−e ) at the entrance ∂oΩ is constant

and the nozzle Ω∩ −1 ≤ xn ≤ −1 + δ = Λ× [−1,−1 + δ] for some δ > 0 like a deLaval nozzle, then the compatibility conditions are automatically satisfied: In fact,in this case, ϕ−(x) = ψ−

e xn is a solution near xn = −1 in the nozzle. When n = 2,condition (3.4.22) for the supersonic upstream flow (ϕ−

e , ψ−e ) on ∂oΩ in Theorem

3.4.1 can be replaced by the C3-condition:

‖ϕ−e − q−0 xn‖C3 + ‖ψ−

e − q−0 ‖C2 ≤ σ, (3.4.30)

which can be achieved by following the arguments in Li-Yu 70. See Chen-Feldman23 for more details.

Transonic Shocks near Flat Shocks in Rn

We now consider perturbations of the uniform transonic shock solution (3.4.12) inthe whole space R

n with n ≥ 3. As in Section 4.1.1, since it is enough to specify thesupersonic perturbation ϕ− only in a neighborhood of the unperturbed shock surfacexn = 0, we introduce domains Ω := R

n−1 × (−1,∞) and Ω2 := Rn−1 × (−1, 1).

Note that we expect the subsonic region Ω+ to be close to the half-space Ω+0 =

xn > 0. We use the norms in (3.4.13) with the weight function δx = 1 + |x| andconsider the following problem:

Problem 4.2. Given a supersonic solution ϕ−(x) of (3.4.1) in Ω2 satisfying that,for some α > 0,

‖ϕ− − ϕ−0 ‖

(n−1)2,α,Ω2

≤ σ (3.4.31)


with σ > 0 small, find a transonic shock solution ϕ(x) in Ω such that Ω− := Ω\Ω+ ⊂Ω2 with Ω+ := x ∈ Ω : |Dϕ(x)| < c∗, ϕ(x) = ϕ−(x) in Ω−, and

ϕ = ϕ−, ∂xnϕ = ∂xnϕ− on xn = −1, (3.4.32)

limR→∞

‖ϕ− ϕ+0 ‖C1(Ω+\BR(0)) = 0. (3.4.33)

Condition (3.4.32) guarantees that the solution has supersonic upstream atxn = −1, while condition (3.4.33) determines, in particular, that the uniform velocitystate at infinity in the downstream direction is equal to the unperturbed downstreamvelocity state. The requirement in (3.4.33) that ϕ→ ϕ+

0 at infinity within Ω+ fixesthe position of shock at infinity. This allows to determine the solution of Problem4.2 uniquely. Then we have the following theorem.

Theorem 3.4.2 Let |(V ′0 , q

+0 )| ∈ (0, c∗) and q−0 ∈ (c∗,

√2/(γ − 1)) satisfy (3.4.11),

and let ϕ0(x) be the transonic shock solution (3.4.12). Then there exist positiveconstants σ0, C1, and C2 depending only on n, γ, α, |V ′

0 |, and q+0 such that, for

every σ ≤ σ0 and any supersonic solution ϕ−(x) of (3.4.1) satisfying the conditionsstated in Problem 4.2, there exists a unique solution ϕ(x) of Problem 4.2 satisfying

‖ϕ− ϕ+0 ‖

(n−2)

2,α,Ω+ ≤ C1σ. (3.4.34)

In addition, there exists f : Rn−1 → R such that

Ω+ = xn > f(x′), ‖f‖(n−2)

2,α,Rn−1 ≤ C2σ, (3.4.35)

that is, the shock surface S = (x′, xn) : xn = f(x′),x′ ∈ Rn−1 is in C2,α and

converges with an appropriate algebraic rate to the hyperplane S0 = xn = 0 atinfinity. Moreover, there exist a nonnegative nondecreasing function Ψ ∈ C(IR+)satisfying Ψ(0) = 0 and a constant σ0 depending only on n, γ, α, |V ′

0 |, and q+0 such

that, if σ < σ0 and smooth supersonic solutions ϕ−(x) and ϕ−(x) of (3.4.1) satisfy(3.4.31), then the unique solutions ϕ(x) and ϕ(x) of Problem 4.2 for ϕ−(x) andϕ−(x), respectively, satisfy

‖fϕ − fϕ‖(n−2)

2,α,Rn−1 ≤ Ψ(‖ϕ− − ϕ−‖(n−1)

2,α,Ω1

), (3.4.36)

where fϕ(x′) and fϕ(x′) are the free boundary functions (3.4.35) of ϕ(x) and ϕ(x),respectively.

This existence result can be extended to the case that the regularity of thesteady perturbation ϕ− is only C1,1, that is, (3.4.31) can be replaced by

‖ϕ− − ϕ−0 ‖

(n−1)1,1,Ω1

≤ σ. (3.4.37)


See Chen-Feldman 22 for more details.

Other transonic problems include the stability of transonic flows past infinitenonsmooth wedges and the existence and stability of regular shock reflection solu-tions, which are now under investigation with the aid of the approaches which willbe discussed in Sections 4.2–4.3.

Free Boundary Problems

The transonic shock problems can be formulated into a one-phase free boundaryproblem for a nonlinear elliptic equation: Given ϕ− ∈ C1,α(Ω), find a function ϕthat is continuous in Ω and satisfies

ϕ ≤ ϕ− in Ω, (3.4.38)

equation (3.4.1), the ellipticity condition in the non-coincidence set Ω+ = ϕ < ϕ−,the free boundary condition (3.4.4) on the boundary S = ∂Ω+ ∩ Ω, as well as theprescribed conditions on the fixed boundary ∂Ω and at infinity. These conditionsare different in different problems, say, conditions (3.4.17)–(3.4.19) for Problem 4.1and (3.4.32)–(3.4.33) for Problem 4.2.

The free boundary is the location of the shock, and the free boundary condi-tions (3.4.3)–(3.4.4) are the Rankine-Hugoniot conditions in (3.4.2). Note that con-dition (3.4.38) is motivated by the similar property (3.4.12) of unperturbed shocks;and, locally on the shock, (3.4.38) is equivalent to the entropy condition (3.4.10).Condition (3.4.38) transforms the transonic shock problem, in which the subsonic re-gion Ω+ is determined by the gradient condition |Dϕ(x)| < c∗, into a free boundaryproblem, in which Ω+ is the non-coincidence set.

In order to solve this free boundary problem, equation (3.4.1) is modifiedto be uniformly elliptic and, correspondingly, modify the free boundary condition(3.4.4). Then this modified free boundary problem is solved. Since ϕ− is a smallC1,α perturbation of ϕ−

0 , the solution ϕ of the free boundary problem is shown tobe a small C1,α perturbation of the given subsonic shock solution ϕ+

0 in Ω+. Inparticular, the gradient estimate implies that ϕ in fact satisfies the original freeboundary problem, hence the transonic shock problem, Problem 4.1 (Problem 4.2,respectively).

The modified free boundary problem does not directly fit into the variationalframework of Alt-Caffarelli 1 and Alt-Caffarelli-Friedman 2 and the regularizationframework of Berestycki-Caffarelli-Nirenberg 6. Also, the nonlinearity of the freeboundary problem makes it difficult to apply the Harnack inequality approach ofCaffarelli 13. In particular, a boundary comparison principle for positive solutions ofnonlinear elliptic equations in Lipschitz domains is not available yet for the equations


that are not homogeneous with respect to D2u,Du, and u, which however is ourcase.

3.4.2 Iteration Approach

The first approach we developed in Chen-Feldman 20; 21; 23 is an iteration schemebased on the non-degeneracy of the free boundary condition: the jump of the normalderivative of a solution across the free boundary has a strictly positive lower bound.Our iteration process is as follows: Suppose the domain Ω+

k is given so that Sk :=∂Ω+

k \ ∂Ω is C1,α. Consider the oblique derivative problem in Ω+k obtained by

rewriting the (modified) equation (3.4.1) and free boundary condition (3.4.4) interms of the function u := ϕ− ϕ+

0 . Then the problem has the following form:

divA(x, Du) = F (x) in Ω+k := u > 0,

A(x, Du) · n = G(x,n) on S := ∂Ω+k \ ∂Ω,

(3.4.39)

plus the fixed boundary conditions on ∂Ω+k ∩ ∂Ω and the conditions at infinity.

The equation is quasilinear, uniformly elliptic, A(x, 0) ≡ 0, while G(x,n) has a

certain structure. Let uk ∈ C1,α(Ω+k ) be the solution of (3.4.39). Then ‖uk‖1,α,Ω+

k

is estimated to be small if the perturbation is small, where appropriate weightedHolder norms are actually needed in the unbounded domains. Then the functionϕk := ϕ+

0 + uk from Ω+k is extended to Ω so that the C1,α norm of ϕk − ϕ+

0 inΩ is controlled by ‖uk‖1,α,Ω+

k. Define Ω+

k+1 := x ∈ Ω : ϕk(x) < ϕ−(x) for the

next step. Note that, since ‖ϕk − ϕ+0 ‖1,α,Ω and ‖ϕ− − ϕ−

0 ‖1,α,Ω are small, we have|Dϕ−|−|Dϕk| ≥ δ > 0 in Ω, and this nondegeneracy implies that Sk+1 := ∂Ω+

k+1\∂Ωis C1,α and its norm is estimated in terms of the data of the problem.

The fixed point Ω+ of this process determines a solution of the free bound-ary problem since the corresponding solution ϕ satisfies Ω+ = ϕ < ϕ− and theRankine-Hugoniot condition holds on S := ∂Ω+ ∩ Ω.

On the other hand, the elliptic estimates alone are not sufficient to get theexistence of a fixed point, because the right-hand side of the boundary condition inthe problem (3.4.39) depends on the unit normal n of the free boundary. One wayis to require the orthogonality of the flat shocks so that

ρ(|Dϕ+0 |2)Dϕ+

0 = ρ(|Dϕ−0 |2)Dϕ−

0 in Ω (3.4.40)

to obtain better estimates for the iteration and to prove the existence of a fixedpoint. Note that (3.4.40) is a vector identity, and the Rankine-Hugoniot condition(3.4.4) is the normal part of (3.4.40) on the unperturbed free boundary S0.

The uniqueness and stability of solutions for the transonic shock problems areobtained by using the regularity and nondegeneracy of solutions.


3.4.3 Partial Hodograph Approach

The second approach we developed in 22; 23 is a partial hodograph procedure, withwhich we can handle the existence and stability of multidimensional transonic shocksthat are not nearly orthogonal to the flow direction. One of the main ingredientsin this new approach is to employ a partial hodograph transform to reduce the freeboundary problem to a conormal boundary value problem for the correspondingnonlinear second-order elliptic equation of divergence form in unbounded domainsand then develop techniques to solve the conormal boundary value problem in theunbounded domain. To achieve this, the strategy is to construct first solutions in theintersection domains between the physical unbounded domain under considerationand a series of half balls with radius R, then make uniform estimates in R, and finallysend R → ∞. It requires delicate apriori estimates to achieve this. A uniform boundin a weighted L∞-norm can be achieved by employing a comparison principle andidentifying a global function with the same decay rate as the fundamental solution ofthe elliptic equation with constant coefficients which controls the solutions. Then, byscaling arguments, the uniform estimates can be obtained in a weighted Holder normfor the solutions, which lead to the existence of a solution in the unbounded domainand some decay rate of this solution at infinity. For such decaying solutions, acomparison principle holds, which implies the uniqueness for the conormal problem.Finally, by the gradient estimate, the limit function can be shown to be a solutionof the multidimensional transonic shock problem, and the existence result can beextended to the case that the regularity of the steady perturbation is only C1,1. Wecan further prove that the multidimensional transonic shock solution is stable withrespect to the C2,α supersonic perturbation.

The approach can also be extended by using the partial hodograph transformin the radial direction in the polar coordinates to establish the existence and stabilityof multidimensional transonic shocks near spheres in IRn, n ≥ 3. The case n = 2can also be handled with similar approaches.

When the regularity of the steady perturbation is C3,α or higher, that is,

‖ϕ− − ϕ−0 ‖

(n−1)3,α,Ω1

≤ σ, (3.4.41)

we have introduced the implicit function approach, the third simpler approach, todeal with the existence and stability problem. For more details, see Chen-Feldman22; 23.

Another approach has also been introduced and can be found in Canic-Keyfitz-Lieberman 14, Canic-Keyfitz-Kim 15, and Zheng 111.

3.5. A THEORY OF DIVERGENCE-MEASURE FIELDS 61

3.5 A Theory of Divergence-Measure Fields

A vector field is called a divergence-measure field if the divergence of the field isa Radon measure; these fields include vector fields in Lp, 1 ≤ p ≤ ∞, and vector-valued Radon measures, whose divergences are Radon measures. The DM -fieldsarise naturally in the study of the behavior of entropy solutions of nonlinear hyper-bolic systems of conservation laws. When the initial data has sufficiently small totalvariation and stays away from vacuum for (3.1.14), the Glimm theorem 57 indicatesthat there exists a global entropy solution in BV satisfying the Clausius inequality:

St ≥ 0 (3.5.1)

in the sense of distributions. On the other hand, when the initial data is allowedto be large, still away from vacuum, the solutions may develop vacuum in finitetime, even instantaneously as t > 0. In this case, the specific volume τ = 1/ρ thenbecomes a Radon measure or an L1 function, rather than a function of boundedvariation. This indicates that solutions of nonlinear hyperbolic conservation lawsare generally either in M(IR+ × IRn), the space of signed Radon measures, or inLp(IR+ × IRn), 1 ≤ p ≤ ∞. On the other hand, the fact that (3.1.14) and (3.5.1)hold in the sense of distributions implies that the divergences of the fields (τ,−v),(v, p), (e+ v2/2, pv), and (S, 0) in the (t, x) variables are Radon measures, in whichthe first three are the trivial null measure and the last one is a nonnegative measureas a consequence of the Schwartz Lemma. This motivates our study of the extendeddivergence-measure fields.

Let D ⊂ IRN be open. For F ∈ Lp(D; IRN), 1 ≤ p ≤ ∞, or F ∈ M(D; IRN),set

|divF |(D) := sup∫D∇ϕ · F : ϕ ∈ C1

0(D; IR), |ϕ(x)| ≤ 1, x ∈ D .

For 1 ≤ p ≤ ∞, we say that F is an Lp-divergence-measure field over D, i.e.,F ∈ DMp(D), if F ∈ Lp(D; IRN) and

‖F‖DMp(D) := ‖F‖Lp(D;IRN ) + |divF |(D) <∞. (3.5.2)

We say that F is an extended divergence-measure field over D, i.e., F ∈ DM ext(D),if F ∈ M(D; IRN) and

‖F‖DMext(D) := |F |(D) + |divF |(D) <∞. (3.5.3)

We say F ∈ DMploc(IR

N) if F ∈ DMp(D) for any open set D IRN , and we sayF ∈ DM ext

loc (IRN) if F ∈ DM ext(D) for any open set D IRN .


It is easy to check that these spaces under the norms (3.5.2) and (3.5.3), re-spectively, are Banach spaces. These spaces are larger than the space of BV -fields.The establishment of the Gauss-Green theorem, traces, and other properties of BVfunctions in the middle of last century (see 4; 55; 56; 102) has significantly advancedour understanding of solutions of nonlinear partial differential equations and non-linear problems in calculus of variations, differential geometry, and other areas. Anatural question is whether the DM -fields have similar properties, especially thenormal traces and the Gauss-Green formula. At a first glance, it seems impossibledue to the Whitney paradox 103.

Whitney paradox. The field F (x, y) = (− yx2+y2 ,

xx2+y2 ) belongs to DM1

loc(IR2).

As remarked in Whitney 103, for Ω = (0, 1) × (0, 1),∫Ω

divF dxdy = 0 =∫

∂Ω

F · n dH1 =π

2,

if one understands F ·n in the classical sense, which implies that the classical Gauss-Green theorem fails.

Motivated by various nonlinear problems from conservation laws, we first re-solved the Whitney paradox and succeeded in using the neighborhood informationvia the Lipschitz deformation to develop a natural notion of normal traces, un-der which a generalized Gauss-Green theorem holds, even for F ∈ DM ext(D) inChen-Frid 24; 26. In particular, we identified an explicit way to calculate the nor-mal traces over any deformable Lipschitz surface, suitable for applications, by usingthe neighborhood information of the fields near the surface and the level set func-tion of the Lipschitz deformation surfaces. We also established product rules andextension theorems for these extended fields. We further extended the theory ofDM -fields over sets of finite perimeters in Chen-Torres 31 and Chen-Torres-Ziemer32. The proofs require some refined properties of Radon measures, the Whitneyextension theory, and geometric measure theory, among others. For more details,see 19; 24; 26; 31; 32.

acknowledgments

The research of Gui-Qiang Chen was supported in part by the National ScienceFoundation under Grants DMS-0244473, DMS-0204225, DMS-0426172, and INT-9987378, and an Alexandre von Humboldt Foundation Fellowship. The authorthanks his collaborators M. Feldman, H. Frid, Ph. LeFloch, M. Torres, D. Wang, Y.Zhang, D. Zhu, and W. Ziemer for their explicit and implicit contributions to thispaper.


References

[1] Alt, H. W. and Caffarelli, L. A., Existence and regularity for a minimum problemwith free boundary, J. Reine Angew. Math. 325 (1981), 105–144.

[2] Alt, H. W., Caffarelli, L. A., and Friedman, A., A free-boundary problem forquasilinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 11 (1984),1–44.

[3] Alt, H. W., Caffarelli, L. A., and Friedman, A., Compressible flows of jets andcavities, J. Diff. Eqs. 56 (1985), 82–141.

[4] Ambrosio, L., Fusco, N., and Pallara, D., Functions of Bounded Variation andFree Discontinuity Problems, The Clarendon Press, Oxford University Press: NewYork, 2000.

[5] Artola, M. and Majda, A., Nonlinear development of instabilities in supersonicvortex sheets, I: the basic kink modes, Phys. D. 28 (1987), 253-281; II: Resonantinteraction among kink modes, SIAM J. Appl. Math. 49 (1989), 1310-1349; Non-linear kink modes for supersonic vortex sheets, Phys. Fluids A, 1 (1989), 583–596.

[6] Berestycki, H., Caffarelli, L., and Nirenberg, L., Uniform estimates for regulariza-tion of free boundary problems, In: Analysis and Partial Differential Equations,pp. 567–619, Lecture Notes in Pure and Appl. Math. 122, Dekker: New York,1990.

[7] Bianchini, S. and Bressan, A., Vanshing viscosity solutions of nonlinear hyperbolicsystems, Preprint, 2001.

[8] Bressan, A., Global solutions of systems of conservation laws by wave-front track-ing, J. Math. Anal. Appl. 170 (1992), 414-432.

[9] Bressan, A., The unique limit of the Glimm scheme, Arch. Rational Mech. Anal.130 (1995), 205-230.

[10] Bressan, A., Hyperbolic Systems of Conservation Laws: The One DimensionalCauchy Problem, Oxford University Press: Oxford, 2000.

[11] Bressan, A. and LeFloch, P., Uniqueness of weak solutions to systems of con-servation laws, Arch. Rational Mech. Anal. 140 (1970), 301-317.

[12] Bressan, A., Liu, T.-P., and Yang, T., L1 stability estimates for n × n conser-vation laws, Arch. Rational Mech. Anal. 149 (1999), 1-22.

[13] Caffarelli, L. A., A Harnack inequality approach to the regularity of free bound-aries, I. Lipschitz free boundaries are C1,α, Rev. Mat. Iberoamericana, 3 (1987),139–162; II. Flat free boundaries are Lipschitz, Comm. Pure Appl. Math. 42(1989), 55–78; III. Existence theory, compactness, and dependence on X, Ann.


Scuola Norm. Sup. Pisa Cl. Sci. (4), 15 (1989), 583–602.

[14] Canic, S., Keyfitz, B. L., and Lieberman, G., A proof of existence of perturbedsteady transonic shocks via a free boundary problem, Comm. Pure Appl. Math.53 (2000), 484-511.

[15] Canic, S., Keyfitz, B. L., and Kim, E. H., Free boundary problems for the un-steady transonic small disturbance equation: transonic regular reflection, MethodsAppl. Anal. 7 (2000), 313–335; A free boundary problem for a quasi-linear degen-erate elliptic equation: regular reflection of weak shocks, Comm. Pure Appl. Math.55 (2002), 71–92.

[16]Chang, T. and Hsiao, L., The Riemann Problem and Interaction of Waves inGas Dynamics, Longman Scientific & Technical: Essex, England, 1989.

[17]Chen, G.-Q., Overtaking of shocks of the same kind in the isentropic steadysupersonic plane flow, Acta Math. Sci. (English Ed.), 7 (1987), 311–327.

[18] Chen, G.-Q., Convergence of the Lax-Friedrichs scheme for isentropic gas dy-namics (III), Acta Math. Sci. 6 (1986), 75-120 (in English); 8 (1988), 243-276 (inChinese).

[19] Chen, G.-Q., Some recent methods for partial differential equations of divergenceform, Bull. Braz. Math. Soc. (N.S.), 34 (2003), 107–144.

[20] Chen, G.-Q. and Feldman, M., Multidimensional transonic shocks and freeboundary problems for nonlinear equations of mixed type, J. Amer. Math. Soc.16 (2003), 461-494.

[21] Chen, G.-Q. and Feldman, M., Steady transonic shocks and free boundary prob-lems in infinite cylinders for the Euler equations, Comm. Pure Appl. Math. 57(2004), 310–356.

[22] Chen, G.-Q. and Feldman, M., Free boundary problems and transonic shocksfor the Euler equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl.Sci. (5), 4 (2004) (to appear).

[23] Chen, G.-Q. and Feldman, M., Existence and stability of multidimensional tran-sonic flows through an infinite nozzle of arbitrary cross-sections, Preprint, Decem-ber 2003; also Preprint No. 2004-069, http://www.math.ntnu.no/conservation/2004.

[24] Chen, G.-Q. and Frid, H., Divergence-measure fields and hyperbolic conservationlaws, Arch. Rational Mech. Anal. 147 (1999), 89-118.

[25] Chen, G.-Q. and Frid, H., Decay of entropy solutions of nonlinear conservationlaws, Arch. Rational Mech. Anal. 146 (1999), 95-127.

[26] Chen, G.-Q. and Frid, H., Extended divergence-measure fields and the Eulerequations for gas dynamics, Commun. Math. Phys. 236 (2003), 251–280.


[27] Chen, G.-Q. and LeFloch, P., Compressible Euler equations with general pres-sure law, Arch. Rational Mech. Anal. 153 (2000), 221-259.

[28] Chen, G.-Q. and LeFloch, P., Entropies and flux-splittings for the isentropicEuler equations, Chinese Ann. Math. 22B (2001), 145-158.

[29] Chen, G.-Q. and LeFloch, P., Existence theory for the isentropic Euler equations,Arch. Rational Mech. Anal. 66 (2003), 81–98.

[30] Chen, G.-Q., Li, B.-H., and Li, T.-H., Entropy solutions in L∞ for the Eulerequations in nonlinear elastodynamics and related equations, Arch. Rational Mech.Anal. 170 (2003), 331–357.

[31] Chen, G.-Q. and Torres, M., Divergence measure fields, sets of finite perimeter,and conservation laws, Arch. Rational Mech. Anal. 175 (2005), 245-267.

[32] Chen, G.-Q., Torres, M., and Ziemer, W., Cauchy flux, balance laws, anddivergence-measure fields, Preprint, Northwestern University, November 2004.

[33] Chen, G.-Q. and Wagner, D., Global entropy solutions to exothermically react-ing, compressible Euler equations, J. Diff. Eqs. 191 (2003), 277-322.

[34] Chen, G.-Q. and Wang, D., The Cauchy Problem for the Euler Equations forCompressible Fluids, Chapter 5, Handbook of Mathematical Fluid Dynamics, Vol.1, pp. 421–543, 2002. Elsevier Science B. V: Amsterdam, The Netherlands.

[35] Chen, G.-Q., Zhang, Y., and Zhu, D., Existence and stability of supersonic Eulerflows past Lipschitz wedges, Preprint, Northwestern University, November 2004.

[36] Chen, G.-Q., Zhang, Y., and Zhu, D., Stability of supersonic vortex sheetsin steady Euler flows past Lipschitz walls, Preprint, Northwestern University,December 2004.

[37] Chen, S., Supersonic flow past a concave wedge, Science in China, 10A (27)(1997), 903-910.

[38] Chen, S., Asymptotic behavior of supersonic flow past a convex combined wedge,Chinese Ann. Math. 19B (1998), 255-264.

[39] Chen, S., Global existence of supersonic flow past a curved convex wedge, J.Partial Diff. Eqs. 11 (1998), 73-82.

[40]Chen, S., Xin, P., and Yin, H., Global shock waves for the supersonic flow pasta perturbed cone, Commun. Math. Phys. 228 (2002), 47–84.

[41] Chorin, A., Random choice solution of hyperbolic systems, J. Comp. Phys. 22(1976), 517–533; Random choice methods with applications to reacting gas flow,J. Comp. Phys. 25 (1977), 253–272.


[42] Courant, R. and Friedrichs, K. O., Supersonic Flow and Shock Waves, WileyInterscience: New York, 1948.

[43] Dafermos, C., Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag: Berlin, 2000.

[44] Dafermos, C. M., Polygonal approximations of solutions of the initial-valueproblem for a conservation law, J. Math. Anal. Appl. 38 (1972), 33-41.

[45] Dafermos, C. M. and Hsiao, L., Hyperbolic systems of balance laws with inho-mogeneity and dissipation, Indiana Univ. Math. J. 31 (1982), 471-491.

[46] Ding, X., Zhang, T., Wang, C.-H., Hsiao, L., and Li, T.-C., A study of theglobal solutions for quasilinear hyperbolic systems of conservation laws, ScienticaSinica, 16 (1973), 317-335.

[47] Ding, X., Chen, G.-Q., and Luo, P., Convergence of the Lax-Friedrichs schemefor isentropic gas dynamics (I)-(II), Acta Math. Sci. 5 (1985), 483-500, 501-540(in English); 7 (1987), 467-480, 8 (1988), 61-94 (in Chinese).

[48] Ding, X., Chen, G.-Q., and Luo, P., Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for isentropic gas dynamics, Commun.Math. Phys. 121 (1989), 63-84.

[49] DiPerna, R., Existence in the large for quasilinear hyperbolic conservation laws,Arch. Rational Mech. Anal. 52 (1973), 244-257.

[50] DiPerna, R., Decay and asymptotic behavior of solutions to nonlinear hyperbolicsystems of conservation laws, Indiana Univ. Math. J. 24 (1975), 1047-1071.

[51] DiPerna, R., Global existence of solutions to nonlinear hyperbolic systems ofconservation laws, J. Diff. Eqs. 20 (1976), 187-212.

[52] DiPerna, R., Decay of solutions of hyperbolic systems of conservation laws witha convex extension, Arch. Rational Mech. Anal. 64 (1977), 1-46.

[53] DiPerna, R., Convergence of approximate solutions to conservation laws, Arch.Rational Mech. Anal. 88 (1985), 223-270.

[54] DiPerna, R., Convergence of viscosity method for isentropic gas dynamics, Com-mun. Math. Phys. 91 (1983), 1-30.

[55] Federer, H., Geometric Measure Theory, Springer-Verlag: New York, 1969.

[56] Giusti, E., Minimal Surfaces and Functions of Bounded Variation, BirkhauserVerlag: Basel, 1984.

[57] Glimm, J., Solution in the large for nonlinear systems of conservation laws,Comm. Pure Appl. Math. 18 (1965), 695-715.


[58] Glimm, J. and Lax, P. D., Decay of Solutions of Systems of Hyperbolic Conser-vation Laws, Mem. Amer. Math. Soc. 101 (1970).

[59] Glimm, J. and Majda, A., Multidimensional Hyperbolic Problems and Compu-tations, Springer-Verlag: New York, 1991.

[60] Gu, C., A method for solving the supersonic flow past a curved wedge, FudanJ.(Nature Sci.), 7 (1962), 11-14.

[61] Holden, H. and Risebro, N. H., Front Tracking for Hyperbolic ConservationLaws, Springer-Verlag: New York, 2002.

[62] Hu, J. and LeFloch, P. G., L1 continuous dependence property for systemsconservation laws, Arch. Rational Mech. Anal. 151 (2000), 45-93.

[63] Huang, F. and Wang, Z., Convergence of viscosity solutions for isothermal gasdynamics, SIAM J. Math. Anal. 34 (2002), 595–610.

[64] Keyfitz, B. and Warneck G., The existence of viscous profiles and admissibilityfor transonic shock, Commun. Partial Diff. Eqs. 16 (1991), 1197-1221.

[65] Lax, P. D., Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math.10 (1957), 537-566.

[66] LeFloch, P. G., Hyperbolic Systems of Conservation Laws : The Theory of Clas-sical and Nonclassical Shock Waves, ETH Lecture Notes Series, Birkhauser, 2002.

[67] Lewicka, M., Well-posedness for hyperbolic systems of conservation laws withlarge BV data, Arch. Ration. Mech. Anal. 173 (2004), 415–445.

[68] Lewicka, M. and Trivisa, K., On the L1 well posedness of systems of conservationlaws near solutions containing two large shocks, J. Diff. Eqs. 179 (2002), 133–177.

[69] Li, T., On a free boundary problem, Chinese Ann. Math. 1 (1980), 351-358.

[70] Li, T.-T. and Yu, W.-C., Boundary Value Problems for Quasilinear Hyper-bolic Systems, Duke University Mathematics Series, V. Durham: Duke University,Mathematics Department. X, 1985.

[71] Lien W. and Liu, T. P., Nonlinear stability of a self-similar 3-dimensional gasflow, Commun. Math. Phys. 204 (1999), 525-549.

[72] Lions, P.-L., Perthame, B., and Souganidis, P., Existence and stability of entropysolutions for the hyperbolic systems of isentropic gas dynamics in Eulerian andLagrangian coordinates, Comm. Pure Appl. Math. 49 (1996), 599-638.

[73] Lions, P.-L., Perthame, B., and Tadmor, E., Kinetic formulation of the isentropicgas dynamics and p-systems, Commun. Math. Phys. 163 (1994), 169-172.

[74] Liu, T. P., The deterministic version of the Glimm scheme, Commun. Math.Phys. 57 (1977), 135-148.


[75] Liu, T. P., Hyperbolic and Viscous Conservation Laws, CBMS-NSF RegionalConf. Series in Appl. Math. 72, SIAM: Philadelphia, 2000.

[76] Liu, T. P., Solutions in the large for the equations of nonisentropic gas dynamics,Indiana Univ. Math. J. 26 (1977), 147-177.

[77] Liu, T. P., Large-time behaviour of initial and initial-boundary value problemsof a general system of hyperbolic conservation laws, Commun. Math. Phys. 55(1977), 163-177.

[78] Liu, T. P., Initial-boundary value problems for gas dynamics, Arch. RationalMech. Anal. 64 (1977), 137–168.

[79] Liu, T. P., Decay to N-waves of solutions of general systems of nonlinear hyper-bolic conservation laws, Comm. Pure Appl. Math. 30 (1977), 585-610.

[80] Liu, T. P., Linear and nonlinear large-time behavior of solutions of hyperbolicconservation laws, Comm. Pure Appl. Math. 30 (1977), 767-796.

[81] Liu, T. P., Quasilinear hyperbolic systems, Commun. Math. Phys. 68 (1979),141-172.

[82] Liu, T.-P. and Yang, T., L1 stability for systems of hyperbolic conservationlaws, In, Nonlinear Partial Differential Equations and Applications, G.-Q. Chenand E. DiBenedetto (eds.), Contemp. Math. 238, pp. 183-192, AMS: Providence,1999.

[83] Liu, T.-P. and Yang, T., Well-posedness theory for hyperbolic conservation laws,Comm. Pure Appl. Math. 52 (1999), 1553-1586.

[84] Morawetz, C. S., On a weak solution for transonic flow problem, Comm. PureAppl. Math. 38 (1985), 797-818.

[85] Murat, F., Compacite par compensation, Ann. Scuola Norm. Sup. Pisa Sci.Fis. Mat. 5 (1978), 489–507.

[86] Nishida, T., Global solution for an initial-boundary value problem of a quasi-linear hyperbolic systems, Proc. Japan Acad. 44 (1968), 642-646.

[87] Nishida, T. and Smoller, J., Solutions in the large for some nonlinear hyperbolicconservation laws, Comm. Pure Appl. Math. 26 (1973), 183-200.

[88] Nishida, T. and Smoller, J., Mixed problems for nonlinear conservation laws, J.Diff. Eqs. 23 (1977), 244-269.

[89] Peng, Y.-J., Solutions faibles globales pour l’equation d’Euler d’un fluide com-pressible avec de grandes donnees initiales, Commun. Partial Diff. Eqs. 17 (1992),161–187.


[90] Perthame, B. and Tzavaras, A., Kinetic formulation for systems of two conser-vation laws and elastodynamics. (English. English summary) Arch. Ration. Mech.Anal. 155 (2000), 1–48.

[91] Risebro, N. H., A front-tracking alternative to the random choice method, Proc.Amer. Math. Soc. 117 (1993), 1125-1139.

[92] Sable-Tougeron, M., Le N-ondes de Lax pour le probleme mixte, Commun.Partial Diff. Eqs. 19 (1994), 1449-1479.

[93] Schaeffer, D. G., Supersonic flow past a nearly straight wedge, Duke Math. J.43 (1976), 637-670.

[94] Schochet, S., Sufficient conditions for local existence via Glimm’s scheme forlarge BV data, J. Diff. Eqs. 89 (1991), 317-354.

[95] Serre, D., Systems of Conservation Laws, I, II, Cambridge University Press:Cambridge, 1999.

[96] Serre, D., La compacite par compensation pour les systemes hyperboliques nonlineaires de deux equations a une dimension d’espace, J. Math. Pure Appl. (9),65 (1986), 423-468.

[97] Smoller, J., Shock Waves and Reaction-Diffusion Equations, Springer-Verlag:New York, 1983.

[98] Tartar L., Compensated compactness and applications to partial differentialequations, In: Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol.IV, pp. 136–212, Res. Notes in Math. 39, Pitman: Boston, Mass.-London, 1979.

[99] Temple, B., Solutions in the large for the nonlinear hyperbolic conservation lawsof gas dynamics, J. Diff. Eqs. 41 (1981), 96–161.

[100] Temple, B. and Young, R., The large time existence of periodic solutions forthe compressible Euler equations, Mat. Contemp. 11 (1996), 171-190.

[101] Temple, B. and Young, R., The large time stability of sound waves, Commun.Math. Phys. 179 (1996), 417-466.

[102] Volpert, A. I. and Hudjaev, S. I., Analysis in Classes of Discontinuous Func-tions and Equations of Mathematical Physics, Martinus Nijhoff Publishers: Dor-drecht, 1985.

[103] Whitham, G. B., Linear and Nonlinear Waves, Wiley-Interscience: New York,1973.

[104] Woodward, P. R., Simulation of the Kelvin-Helmholtz instability of a su-personic slip surface with the piecewise-parabolic method (PPM), In: Numeri-cal Methods for the Euler Equations of Fluid Dynamics (Rocquencourt, 1983),


493–508, SIAM, Philadelphia, PA, 1985.

[105] Yong, W.-A., A simple approach to Glimm’s interaction estimates, Appl. Math.Let. 12 (1999), 29-34.

[106] Young, R., Talk at the Conference on Analysis, Modelling and Computationof PDE and Multiphase Flow, August 3-5, 2004, Stony Brook.

[107] Zhang, T. and Guo, Y.-F., A class of initial-value problem for systems ofaerodynamic equations, Acta Math. Sinica, 15 (1965), 386-396.

[108] Zhang, Y., Global existence of steady supersonic potential flow past a curvedwedge with piecewise smooth boundary, SIAM J. Math. Anal. 31 (1999), 166-183.

[109] Zhang, Y., Steady supersonic flow past an almost straight wedge with largevertex angle, J. Diff. Eqs. 192 (2003), 1-46.

[110] Zheng, Y., Systems of Conservation Laws: Two-Dimensional Riemann Prob-lems, Birkhauser: Boston, 2001.

[111] Zheng, Y., A global solution to a two-dimensional Riemann problem involvingshocks as free boundaries, Acta Math. Appl. Sin. 19 (2003), 559–572.

Chapter 4

Numerical Simulation ofNonisothermal MultiphaseMulticomponent Flow in PorousMedia

Zhangxin ChenCenter for Scientific ComputationBox 750156, Southern Methodist UniversityDallas, TX 75275-0156, [email protected]

Yuanle MaINET, Tsinghua UniversityBeijing 100084, P. R. [email protected]

Abstract: Thermal methods, particularly steam drive and soak, occupy avery large share of the enhanced oil recovery projects in petroleum industry. How-ever, they involve very complex nonisothermal multiphase, multicomponent flow andtransport processes in porous media. These processes deal with not only the transferof mass but also the transfer of energy between phases. Numerical simulation can beutilized to conduct mechanism study, feasibility evaluation, pilot plan optimization,and performance prediction for the thermal methods to improve recovery efficiencyand reduce operational costs. In this paper, we develop a multi-dimensional numer-ical simulator for nonisothermal multiphase, multicomponent flow and transport in

71

72 CHAPTER 4. NUMERICAL SIMULATION OF NONISOTHERMAL FLOW

porous media. This numerical simulator is based on a fully implicit, coupled solu-tion approach for reservoir domains. Numerical experiments are reported for thebenchmark problems of the fourth comparative solution project organized by thesociety of petroleum engineers.

Keywords: Multiphase flow, porous media, petroleum reservoir, numerical sim-ulation, thermal recovery, numerical experiments

Introduction

Most of numerical reservoir simulators have been thus far based on the black oil andcompositional models [2] [10]. These two models describe mass conservation, Darcy’slaw, and mass interchange between fluid phases; they do not involve nonisothermalflow and transport processes in porous media. The nonisothermal multiphase, multi-component flow and transport processes involve more physics and are more complexthan the isothermal processes in the black oil and compositional models. Not onlydo they introduce one more unknown (temperature or energy), they also introducegreater nonlinearity and coupling in the governing equations. Hence numerical sim-ulation for nonisothermal flow and transport is far more complicated than that forthe isothermal case.

Thermal methods (particularly steam drive and soak) involve nonisothermalmultiphase, multicomponent flow and transport processes in petroleum reservoirs,and occupy a very large share of the enhanced oil recovery (EOR) projects in petro-leum industry. They have experienced rapid growth since the early 1970s. Steammethods currently account for nearly 80% of the EOR oil in USA [7], for exam-ple. Thermal flooding has been commercially successful for the past over 30 years.Therefore, the development of numerical thermal simulators has become increasinglyimportant.

The thermal methods rely on several displacement mechanisms to recover oil,such as viscosity reduction, distillation, miscible displacement, thermal expansion,wettability changes, cracking, and lowered oil-water interfacial tension. For manyapplications, the most important is the reduction of crude viscosity with increasingtemperature. The four basic approaches to achieve this mechanism are hot waterflooding, steam soak, steam drive, and in situ combustion. In a steam soak (stim-ulation or huff’n puff), for example, steam is introduced into a well, and then thewell is returned to production after a brief shut-in period.

Fluid flow models in porous media involve large systems of nonlinear, coupled,time-dependent partial differential equations. An important problem in reservoirsimulation is to develop stable, efficient, robust, and accurate solution approaches

4.1. GOVERNING DIFFERENTIAL EQUATIONS 73

for solving these coupled equations. Essentially, there are three types of solutionapproaches in reservoir simulation: the IMPES (implicit in pressure and explicit insaturation), the sequential, and the fully implicit. The sequential approach [9] splitsthe coupled system of nonlinear governing equations up into individual equationsand solves each of these equations separately and implicitly. The fully implicitapproach, which is also called the simultaneous solution method [6], solves all of thecoupled nonlinear equations simultaneously and implicitly. In thermal methods, inaddition to the above-mentioned strong nonlinearity and coupling in the governingequations, pressure and temperature greatly vary, and the mass and energy transferbetween the liquid and vapor phases frequently occur. When applied to the thermalmethods, the IMPES and sequential solution approaches are far less stable thanthe fully implicit approach. The latter approach can take large time steps, whileits stability is maintained. Therefore, in the present paper, by a careful choiceof the primary unknowns this approach is employed to solve the system of thegoverning equations that describe nonisothermal multiphase, multicomponent flowand transport in porous media. Nonlinearity is handled with Newton-Raphson’siterations.

In this paper, we develop a multi-dimensional numerical simulator for non-isothermal multiphase, multicomponent flow and transport in porous media. Thespatial discretization scheme of this simulator is based on the block-centered finitedifference method with the coefficients of differential equations harmonically aver-aged (equivalently, a mixed finite element method on rectangular parallelepipeds[13]). Numerical results are reported for the benchmark problems of the fourthcomparative solution project (CSP) organized by the society of petroleum engineers(SPE) [1], and show that this numerical simulator performs very well for large-scalethermal problems.

The rest of this paper is organized as follows. In the next section, we reviewthe differential equations governing thermal recovery simulation. Then, in the thirdsection, we choose the primary unknowns and present a fully implicit approach usingthese unknowns. Numerical experiments are presented in the fourth section.

4.1 Governing Differential Equations

The basic equations for nonisothermal multiphase, multicomponent flow and trans-port in a porous medium Ω involve mass conservation, Darcy’s laws, energy conser-vation, and mole fraction, saturation, and capillary pressure constraint equations.These equations are based on the displacement mechanisms of thermal methods:(a) reduction of crude viscosity with increasing temperature, (b) change of relativepermeabilities for greater oil displacement, (c) vaporization of connate water and


of portion of crudes for a miscible displacement of light components, and (d) hightemperatures of fluids and rock to maintain high reservoir pressure. They can modelthe important physical factors and processes:

• viscosity, gravity, and capillary forces,

• heat conduction and convection processes,

• heat losses to overburden and underburden of a reservoir,

• mass transfer between phases,

• effects of temperature on physical property parameters of oil, gas, and water,

• rock compression and expansion.

We assume that the chemical components form at most three phases (e.g.,water, oil, and gas), there are Nc chemical components that may exist in all threephases, and the diffusive effects are neglected.

Let φ and k denote the porosity and permeability of the porous medium Ω ⊂IR3, and Sα, µα, pα, uα, and krα be the saturation, viscosity, pressure, volumetricvelocity, and relative permeability of the α phase, α = w, o, g, respectively. Also,let ξiα represent the molar density of component i in the α phase, i = 1, 2, . . . , Nc,α = w, o, g. The molar density of phase α is given by

ξα =Nc∑i=1

ξiα, α = w, o, g. (4.1.1)

The mole fraction of component i in phase α is then defined by

xiα = ξiα/ξα, 1, 2, . . . , Nc, α = w, o, g. (4.1.2)

The total mass is conserved for each component:

∂

∂t

g∑α=w

xiαξαSα + ∇ ·g∑

α=w

xiαξαuα

=

g∑α=w

xiαqα, i = 1, . . . , Nc,

(4.1.3)

where qα stands for the flow rate of phase α at wells. In equation (4.1.3), thevolumetric velocity uα is given by Darcy’s law:

uα = −krα

µα

k (∇pα − ρα℘∇z) , α = w, o, g, (4.1.4)


where ρα is the mass density of the α-phase, ℘ is the magnitude of the gravitationalacceleration, and z is the depth. The energy conservation equation takes the form

∂

∂t

(φ

g∑α=w

ραSαUα + (1 − φ)ρsCsT

)

+∇ ·g∑

α=w

ραuαHα −∇ · (kT∇T ) = qc − qL,

(4.1.5)

where T is temperature, Uα and Hα are the specific internal energy and enthalpy ofthe α-phase (per unit mass), ρs and Cs are the density and the specific heat capacityof the solid, kT represents the total thermal conductivity, qc denotes the heat sourceitem, and qL indicates the heat loss to overburden and underburden. In (4.1.5), thespecific internal energy Uα and enthalpy Hα of phase α can be computed as follows:

Uα = CV αT, Hα = CpαT,

where CV α and Cpα represent the heat capacities of phase α at constant volume andpressure, respectively.

In addition to the differential equations (4.1.3)–(4.1.5), there are also algebraicconstraints. The mole fraction balance implies

Nc∑i=1

xiα = 1, α = w, o, g. (4.1.6)

In the transport process, the saturation constraint reads

Sw + So + Sg = 1. (4.1.7)

Finally, the phase pressures are related by capillary pressures

pcow = po − pw, pcgo = pg − po. (4.1.8)

The equilibrium relations describing the distribution of hydrocarbon componentsinto the phases are given by

fiw(pw, T, x1w, x2w, . . . , xNcw) = fio(po, T, x1o, x2o, . . . , xNco),

fio(po, T, x1o, x2o, . . . , xNco) = fig(pg, T, x1g, x2g, . . . , xNcg),(4.1.9)

where fiα is the fugacity function of the ith component in the α phase, i = 1, 2, . . . , Nc,α = w, o, g.

In thermal methods, heat is lost to the adjacent strata of a reservoir or theoverburden and underburden, which is included in qL of (4.1.5). We assume that the


reservoir

underburden

overburden

Fig. 1. Reservoir, overburden, and underburden.

overburden and underburden extend to infinity along both the positive and negativex3-axis (the vertical direction); see Fig. 1. If the overburden and underburdenare impermeable, heat is entirely transferred through conduction. With all fluidvelocities and convective fluxes being zero, the energy conservation equation (4.1.5)reduces to

∂

∂t(ρobCp,obTob) = ∇ · (kob∇Tob), (4.1.10)

where the subscript ob indicates that the variables are associated with the overburdenand Cp,ob is the heat capacity at constant pressure. The initial condition is theoriginal temperature Tob,0 of the overburden:

Tob(x, 0) = Tob,0(x).

The boundary condition at x3 = 0 (the top of the reservoir) is

Tob(x1, x2, 0, t) = T (x1, x2, 0, t).

At infinity, Tob is fixed:Tob(x1, x2,∞, t) = T∞.

On other boundaries, we can use the impervious boundary condition

kob∇Tob · ν = 0,

where ν represents the outward unit normal to these boundaries. Now, the rateof heat loss to the overburden can be calculated by kob∇Tob · ν, where ν is theunit normal to the interface between the overburden and reservoir (pointing to theoverburden). For the underburden, the heat conduction equation is given by

∂

∂t(ρubCp,ubTub) = ∇ · (kub∇Tub), (4.1.11)


and similar initial and boundary conditions can be developed as for the overburden.

Equations (4.1.3)–(4.1.9) provide 3Nc+10 independent relations, differential oralgebraic, for the 3Nc + 10 dependent variables: xiα, uα, pα, T , and Sα, α = w, o, g,i = 1, 2, . . . , Nc. If equations (4.1.10) and (4.1.11) are included, two more unknownsTob and Tub are added. With proper initial and boundary conditions, there is aclosed differential system for these unknowns.

4.1.1 Rock properties

The rock properties for nonisothermal flow are similar to those for the isothermalblack oil and compositional models; now, these properties depend on temperature.In particular, the capillary pressures are of the form

pcw(Sw, T ) = pw − po, pcg(Sg, T ) = pg − po, (4.1.12)

where pcw = −pcow and pcg = pcgo. For notational convenience, set pco = 0. Similarly,the relative permeabilities for water, oil, and gas are

krw = krw(Sw, T ), krow = krow(Sw, T ),

krg = krg(Sg, T ), krog = krog(Sg, T ),

kro = kro(Sw, Sg, T ).

(4.1.13)

Stone’s models [15] [16] can be adopted for the oil relative permeability kro, forexample.

As an example, the relative permeability functions krw and krow for a water-oilsystem can be defined by

krw = krwro(T )

(Sw − Swir(T )

1 − Sorw(T ) − Swir(T )

)nw

,

krow = krocw(T )

(1 − Sw − Sorw(T )

1 − Sorw(T ) − Swc(T )

)now

,

(4.1.14)

and for a gas-oil system, krg and krog by

krg = krgro(T )

(Sg − S∗

gr

1 − Swc(T ) − Soinit − S∗gr

)ng

,

krog = krocw(T )

(1 − Sg − Swc(T ) − Sorg(T )

1 − Swc(T ) − Sorg(T )

)nog

,

(4.1.15)

where nw, now, ng, and nog are nonnegative real numbers, Swc, Swir, Sorw, Sorg,and S∗

gr are the connate water saturation, irreducible water saturation, residual oil


saturation in the water-oil system, residual oil saturation in the gas-oil system, andresidual gas saturation, krwro, krocw, and krgro are the water relative permeability atthe residual oil saturation for the water-oil system, the oil relative permeability atthe connate water saturation, and the gas relative permeability at the residual oilsaturation for the gas-oil system, respectively, and Soinit is the initial oil saturation inthe gas-oil system. Finally, for the rock properties, one needs to consider the thermalconductivity and heat capacity of the reservoir, overburden, and underburden.

4.1.2 Fluid properties

Several EOS (equations of state) can be used to define the fugacity functions fiα

in equation (4.1.9), such as the Redlich-Kwong, Redlich-Kwong-Soave, and Peng-Robinson EOS [5] [11]. Because of complexity of nonisothermal flow, however, anequilibrium K-value approach is often used to describe the equilibrium relations:

xiw = Kiw(p, T )xio, xig = Kig(p, T )xio, i = 1, 2, . . . , Nc. (4.1.16)

One of the examples for evaluating the K-values Kiα uses the empirical formula

Kiα =

(κ1

iα +κ2

iα

p+ κ3

iα p

)exp

(− κ4

iα

T − κ5iα

), (4.1.17)

where the constants κjiα may be obtained in laboratory, i = 1, 2, . . . , Nc, j =

1, 2, 3, 4, 5, α = w, g, and p and T are pressure and temperature. For the nota-tional convenience, we use Kio = 1, i = 1, 2, . . . , Nc.

Water properties

Physical properties of water and steam, such as density, internal energy, enthalpy,and viscosity, can be found from a water-steam table [7]. Such a table is given interms of the independent variables: pressure and temperature. In the case whereall three phases co-exist, a reservoir is referred to as in the saturated state. In thiscase, there is free gas; pressure and temperature are related, and only one of themis employed as an independent variable.

Oil properties

While any number of hydrocarbon components can be treated in the differentialsystem describing the nonisothermal multiphase, multicomponent flow consideredin this section, computational work and time significantly increases as the number

4.2. SOLUTION APPROACHES 79

of components increases. It is often computationally convenient (or necessary) togroup several similar chemical components into one mathematical component. Inthis way, only several components (or pseudo-components) need to be simulated inpractical applications.

The oil phase is a mixture of hydrocarbon components, and these componentsrange from the lightest component, methane (CH4), to the heaviest component, bitu-men. A way to reduce the number of components is to introduce pseudo-components,as noted. According to the compositions of each pseudo-component, one can deduceits physical properties, such as its pseudo-molecular weight (which may not be a con-stant), critical pressure and temperature, compressibility, density, viscosity, thermalexpansion coefficient, and specific heat. Apparently, these properties are functionsof pressure and temperature.

The most important property is the oil and gas phase viscosity dependence ontemperature:

µio = exp(a1T

b1)

+ c1, µig = a2Tb2 ,

where T is in absolute degree, a1, b1, c1, a2, and b2 are empirical parameters and canbe measured in laboratory, and µio and µig are the viscosities of the ith componentin the oil and gas phases, respectively.

4.2 Solution Approaches

In simulation of nonisothermal flow, three parts must be treated: the oil reservoir,overburden, and underburden. Because of weak coupling between the reservoir andthe overburden and underburden, the equations in these three parts can be de-coupled; that is, they are solved in a sequential manner. As noted, in the reservoirpart, the IMPES, sequential, and fully implicit solution approaches can be employed.Because there exists strong nonlinearity and coupling in the governing equations,pressure and temperature greatly vary, and the mass and energy transfer betweenthe oil and gas phases frequently occur, the fully implicit approach is an appro-priate approach for the reservoir system [4]. The heat conduction equations overoverburden and underburden are simple enough that a fully implicit scheme in timeapproximation can be also applied. We concentrate on the discussion of this solutionapproach for the reservoir domain.

4.2.1 Undersaturated state

As noted, if all three phases co-exist, a reservoir is referred to as in the saturatedstate. When all gas dissolves into the oil phase (i.e., there is no free gas; Sg = 0), the


reservoir is said to be in the undersaturated state. The choice of primary unknownsdepends on the state of a reservoir.

We introduce the potentials

Φα = pα − ρα℘z, α = w, o, g. (4.2.1)

Also, we define the transmissibilities

Tα =ραkrα

µα

k,

Tiα =xiαξαkrα

µα

k, i = 1, 2, . . . , Nc, α = w, o, g.(4.2.2)

Moreover, we use the total mole fraction

xi =

g∑α=w

xiα, i = 1, 2, . . . , Nc. (4.2.3)

Using (4.1.16), equation (4.2.3) becomes

xio =1

Kiwog(p, T )xi, i = 1, 2, . . . , Nc, (4.2.4)

where Kiwog(p, T ) = Kiw + 1 +Kig. As a result, we see that

xiw =Kiw

Kiwog

xi, xig =Kig

Kiwog

xi, i = 1, 2, . . . , Nc. (4.2.5)

Thus xi should be used as a primary unknown, i = 1, 2, . . . , Nc. Because of (4.1.6),only Nc − 2 of them are independent. In the undersaturated state, we choose theprimary unknowns (p, S, x1, x2, . . . , xNc−2, T ), where p = po and S = Sw. The differ-ential system for these unknowns consists of the Nc component mass conservationequations

∂(φFixi)

∂t=

g∑α=w

∇ · (Tiα∇Φα) +

g∑α=w

xiαqα, i = 1, 2, . . . , Nc, (4.2.6)

and the energy conservation equation

∂

∂t

(φ

g∑α=w

ραSαCV αT + (1 − φ)ρsCsT

)

−∇ ·g∑

α=w

CpαTTα∇Φα −∇ · (kT∇T ) = qc − qL,

(4.2.7)

where

Fi =

g∑α=w

Kiα

Kiwog

ξαSα.

4.3. NUMERICAL EXPERIMENTS 81

4.2.2 Saturated state

In the saturated state, there is free gas. In this case, pressure p and temperatureT are related; in general, their relationship can be given through a saturated steamtable. Thus only one of them can be used as a primary unknown. In the saturatedcase, we choose the primary unknowns (p, Sw, So, x1, x2, . . . , xNc−2), where p = po.The system of differential equations are composed of the Nc component mass con-servation equations in (4.2.6) and the energy conservation equation (4.2.7).

4.3 Numerical Experiments

The experimental problems are chosen from the benchmark problems of the fourthCSP [1]. Six organizations participated in that comparative project. Two relatedsteam injection problems were numerically studied. The first problem deals withcyclic steam injection in a non-distillable petroleum reservoir with two-dimensionalradial cross-sectional grids, and the second problem deals with non-distillable oildisplacement by steam in an inverted nine-spot pattern by considering one-eighthof the full pattern (see Fig. 2). Standard conditions for these problems are 14.7psia and 60F. The problems were chosen to exercise features of the models thatare important in practical applications, though they may not represent a real fieldanalysis.

Table 1. Rock properties.

kh starting with the top layer: 2,000, 500, 1,000, and 2,000 mdkv: 50% of kh

Porosity: 0.3 for all layersThermal conductivity: 24 BTU/(ft.-day-F)Heat capacity: 35 BTU/(ft3 of rock-F)Effective rock compressibility: 5.0E-4 psi−1

Table 2. Oil properties.

Density at standard conditions: 60.68 lb/ft3

Compressibility: 5.0E-6 psi−1

Molecular weight: 600Thermal expansion coefficient: 3.8E-4 1/RSpecific heat: 0.5 BTU/(lb.-R)


Injector330ft

14.585ft

29.17ft

Near producer

Far producer

Fig. 2. Element of symmetry in an inverted nine-spot.

4.3.1 The first problem

The aim is to simulate cyclic steam injection in a two-dimensional reservoir (closedsystem) that has four layers. The rock properties are stated in Table 1, where kh andkv denote the horizontal and vertical permeabilities, respectively, and the thermalconductivity and heat capacity are for the reservoir, overburden, and underburden.Water is assumed to be pure water with standard properties. Oil properties arelisted in Table 2, and the viscosity dependence on temperature is given in Table 3.The capillary pressures are zero. The relative permeability functions are definedby (4.1.14) and (4.1.15) with the data nw = 2.5, now = nog = 2, ng = 1.5,Swc = Swir = 0.45, Sorw = 0.15, Sorg = 0.1, S∗

gr = 0.06, krwro = 0.1, krocw = 0.4,and krgro = 0.2. The initial conditions are presented in Table 4, where the pressuredistribution is according to the gravity head.

Table 3. Oil viscosity dependence on temperature.

Temp (F) 75 100 150 200 250 300 350 500Viscosity (cp) 5,780 1,389 187 47 17.4 8.5 5.2 2.5

Table 4. Initial conditions.

Oil saturation: 0.55Water saturation: 0.45Reservoir temperature: 125FPressure at the center of the top layer: 75 psia

The computational grid uses a cylindrical grid with 13 grid points in the radialdirection. The well radius is 0.3 ft, and the exterior radius is 263.0 ft. The block


Fig. 3. Cumulative oil production (MSTB) vs. time (days).

Fig. 4. Oil production rate (STB/day).

boundaries in the radial direction are at 0.30, 3.0, 13.0, 23.0, 33.0, 43.0, 53.0, 63.0,73.0, 83.0, 93.0, 103.0, 143.0, and 263.0 ft, and the block boundaries in the verticaldirection are at 0.0 (top of pay), 10.0, 30.0, 55.0, and 80.0 ft. The depth to the topof pay is 1,500 ft subsea.

Finally, the operating conditions are summarized as follows: All layers areopen to flow during injection and production (zero skin factor). The energy contentof the injected steam is based on 0.7 quality and 450F. Steam quality at bottomhole conditions is fixed at 0.7. Three cycles are simulated: Each cycle is of 365days with injection for 10 days followed by a 7 day soak period, and the cycle iscompleted with 348 days of production. Steam is injected at capacity subject to thefollowing conditions: The maximum bottom-hole pressure is 1,000 psia at the center


Fig. 5. Cumulative oil production for the full pattern (MSTB vs. days).

of the top layer, and the maximum injection rate is 1,000 STB/day. The productioncapacity is subject to the following constraints: The minimum bottom-hole pressureis 17 psia at the center of the top layer, and the maximum production rate is 1,000STB/day of liquids.

Figs. 3 and 4 show the cumulative oil production and oil production rate,respectively. Compared with the results presented in [1], the two quantities inFigs. 3 and 4 are close to the respective averaged values of those provided by thesix companies for the first problem.

4.3.2 The second problem

The objective is to simulate one-eighth of an inverted nine-spot pattern via sym-metry. The total pattern area is 2.5 acres. The rock and fluid properties, relativepermeability data, and initial conditions are the same as those for the first problem.The grid dimensions are 9× 5× 4 (uniform in the horizontal direction). The radiusof all wells is 0.3 ft.

The operating conditions are given as follows: Injection occurs only in thebottom layer, and production occurs from all four layers. Steam conditions arethe same as in the first problem. Steam is injected at capacity subject to thefollowing conditions: The maximum bottom-hole pressure is 1,000 psia at the centerof the bottom layer, and the maximum injection rate is 1,000 STB/day on a full-well basis. The production capacity is subject to the following constraints: Theminimum bottom-hole pressure is 17 psia at the center of the top layer, the maximumproduction rate is 1,000 STB/day of liquids, and the maximum steam rate is 10


Fig. 6. Oil production rate for the far producer (STB/day).

Fig. 7. Oil production rate for the near producer (STB/day).


STB/day. The simulation time is 10 years of injection and production.

Figs. 5–7 indicate the cumulative oil production for the full pattern, the oilproduction rate for the far producer, and the oil production rate for the near pro-ducer, respectively. All well data presented are on a full-well basis, and the patternresults are for the full pattern consisting of four quarter (far) producers and fourhalf (near) producers. Again, compared with the results presented in [1], the threequantities are close to the respective averaged values of those provided by the sixcompanies for the second problem.

References

[1] Aziz, K., B. Ramesh, and P. T. Woo, Fourth SPE comparative solution project: Acomparison of steam injection simulators, SPE 13510, SPE Reservoir SimulationSymposium Dallas, Texas, February 10–13, 1985, pp. 441–454.

[2] Aziz, K. and A. Settari, Petroleum Reservoir Simulation, Applied Science Pub-lishers Ltd, London, 1979.

[3] Chen, Z., G. Huan, and B. Li, An improved IMPES method for two-phase flowin porous media, Transport in Porous Media 54 (2004), 361–376.

[4] Chen, Z. and Y. Ma, Parallel computation for reservoir thermal simulation ofmulticomponent and multiphase fluid flow, Journal of Computational Physics 201(2004), 224-237.

[5] Coats, K. H., An equation of state compositional model, Soc. Pet. Eng. J. 20(1980), 363–376.

[6] Douglas, Jr., J., D. W. Peaceman, and H H. Rachford, Jr. A method for calculat-ing multi-dimensional immiscible displacement, Trans. SPE of AIME 216 (1959),297–306.

[7] Lake, L. W., Enhanced Oil Recovery, Prentice Hall, Englewood Cliffs, New Jersey,1989.

[8] Li, B., Z. Chen, and G. Huan, Comparison of solution schemes for black oil reser-voir simulations with unstructured grids, Computer Methods in Applied Mechanicsand Engineering 193 (2004), 319–355.

[9] MacDonald, R. C. and K. H. Coats, Methods for numerical simulation of waterand gas coning, Trans. SPE of AIME 249 (1970), 425–436.

[9] Peaceman, D. W., Fundamentals of Numerical Reservoir Simulation, Elsevier,New York, 1977.


[10] Peng, D. Y. and D. B. Robinson, A new two-constant equation of state, Ind.Eng. Chem. Fund. 15 (1976), 59–64.

[11] Price, H. S. and K. H. Coats, Direct methods in reservoir simulation, Soc. Pet.Eng. J. June (1974), 295–308.

[12] Russell, T. F. and M. F. Wheeler, Finite element and finite difference methodsfor continuous flows in porous media, the Mathematics of Reservoir Simulation,R. E. Ewing, ed., SIAM, Philadelphia, pp. 35–106, 1983.

[13] Saad, Y. and M. H. Schultz, GMRES: A generalized minimal residual algorithmfor solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 7 (1986),856–869.

[14] Stone, H. L., Probability model for estimating three–phase relative permeability,J. Pet. Tech. 11 (1970), 214–220.

[15] Stone, H. L., Estimation of three–phase relative permeability and residual oildata, J. Can. Pet. Tech. 12 (1973), 53–67.

[16] Vinsome, P. K. W., ORTHOMIN, an iterative method for solving sparse sets ofsimultaneous linear equations, in Proc. Fourth Symposium on Reservoir simula-tions, Society of Petroleum Engineers of AIME, 1976, 149–157.

Chapter 5

Numerical Simulation of HighMach Number Astrophysical Jets

Carl L. GardnerDepartment of Mathematics and StatisticsArizona State University, Tempe AZ 85287

Youngsoo HaDivision of Applied MathematicsKorean Advanced Institute of Science and TechnologyTaejon, South Korea 305-701

J. Jeff HesterDepartment of Physics and AstronomyArizona State University, Tempe AZ 85287

John E. KristJet Propulsion Laboratory, Pasadena CA 91109

Chi-Wang ShuDivision of Applied MathematicsBrown University, Providence RI 02912

Karl R. StapelfeldtJet Propulsion LaboratoryPasadena CA 91109

Abstract: Computational fluid dynamics simulations using the WENO-LFmethod are applied to high Mach number nonrelativistic astrophysical jets, includingthe effects of radiative cooling. Our numerical methods have allowed us to simulate

89

90 CHAPTER 5. NUMERICAL SIMULATION OF ASTROPHYSICAL JETS

astrophysical jets at much higher Mach numbers than have been attained (Mach 20)in the literature. Mach 80 simulations of the HH 1-2 astrophysical jets and the XZTauri proto-jet are presented.

5.1 Introduction

A new wealth of detail in astrophysical jet gas flows and shock wave patterns hasbeen revealed in recent Hubble Space Telescope images. Simulating the imaged fluidflows and shock waves will help us understand the astrophysical processes at workin these jets.

We apply [1] the WENO-LF method—a modern high-order upwind method—to simulate high Mach number nonrelativistic astrophysical jets from young starsincluding the effects of radiative cooling. In the astrophysical setting, the jet gas ison the order of ten times the density of the ambient gas. Simulations at high Machnumbers and with radiative cooling are essential for achieving detailed agreementwith the astrophysical observations. For example, the gas flows in the HH 1–2astrophysical jets [2] are at about Mach 80. The WENO-LF method allows us tosimulate astrophysical jets at much higher Mach numbers than have been attained(Mach 20) in the literature (see [1] and references therein). Note that the conventionis to specify the Mach number of the jet with respect to the jet gas.

Computer simulations and astrophysical theory will allow us to analyze thedetailed properties of the astrophysical flows including the shock waves that developin and around the jet and the temperatures, densities, velocities, and chemical com-positions of the jets. We are especially interested in how radiative cooling affectsmorphology and propagation of the jets.

Here we describe our implementation of the two-dimensional “slab” jet problemusing the WENO-LF method, including a realistic model for radiative cooling. Thesimulations of the basic jet flows agree well with the Hubble Space Telescope imagesof HH 1–2. In addition, we present a preliminary model of the XZ Tauri proto-jet [3]including the pair of expanding bow shock “bubbles”.

5.2 Gas Dynamics with Radiative Cooling

The equations of gas dynamics with radiative cooling take the form

∂ρ

∂t+

∂

∂xi

(ρui) = 0 (5.2.1)

5.3. NUMERICAL METHODS 91

∂

∂t(ρuj) +

∂

∂xi

(ρuiuj) +∂P

∂xj

= 0 (5.2.2)

∂E

∂t+

∂

∂xi

(ui(E + P )) = −n2Λ(T ) (5.2.3)

where ρ = mHn is the density of the the gas (here assumed to be H), mH is themass of H, n is the number density, ui is the velocity, ρui is the momentum density,P = nkBT is the pressure, kB is Boltzmann’s constant, T is the temperature, and

E =3

2nkBT +

1

2ρu2 (5.2.4)

is the energy density. Indices i, j equal 1, 2, 3, and repeated indices are summedover.

Radiative cooling of the gas is incorporated through the right-hand side ofEq. (5.2.3), with the model for Λ(T ) taken from Fig. 8 of [4].

5.3 Numerical Methods

We use a third-order WENO [5] (weighted essentially non-oscillatory) method basedon the scheme and gas dynamics code of Shu for our supersonic astrophysical flowsimulations. We have extended and adapted [1] the code for simulating very highMach number flows with radiative cooling.

ENO and WENO schemes are high-order finite difference schemes designed fornonlinear hyperbolic conservation laws with piecewise smooth solutions containingsharp discontinuities like shock waves and contacts. Locally smooth stencils arechosen via a nonlinear adaptive algorithm to avoid crossing discontinuities wheneverpossible in the interpolation procedure. The weighted ENO (WENO) schemes use aconvex combination of all candidate stencils, rather than just one as in the originalENO method.

We now describe the computational procedure for the third-order WENOscheme in more detail. Spatial discretization is discussed first. We start with thesimple case of a scalar equation

ut + f(u)x = 0 (5.3.1)

and assume ∂f(u)/∂u ≥ 0, i.e., that the “wind direction” is positive. More generalcases will be described later. A conservative numerical approximation uj(t) to theexact solution u(xj, t) of (5.3.1) satisfies the following ODE system:

duj(t)

dt+

1

∆x

(fj+1/2 − fj−1/2

)= 0 (5.3.2)


where fj+1/2 is called the numerical flux, the design of which is the key ingredient for

a successful scheme. For the third-order WENO scheme, the numerical flux fj+1/2

is defined as follows:

fj+1/2 = ω1f(1)j+1/2 + ω2f

(2)j+1/2 (5.3.3)

where f(m)j+1/2, for m = 1, 2, are the two second-order accurate fluxes on two different

stencils given by

f(1)j+1/2 = −1

2fj−1 +

3

2fj, f

(2)j+1/2 =

1

2fj +

1

2fj+1. (5.3.4)

The nonlinear weights ωm are given by

ωm =ωm∑2l=1 ωl

, ωl =γl

(ε+ βl)2(5.3.5)

with the linear weights γl given by

γ1 =1

3, γ2 =

2

3(5.3.6)

and the smoothness indicators βl by

β1 = (fj − fj−1)2 , β2 = (fj+1 − fj)

2 . (5.3.7)

Finally, the parameter ε insures that the denominator in Eq. (5.3.5) never becomes0, and is fixed at ε = 10−6 in the computations presented here. The choice of ε doesnot affect accuracy: the numerical errors can be much lower than ε, approachingmachine zero. Note that we have used the short-hand notation fj to denote f(uj(t)),and that the stencil for the scheme is biased to the left because of the positive winddirection.

The main reason that WENO works well, both for smooth solutions and for so-lutions containing shocks or other discontinuities or high gradient regions, is that thenonlinear weights, determined by the smoothness indicators, automatically adjustthemselves based on the numerical solution to use the locally smoothest informationgiven by the solution.

If the wind direction ∂f(u)/∂u ≤ 0, the method for computing the numericalflux fj+1/2 is the exact mirror image with respect to the point xj+1/2 of the descrip-tion above. The stencil would then be biased to the right. If ∂f(u)/∂u changes sign,we use a smooth flux splitting

f(u) = f+(u) + f−(u) (5.3.8)

5.4. ASTROPHYSICAL JET SIMULATIONS 93

where ∂f+(u)/∂u ≥ 0 and ∂f−(u)/∂u ≤ 0, and apply the above procedure sepa-rately on each of them. There are many choices of such flux splittings; the mostpopular one is the Lax-Friedrichs flux splitting where

f±(u) =1

2(f(u) ± αu) (5.3.9)

with α = maxu |∂f(u)/∂u|.For hyperbolic systems of conservation laws (5.3.1), the eigenvalues of the Jaco-

bian ∂f(u)/∂u are all real, and there is a complete set of right and left eigenvectors.This allows us to apply the nonlinear WENO procedure in each of the local charac-teristic fields, obtained by using the left eigenvectors of the Jacobian. For multiplespatial dimensions, the finite difference version of WENO schemes simply appliesthe WENO procedure in each direction to obtain high order approximations to therelevant spatial derivatives. Unlike dimensional splitting, such a dimension by di-mension method allows us to obtain high order accuracy without the computationalcost of truly multidimensional reconstructions.

5.4 Astrophysical Jet Simulations

We first consider the fully developed jets in HH 1–2. Then we model the early stagesof a jet—the XZ Tauri proto-jet.

jet ambientγ = 5/3 γ = 5/3ρj = 500 H/cm3 ρa = 50 H/cm3

uj = 300 km/s ua = 0Tj = 1000 K Ta = 10,000 Kcj = 3.8 km/s ca = 12 km/s

Table 5.4.1: Parameters for the jets in HH 1–2.

The jets in HH 1–2 have the parameters listed in Table 2. The simulationswere performed with the WENO-LF method on a 500∆x × 250∆y grid. The jetwidth is 1010 km and the evolution time is 7 × 108 s ≈ 22 yr. The jet inflow isMach 25 with respect to the soundspeed in the light ambient gas and Mach 80 withrespect to the soundspeed in the heavy jet gas.

Our simulations with radiative cooling accurately reproduce the morphologyand physics of the cylindrically symmetrical jet in HH 1–2, including the bow shockahead of the jet, the terminal Mach disk just inside the tip of the jet, and theKelvin-Helmholtz rollup of the jet tip. With radiative cooling, the jet has a much


higher density contrast near the jet tip (as the shocked, heated gas cools radiatively,it compresses), a much thinner bow shock, reduced Kelvin-Helmholtz rollup of thejet tip, and a lower average temperature.

We also analyze the “turn-on” of the XZ Tauri proto-jet and its associatedexpanding pair of bow shock “bubbles”.

We used the same parameters as for the HH 1–2 jets. The simulations wereperformed with the WENO-LF method on a 250∆x× 250∆y grid. The jet width is1010 km and the evolution time is 3.5 × 108 s ≈ 11 yr.

The Hubble images show a brightening (due to radiative cooling and compres-sion of the gas) of the limb of the leading bow shock only at intermediate stages. Ifcooling is significant in the early stages, the simulated bow shocks are too thin tomatch the images, and the limb of the outer bow shock brightens too early. Thusin these preliminary simulations we have turned off radiative cooling. To obtaina realistic match with the astrophysical images, radiative cooling must be presentbut important only in the intermediate stages. This can be achieved by setting theparameters of the jet and ambient gas so that cooling is initially suppressed, butimportant later on as the bow shock advances.

We pulsed the jet outflow as follows: an initial jet pulse of 108 s, which isthen turned off for 2 × 108 s, followed by a jet pulse of 4 × 108 s. The width of thesimulated bow shocks does agree with the Hubble images: roughly 6 × 1010 km, sothat the proto-jet plus bow shock “bubbles” can be modeled as a single outflow ofa pulsed jet with two bow shocks. No extraneous wind is necessary to match themorphology of the imaged flows.

5.5 Conclusion

In order to make a detailed comparison of the simulations and the astrophysicalimages of the HH 1–2 and XZ Tauri jets, including reproducing the morphology,shock structure, and temperature/ionization profiles of the jets, we plan to extendthe numerical code to a parallel version in three dimensions. Three-dimensionalsimulations with moderate resolution are feasible on modern workstations. Theparallel version is needed to achieve high resolution of fully 3D flows and shockstructures.

5.5. CONCLUSION 95

−3

−2

−1

0

1

2

3

4

−1

0

1

2

3

4

5

−2

−1

0

1

2

3

4

DENSITY

TEMPERATURE

PRESSURE

Figure 5.4.1: Simulation of Mach 80 jet with radiative cooling. Scales are logarith-mic.


Figure 5.4.2: Evolution of XZ Tauri proto-jet.

Log(Density)

5.5. CONCLUSION 97

Log(Pressure)

Temperature

Figure 5.4.3: Numerical simulation of XZ Tauri jet.


Acknowledgement

Research supported in part by the Space Telescope Science Institute under grantHST-GO-09863.06.

References

[1] Y. Ha, C. L. Gardner, A. Gelb, and C.-W. Shu, “Numerical Simulation of HighMach Number Astrophysical Jets with Radiative Cooling,” Journal of ScientificComputing, to appear.

[2] J. J. Hester, K. R. Stapelfeldt, and P. A. Scowen, “Hubble Space Telescope WideField Planetary Camera 2 observations of HH 1–2,” Astronomical Journal 116,372–395, 1998.

[3] J. E. Krist, K. R. Stapelfeldt, C. J. Burrows, J. J. Hester, A. M. Watson, G. E.Ballester, J. T. Clarke, D. Crisp, R. W. Evans, J. S. Gallagher, R. E. Griffiths, J.G. Hoessel, J. A. Holtzman, J. R. Mould, P. A. Scowen, and J. T. Trauger, “HubbleSpace Telescope WFPC2 imaging of XZ Tauri: Time evolution of a Herbig-Harobow shock,” Astrophysical Journal 515: L35–L38, 1999.

[4] T. Schmutzler and W. M. Tscharnuter, “Effective radiative cooling in opticallythin plasmas,” Astronomy and Astrophysics 273, 318–330, 1993.

[5] C.-W. Shu, “High order ENO and WENO schemes for computational fluid dy-namics,” in High-Order Methods for Computational Physics, Lecture Notes inComputational Science and Engineering vol. 9, 439–582. New York: Springer Ver-lag, 1999.

Chapter 6

A residual-based error estimatorfor Lagrangian hydrocodes

M. Laforest 1

Departement de mathematiques et de genie industrielEcole Polytechnique de MontrealC.P. 6079, succ. Centre-villeMontreal, Quebec, Canada H3C [email protected]

Abstract: An error estimate is introduced for finite element solutions to tran-sient compressible fluid flow problems in Lagrangian coordinates. The estimate isconstructed by combining a piecewise linear reconstruction of density and pressurewith an approximate solution of the error equation for acceleration in the space ofcontinuous piecewise quadratic polynomials. This results in an inexpensive estima-tor that is straightforward to implement into existing hydrocodes.

Keywords: error estimation, finite element, Lagrangian, compressible flow,Euler’s equations

Introduction

Given the complexity of fluid flow problems, there is a need for adaptive methodolo-gies that make an optimal use of computer resources. The key to such algorithms

1Work performed under contracts A0343 and P2984 with Sandia National Laboratory.

99

100 CHAPTER 6. A RESIDUAL-BASED ERROR ESTIMATOR

is an efficient, accurate, and reliable means of estimating the errors, and therebycontrol mesh or model refinement [Barth and Deconinck, 2003]. For Lagrangianformulations of the finite element method, researchers currently only have heuristicjump indicators and even standard Richardson extrapolation cannot be applied tothese moving mesh approximations. The aim of this work is to take a first steptowards error estimation for Lagrangian methods applied to Euler’s equations byproposing a simple estimator that could be implemented into existing hydrocodesof the type reviewed in [Benson, 1992].

The error estimator is relatively inexpensive to compute since it combines alocal estimate of the gradient of density and pressure, often already available in ahydrocode, with a local calculation, in space and time, of the error in acceleration.This error estimator is not, as of yet, a demonstrated upper or lower bound forthe error. The localization and the discretization of the error equation is similar toearlier work [Prudhomme and Oden, 1999]. Numerical experiments indicate thatthe error estimator has a range of applicability limited to problems with shocks andrelatively large variations in the regions of smooth flows.

6.1 Lagrangian methods for Euler’s equations

In a Lagrangian model of fluid flow, the quantities are expressed in the coordinatesof the initial configuration. Let Ω = [A,B] ⊂ IR be the initial domain, let X ∈ Ωbe the initial coordinates, and let x = x(X, t) be the spatial coordinates at timet = 0 of the particle initially at X. The map x = x(X, t) satisfies ∂x/∂t = v(X, t)where v(X, t) is the velocity of particle X. We assume that we know for all timesboth the forces b acting in the interior of Ω and the forces pb along the boundary∂Ω = A,B. If the density ρ, pressure p, specific internal energy e, and velocity vare known functions of X at time t = 0 then the evolution of a compressible fluidcan be modeled by Euler’s equations

ρ+ ρ∂xv = 0, (6.1.1)

ρv + ∂xp− ρb = 0, (6.1.2)

ρe+ p∂xv = 0, (6.1.3)

and an equation of state p = (γ − 1)ρe. The dot represents the material deriva-tive. The acceleration will often be identified explicitly as v = a. To characterizediscontinuous solutions to these equations would require us to introduce a weakformulation and impose an additional entropy condition. We will take such a char-acterization for granted and assume that the resulting problem is well-posed, see[Smoller, 1983] .

6.1. LAGRANGIAN METHODS FOR EULER’S EQUATIONS 101

A numerical implementation of a Lagrangian finite element method begins bydiscretizing the domain Ω = [A,B] into N subintervals A = x1 < x2 < . . . <xN+1 = B each denoted by κi ≡ [xi, xi+1]. Let T denote this discretization. Theunknowns are then approximated in the piecewise polynomials spaces associated tothe discretization T . If Qp(κi) is the set of all polynomials of degree p over theinterval κi, then define

Qp(T ) =u ∈ C0(Ω)

∣∣u|κi∈ Qp(κi), i = 1, . . . , N

, for p > 0, and

Q0(T ) =u∣∣u|κi

∈ Q0(κi), i = 1, . . . , N,

without the restriction on continuity.

With these preliminaries, we can now introduce the Lagrangian numericalmethod for Euler’s equations to which our error estimator will apply. We refer to[Benson, 1992] for a detailed description of this scheme. Writing

⟨f, g

⟩=

∫Ωfg dx,

it is straightforward to verify that the equations (6.1.1)-(6.1.3) and the boundaryconditions are equivalent to solving

⟨ρ, q

⟩+

⟨ρ∂xv, q

⟩= 0, (6.1.4)⟨

ρv, u⟩−

⟨p, ∂xu

⟩=

⟨ρb, u

⟩+

⟨pb, u

⟩∂Ω, (6.1.5)⟨

ρe, q⟩

+⟨p∂xv, q

⟩= 0, (6.1.6)

for all q ∈ C∞0 (]A,B[), u ∈ C∞(Ω), and where

⟨pb, u

⟩∂Ω

= pb(X1)u(X1)−pb(XN+1)u(XN+1).The approximate solution is essentially found by replacing the exact values in (6.1.4)-(6.1.6) by approximations ρh, ph, eh ∈ Q0(T ) and vh ∈ Q1(T ) and then solving theequations for all test functions q ∈ Q0(T ) and u ∈ Q1(T ).

The result is the following algorithm. Discretize time 0 = t0 < t1 < . . . <tM = T and introduce the notation ∆tn+1/2 = tn+1 − tn, ∆xi+1/2 = xi+1 − xi. Thequantities in the space of piecewise constant approximations Q0(T ) are completelydetermined by their values at cell centers ρh(x, tn)|κi

= ρh(xi+1/2, tn) ≡ ρi+1/2,n.The velocity and acceleration, approximated in Q1(T ), are completely defined byinterpolating the values at the nodes vh(xi, tn+1/2) ≡ vi,n+1/2. Assuming thatρi+1/2,n, pi+1/2,n, ei+1/2,n, ai,n and vi,n−1/2 are known, then our goal is to find thecorresponding quantities at the staggered times tn+1/2 and tn+1. The particularityof a Lagrangian algorithm is that the positions of the nodes are also a function oftime xi(tn) ≡ xi,n. The first step in this algorithm is to compute, at each node i,the updates

vi,n+1/2 = vi,n−1/2 + ∆tnai,n, (6.1.7)

xi,n+1 = xi,n + ∆tn+1/2vi,n+1/2. (6.1.8)


Conservation of mass (6.1.4) is then discretized at time tn+1/2 with the help of a onepoint quadrature leading to

ρi+1/2,n+1 = ρi+1/2,n

∆xi+1/2,n

∆xi+1/2,n+1

. (6.1.9)

Pressure and energy are then updated using an algorithm of Wilkins based on a dis-cretization around (xi+1/2, tn+1/2), see [Benson, 1992]. Define ∆si+1/2 = ρ−1

i+1/2,n+1 −ρ−1

i+1/2,n, and compute energy using

ei+1/2,n+1 =(ei+1/2,n − 1

2∆si+1/2pi+1/2,n

)/

(1 +

1

2(γ − 1)∆si+1/2ρi+1/2,n+1

), (6.1.10)

and pressure using pi+1/2,n+1 = (γ − 1)ρi+1/2,n+1ei+1/2,n+1. Using one-point quadra-ture on the momentum equation at time tn+1 gives us Newton’s law

ai,n+1 = −2pi+1/2,n+1 − pi−1/2,n+1

∆xi+1/2,n+1ρi+1/2,n+1 + ∆xi−1/2,n+1ρi−1/2,n+1

. (6.1.11)

To properly handle shock waves, we use Kuropatenko’s form of artificial viscositywhen compression is present. In conclusion, the equations (6.1.7)-(6.1.11) providean updated distribution at times tn+1/2 and tn+1 that is 2nd order in space and time.

6.2 Error estimation

6.2.1 Reconstruction

The first assumption in our approach is that the error in the density and pressurescan be effectively captured by a reconstruction procedure. If density ρh and pressureph are post-processed to obtain more accurate quantities ρ∗ and p∗, then we alsoobtain error estimates

ερ ≡ ρ− ρh ≈ ρ∗ − ρh ≡ ερr , εp ≡ p− ph ≈ p∗ − ph ≡ εpr. (6.2.1)

Since the update of density in equation (6.1.9) conserves mass, we reconstruct thedensity in the space of discontinuous piecewise linear elements as

ρ∗(x) = ρh + dρi+1/2(x− xi+1/2), x ∈ κi.

For convenience, we take the same approach for the reconstruction of pressure. Theaccuracy of such reconstructions has never been established for compressible flowsalthough they are commonly used in conjunction with flux-limiting algorithms.

6.2. ERROR ESTIMATION 103

The first reconstruction considered in this report is the minmod slope limiter.Given a quantity q whose values are known at cell centers

dqi+1/2 =

⎧⎪⎨⎪⎩

min

∆qi

∆xi, ∆qi+1

∆xi+1

, if ∆qi > 0 and ∆qi+1 > 0,

max

∆qi

∆xi, ∆qi+1

∆xi+1

, if ∆qi < 0 and ∆qi+1 < 0,

0, if ∆qi∆qi+1 < 0,

(6.2.2)

where we have used the shorthand ∆fi = fi+1/2−fi−1/2. The error estimators (6.2.1)associated to this reconstruction will be identified as ερr,1, ε

pr,1.

The second estimator is given by the usual least-squares estimate of the deriv-ative over 3 adjacent cells and denoted ερr,2, ε

pr,2.

6.2.2 The error equation

In this section, we derive a continuous equation for the error in acceleration εa

assuming that the true errors ερ and εp are known. We will postpone until Section6.2.3 a discussion of the discretization of this error equation and of the resultingcomputable error estimate for acceleration.

We start by observing that the exact acceleration v = a satisfies equation(6.1.5) and then subtract

⟨ρhah, u

⟩−

⟨ph, ∂xu

⟩from both sides to find⟨

ερεa + ρhεa + ερah, u⟩−

⟨εp, ∂xu

⟩−

⟨ερb, u

⟩(6.2.3)

=⟨ρhb− ρhah − ∂xp

h, u⟩

+⟨[ph], u

⟩Γint

+⟨pb − ph, u

⟩∂Ω,

where Γint = x2, . . . , xN is the set of all interior nodes, [ph] = ph(xi−) − ph(xi+),and

⟨[ph], u

⟩Γint

=∑

i

[ph](xi)u(xi). Conceptually, the right hand side of (6.2.3) is

formed of the so-called interior and boundary residuals of the approximation:

rint = ρhb− ρhah − ∂xph, (6.2.4)

rbnd(X) =

[ph], if X ∈ Γint,

n(pb − ph), if X ∈ ∂Ω,(6.2.5)

and where n = +1 at X = A,n = −1 at X = B. If Γ = ∂Ω ∪ Γint, then we cansimplify the error equation (6.2.3) to a global problem: find εa such that⟨

ερεa+ρhεa + ερah, u⟩−

⟨εp,∇ · u

⟩−

⟨ερb, u

⟩(6.2.6)

=⟨rint, u

⟩+

⟨rbnd, u

⟩Γ, ∀u ∈ Q2(T ).

The second assumption in this method is that the restriction of the global errorequation (6.2.6) to a family of local problems over patches of elements maintains


the accuracy of the error equation. One way to localize (6.2.6) is to restrict theintegration in (6.2.6) to impose Ωi ≡ κi−1∪κi while imposing homogeneous Dirichletboundary conditions along Ωi. The problem is then to find εa,i ∈ H1

0 (Ωi) satisfying⟨ρεa,i+ερah, u

⟩Ωi

−⟨εp, ∂xu

⟩Ωi

−⟨ερb, u

⟩Ωi

(6.2.7)

=⟨rint, u

⟩Ωi

+⟨rbnd, u

⟩Γint,∀u ∈ H1

0 (Ωi).

6.2.3 The discrete error equation

The error estimator for acceleration is computed by combining the error estimatorsfor ερr and εpr from Section 6.2.1 with an approximate solution εa,i

h ∈ Q2(T ) for εa,i inequation (6.2.7). The quality of the final error estimator εa,i

h therefore depends on i)the accuracy of the reconstruction, ii) the localization of the error equation (6.2.6),and on iii) the accuracy of an approximate solution to (6.2.7) in Q2(T ). Given ερrand εpr, the local error estimator over a patch Ωi is the solution εa,i

h of the equation⟨ρ∗εa,i

h , uh⟩

Ωi= −

⟨ερra

h − ερrb, uh⟩

Ωi+

⟨εpr, ∂xu

h⟩

Ωi(6.2.8)

+⟨rint, u

h⟩Ωi

+⟨rbnd, u

h⟩

Γint,∀uh ∈ Q2(T ) ∩H1

0 (Ωi).

In 1-D, this provides 3 degrees of freedom over every patch Ωi

6.3 Numerical Results

This section presents numerical results for the error estimators that focuses on theimpact of i) the choice of reconstruction for density and pressure and ii) the local-ization of the error equation.

6.3.1 Shock tube problem

The first experiment is a shock tube problem for which there exists an ana-lytical entropy solution formed of a rarefaction wave, a contact discontinuity and ashock wave, see [Smoller, 1983]. The initial data u(x, 0) = 0,

ρ(x, 0) =

1.0 if x < 0,

0.125 if x ≥ 0,, and p(x, 0) =

1.0e+ 5 if x < 0,

1.0e+ 4 if x ≥ 0,

are defined over the domain [−2, 2]. Pressure boundary conditions are pb(X = −2) =105 and pb(X = 2) = 104 where the boundaries X = −2, 2 are given in Lagrangian

6.3. NUMERICAL RESULTS 105

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1x 10

5

true error

local EE with R#1

global EE with R#1

local EE with R#2

global EE with R#2

Figure 6.3.1: Acceleration: true error, local and global error estimators for tworeconstruction methods.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

0

2

4

6

8

10

12

14x 10

5

a

ah

Figure 6.3.2: a and ah

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1x 10

5

true error

local EE with R#1

local EE with R#2

Figure 6.3.3: Non-uniform grids.


−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 10.5

1

1.5

2

2.5

3

3.5

4

4.5

5

converged ρ

ρh

Figure 6.3.4: ρ∗ and ρh

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

Figure 6.3.5: ||εah||L1/||ah/8 − ah||L1

coordinates. The results are presented at time T = 0.002 using 100 cells in ourinitial spatial discretization.

The approximate acceleration ah is non-trivial only within the rarefaction andclose to the shocks, as can be observed in Figure 6.3.2. Figure 6.3.1 presents thelocal and global estimators for both reconstruction methods. First of all, the figureindicates that the local and global error estimators εa,i

h based on the minmod recon-struction (6.2.2) (solid and dashed lines) are more effective than the error estimatorsbased on the least-squares reconstruction (dash-dotted and dotted lines). For bothreconstruction methods, the local error estimator is comparable to the global esti-mator. Remark that the estimator based on the 1st reconstruction identifies, up toa sign, the order of magnitude of the error in the rarefaction wave [−1.25, 0]. Thiserror would be ignored by a standard error estimator. At the right-moving shock,the local error estimator based on the minmod limiter also captures about 2/3 ofthe size of the error (not shown). Figure 6.3.3 shows the predictions of the localerror estimators (6.2.7) for each reconstruction method for an initially non-uniformdistribution of nodes, in contrast to Figure 6.3.1 for an initially uniform distributionof nodes. Figure 6.3.3 shows that that the error estimator continues to performadequately for non-uniform grids (largest cell/smallest cell = 5).

6.3.2 Strong and weak waves

The second experiment models an acoustic wave moving across a shock wave[Karni et al., 2002] for which there does not exist an analytic solution. The experi-ments are done using 200 cells and the reference exact solution will be a converged

6.3. NUMERICAL RESULTS 107

−10 −9 −8 −7 −6 −5 −4 −3 −2−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

true error

local EE with R#1

global EE with R#1

local EE with R#2

global EE with R#2

Figure 6.3.6: Acceleration: true error, local and global error estimators for tworeconstruction methods.

solution ρ∗, p∗, and v∗ obtained using 3200 cells. The initial data

p(x, 0) =

10.33333 if x < 0,

1.0 if x ≥ 0,and u(x, 0) =

−0.920279 if x < 0,

−3.549648 if x ≥ 0,

ρ(x, 0) =

3.857143 if x < 0,

1 + 0.2 sin(5x) if x ≥ 0,

are defined over the domain [−10, 10] and pressure boundary conditions are pb(X =−10) = 10.33333 and pb(X = 10) = 1.0. The errors and the error estimators arecomputed at time T = 2.5.

The main characteristics of the solution in Figure 6.3.4 are a highly oscillatoryregion immediately to the left of the shock, followed by a region of lower frequencywaves developing into shocks. These secondary shocks lead to sharp jumps in theacceleration of both the approximate and the converged solution although thesejumps are more pronounced in a∗. In fact, these spikes in the acceleration occureven though the acceleration appears to converge to a constant value throughoutthis secondary region [−7,−2]. Unfortunately these spikes in a∗ generate artificialjumps in the error estimate a∗−ah that would not be present in the true error a−ah.These spikes occur at approximately x = −6,−5,−4, and −3 and must be ignoredwhen analyzing the results of the error estimators.


Focusing on the region of smooth flow [−10,−2], Figure 6.3.6 indicates thatthe error estimator appears off by a constant amount of 0.5 but that it’s profile issuggestive of the variation of the true error. The global and local estimators areessentially equivalent but again the reconstruction based on the minmod limiter ismore accurate than the one using the least-squares estimate. Figure 6.3.5 shows theratio of ||εa,i

h ||L1 to a reference estimate of the error ||ah/8 − ah||L1 during the timeinterval [0, 2.5] with both norms taken over space. Although the order of magnitudeis roughly correct over time, it is difficult to judge the accuracy from this graphsince the reference solution ah/8 has spikes in acceleration that lead to jumps in thenorm ||ah/8 − ah||L1 .

6.4 Conclusions

The error estimator proposed here is local, inexpensive, easy to implement, constantfree, and captures parts of the error in smooth regions that a jump indicator wouldignore. Moreover, this estimator is, to our knowledge, the only alternative to aheuristic jump indicator. We have shown that the error estimator has a range ofapplicability that includes problems with shocks and relatively large variations in theregions of smooth flows. Unfortunately the error estimator is not accurate althoughit does appear to predict the order of magnitude and the spatial profile of the error.The choice of reconstruction appears to be critical but the localization of the errorequation does not have a significant impact on the error estimator. Future work willbe carried out to understand the impact of the reconstruction and the choice of thediscretization of the error equation on the accuracy of the errror estimator.

Acknowledgments

The author thanks Thomas Voth for sharing his insights into hydrocodes and hisactive interest in this work.

References

Barth, T. J. and Deconinck, H., editors (2003). Error Estimation and AdaptiveDiscretization Methods in Computational Fluid Dynamics, volume 25 of LectureNotes in Computational Sciences and Engineering. Springer Verlag, Berlin.

Benson, D. J. (1992). Computational methods in Lagrangian and Eulerian hy-drocodes. Comp. Meth. Appl. Mech. Engrg., 99:235–394.

6.4. CONCLUSIONS 109

Karni, S., Kurganov, A., and Petrova, G. (2002). A smoothness indicator for adaptivealgorithms for hyperbolic systems. J. Comp. Phy., 178:323–341.

Prudhomme, S. and Oden, J. T. (1999). A posteriori error estimation and error con-trol for Finite element approximations of time-dependent Navier-Stokes equations.Finite Elements in Analysis and Design, 33:247–262.

Smoller, J. (1983). Shock Waves and Reaction-Diffusion Equations. Springer-Verlag,New York.

Chapter 7

Central Schemes and CentralDiscontinuous Galerkin Methodson Overlapping Cells

Yingjie LiuSchool of MathematicsGeorgia Institute of [email protected]

Abstract: The central scheme of Nessyahu and Tadmor (J. Comput. Phys,87(1990)) has the benefit of not having to deal with the solution within the Rie-mann fan for solving hyperbolic conservation laws and related equations. But thestaggered averaging causes large dissipation when the time step size is small com-paring to the mesh size. The recent work of Kurganov and Tadmor (J. Comput.Phys, 160(2000)) overcomes the problem by use of a variable control volume andobtains a semi-discrete non-staggered central scheme. Motivated by this work, weintroduce overlapping cell averages of the solution at the same discrete time level,and develop a simple alternative technique to control the O(1/∆t) dependence ofthe dissipation. Semi-discrete form of the central scheme can also be obtained. Thistechnique is essentially independent of the reconstruction and the shape of the mesh,thus could also be useful for Voronoi mesh. The overlapping cell representation ofthe solution also opens new possibilities for reconstructions. Generally more com-pact reconstruction can be achieved. We demonstrate through numerical examplesthat combining two classes of the overlapping cells in the reconstruction can achievehigher resolution. Overlapping cells create self similarity in the grid and enable the

111

112 CHAPTER 7. CENTRAL SCHEMES

development of central type discontinuous Galerkin methods for convection diffu-sion equations and elliptic equations with convection, following the series works ofCockburn and Shu (Math. Comp. 52(1989)).

Keywords: Central Scheme, discontinuous Galerkin Method, ENO scheme,MUSCL scheme, TVD scheme.

7.1 Introduction

Godunov scheme first captures the shock wave in a narrow transition layer. It isbased on evolving piece-wise cell average representations of the solution by evaluat-ing the flux at the cell boundary which is obtained from solving a Riemann prob-lem. Various higher resolution schemes has been developed such as FCT, MUSCL,TVD schemes, PPM, ENO, WENO, etc. Unlike Godunov scheme, Lax-Friedrichscheme does not need to solve a Riemann problem. The central scheme of Nessyahuand Tadmor (NT) ([NeTa90]) provides the higher order generalization of the Lax-Friedrich scheme and is based on a staggered average of the piece-wise polynomialrepresentation of the solution, thus avoids dealing with the Riemann fan originatedfrom the jump values at the cell edges. Further developments on central schemescan be found in e.g. [SaWe92; JiTa98; JiLeLiOsTa98; LiTa98; BiPuRu99; KuLe00;AeSt03; KuNoPe01; KuTa00; LePuRu02], etc. The relaxation scheme of Jin andXin ([JiXi95]) provides another approach to nonlinear conservation laws.

Central schemes provide a black box type solution to nonlinear hyperbolicconservation laws and other closely related equations since essentially one only needsto supply the flux function. Similar approaches can also be achieved with upwindschemes with a Lax-Friedrich type flux function or building block, see e.g. Shu andOsher ([ShOs88; ShOs89]), Liu and Osher ([LiOs98]). Since the central schemesusually use staggered average, the time step size cannot be passed to zero. Similarsituation occurs in the 2D conservative front tracking and is overcome by use ofspace-time cells in Glimm et. al. ([GlLiLiXuZh03]). In [KuTa00], Kurganov andTadmor introduce a new kind of central scheme without the large dissipation errorrelated to the small time step size by use of a variable control volume whose sizedepends on time step size. By passing to the limit as the time step size goes to zero,the non-staggered semi-discrete central Godunov type scheme can be developed towhich standard Runge-Kutta methods or the TVD Runge-Kutta methods (Shu andOsher, [ShOs88]) can be applied. This allows the central scheme to be used for alarger class of equations where time step size could be small comparing to the meshsize.

In Liu ([Li04]), an alternative technique is introduced to control the dissipative

7.2. CENTRAL SCHEMES ON OVERLAPPING CELLS 113

error of central schemes. The major idea is to introduce an overlapping cell repre-sentation of the solution which allows a convex combination of the overlapping cellaverages. An immediate advantage is that the time discretization becomes simpleand more robust by use of the TVD Runge-Kutta method ([ShOs88]) due to the selfsimilarity of overlapping cells over time. Also by use of a time step size dependentconvex combination of the overlapping cell averages, the O(1/∆t) dependent dissipa-tive error can be easily controlled. Various reconstruction methods (e.g., MUSCL,ENO, WENO etc) can be applied to the overlapping cells in a standard way byseparating them into two classes and applying the reconstruction method to eachclass. More efficient application of the reconstruction methods using the combinedinformation from the overlapping cell averages has also been explored and requirefurther study particularly in higher space dimensions. The use of overlapping cellsopens many new possibilities. For example, central discontinuous Galerkin type ap-proach on overlapping cells becomes feasible due to the self similarity of the cells,following the works of Cockburn and Shu ([CoSh89; CoSh91; CoSh98], etc). Also thesemi-discrete form on overlapping cells results in a central type locally conservativeelliptic solver which could be suitable for elliptic equations with large advection.

7.2 Central Schemes on Overlapping Cells

Consider 1D conservation law

∂u

∂t+∂f(u)

∂x= 0, (x, t) ∈ R× (0, T ). (7.2.1)

Let xi be a uniform partition in R, with ∆x = xi+1 − xi. Denote xi+1/2 =12(xi + xi+1). Let Ui approximate the cell average

∫ xi+1/2

xi−1/2u(x, t)dx and Ui+1/2 ap-

proximate the cell average∫ xi+1

xiu(x, t)dx. Denote Un

i = Ui(tn), Uni+1/2 = Ui+1/2(tn).

By applying a MUSCL or ENO reconstruction for the two sets of cell averages, oneobtains a function µn(x) which is a piece-wise polynomial for cells (xi−1/2, xi+1/2) :i = 0,±1,±2, · · · and a function νn(x) which is a piece-wise polynomial for cells(xi, xi+1) : i = 0,±1,±2, · · · . For conservation purpose, they should satisfy1

∆x

∫ xi+1/2

xi−1/2µn(x)dx = Un

i and 1∆x

∫ xi+1

xiνn(x)dx = Un

i+1/2. Let ∆tn = tn+1 − tn be

the current time step size, following Nessyahu and Tadmor ([NeTa90]), the centralscheme with forward Euler time discretization can be written on overlapping cellsas follows

Un+1i = 1

∆x

∫ xi+1/2

xi−1/2νn(x)dx− ∆tn

∆x[f(νn(xi+1/2)) − f(νn(xi−1/2))],

Un+1i+1/2 = 1

∆x

∫ xi+1

xiµn(x)dx− ∆tn

∆x[f(µn(xi+1)) − f(µn(xi))].

(7.2.2)


The higher order time discretization can be obtained by applying the TVD Runge-Kutta time discretization procedure ([ShOs88]). Kurganov and Tadmor ([KuTa00])point out that since the numerical dissipation from 1

∆x

∫ xi+1/2

xi−1/2νn(x)dx does not de-

pend on ∆tn, the cumulative error will depend on O(1/∆t), the total number oftime steps in the computation. Therefore when ∆t is very small, e.g. ∆t = O(∆x2)for convection diffusion equations, the numerical dissipation becomes large. This iseasily seen if f(u) ≡ 0, then what the central scheme does is conservative rezoningat every time step, which will smear out the solution with the number of itera-tions increasing. By choosing the size of the control volume (xi, xi+1) proportionalto ∆t, this O(1/∆t) dependence can be removed and by passing to the limit as∆t→ 0, semi-discrete Godunov type central schemes can be developed ([KuTa00]).Liu ([Li04]) introduces another easy modification of the NT scheme to remove theO(1/∆t) dependence of the error taking advantage of the overlapping cell repre-sentation Un

i and Uni+1/2. The idea is to use a time dependent weighted average of

1∆x

∫ xi+1/2

xi−1/2νn(x)dx and Un

i in (7.2.2), which does not change the order of accuracy

of the scheme. In fact the difference between them is the local dissipation error.Suppose ∆tn ≤ ∆τn and ∆τn is an upper bound for the current time step size dueto the CFL restriction. The forward Euler form of the new central scheme can beformulated as follows

Un+1i = θ( 1

∆x

∫ xi+1/2

xi−1/2νn(x)dx) + (1 − θ)Un

i − ∆tn∆x

[f(νn(xi+1/2)) − f(νn(xi−1/2))],

Un+1i+1/2 = θ( 1

∆x

∫ xi+1

xiµn(x)dx) + (1 − θ)Un

i+1/2 − ∆tn∆x

[f(µn(xi+1)) − f(µn(xi))],

(7.2.3)where θ = ∆tn/∆τn. Note that when θ = 1, it becomes the scheme (7.2.2). Thecomparison of schemes (7.2.2) and (7.2.3) for Burgers equation with very small timestep size can be found in Fig. 7.2.2 (a) and (b). One can also obtain the followingsemi-discrete form by moving Un

i and Uni+1/2 to the left hand side and multiplying

both side by 1∆tn

, then passing to the limit as ∆tn → 0

ddtUi(tn) = 1

∆τn∆x

∫ xi+1/2

xi−1/2νn(x)dx− 1

∆τnUn

i − 1∆x

[f(νn(xi+1/2)) − f(νn(xi−1/2))],ddtUi+1/2(tn) = 1

∆τn∆x

∫ xi+1

xiµn(x)dx− 1

∆τnUn

i+1/2 − 1∆x

[f(µn(xi+1)) − f(µn(xi))].

(7.2.4)

Note that this semi-discrete form doesn’t need to explicitly evaluate the jumpvalues of νn(x) and µn(x) across their respective cell edges (which is one of thefeatures of the NT scheme). See Fig. 7.2.1. We have the following theorem.

Theorem 1 Let the schemes (7.2.2) and (7.2.3) start from the same time tn withthe same initial values Un

i , Uni+1/2. If the scheme (7.2.2) is TVD from time step tn

to tn + ∆τn, then the scheme (7.2.3) is also TVD from time tn to tn + ∆tn, for any∆tn ∈ [0,∆τn].

7.2. CENTRAL SCHEMES ON OVERLAPPING CELLS 115

A B C

Figure 7.2.1: (A) NT scheme; (B) 1D overlapping cells; (C) overlapping cells cre-ate self similarity for the grid over time and allow a convex combination of theoverlapping cell averages to control the dissipation.

∆x 1/10 1/20 1/40 1/80 1/160 1/320l1 error E1 0.0117 0.00147 0.000184 2.30e-05 2.88e-06 3.60e-07

order - 2.99 3.00 3.00 3.00 3.00l1 error E2 0.00406 0.000506 6.32e-05 7.89e-06 9.86e-07 1.23e-07E1/E2 2.88 2.91 2.91 2.92 2.92 2.93

Table 7.2.1: E1: reconstruction done for two classes of cells separately; E2: recon-struction on combined overlapping cells.

The theorem provides some insights into two reconstruction procedures: one isstandard to reconstruct for the two classes of cell averages Un

i : i = 0,±1,±2, · · · and Un

i+1/2 : i = 0,±1,±2, · · · separately; the other mixes the two classes in thereconstruction. In Table 7.2.1 the comparison of errors is shown for a 1D lineartranslation (ut + ux = 0) of sin(πx) computed by central scheme on overlappingcells (7.2.4) by use of the ENO quadratic reconstruction on two classes of cellsseparately (E1) and on combined overlapping cells (E2). Comparison of the twokinds of reconstructions are shown for the Shu-Osher problem ([ShOs89]) in Fig.7.2.2 (c) and (d); for Lax problem in (e) and (f). Note that without characteristicdecomposition, they achieve high resolution in the Shu-Osher problem while keepinga non-oscillatory profile for the Lax problem even with quite large cell size.

Note that in (7.2.3),

θ(1

∆x

∫ xi+1/2

xi−1/2

νn(x)dx) + (1 − θ)Uni = Un

i +∆tn∆τn

(1

∆x

∫ xi+1/2

xi−1/2

νn(x)dx− Uni ).

and ∆τn = O(∆x) is due to the CFL restriction for the scheme (7.2.2). Thereforethe local dissipative error now has a factor of ∆tn and the cumulative error will notbe degenerated by choosing very small ∆tn. In the lowest order case, scheme (7.2.3)can be viewed as a Godunov type scheme with a Lax-Friedrich flux.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

−5 −4 −3 −2 −1 0 1 2 3 4 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

−5 −4 −3 −2 −1 0 1 2 3 4 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Density

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Density

x

y

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 7.2.2: (a) Central scheme for Burgers equation without dissipation con-trol, ∆t = ∆x2/16, (b) with dissipation control (7.2.3); (c) Shu-Osher problem ([ShOs89]), reconstruction done for two classes of cells separately, ∆x = 1/40, (d)reconstruction done for combined overlapping cells; (e)Lax problem, reconstructiondone for two classes of cells separately, ∆x = 1/100, (f) reconstruction done forcombined overlapping cells. (g)2D overlapping cells by collapsing the staggereddual cells on two adjacent time levels to one time level; (h) 2D Riemann Problem ([LaLi98]) computed by DLM, ∆x = 1/200. (a), (b) 2nd order; (c)-(f), (h) 3rd order.All without characteristic decomposition.

7.3 Central Schemes on Overlapping Cells for Con-

vection Diffusion Equations

Consider the convection diffusion equation

∂u

∂t+∂f(u)

∂x=

∂

∂x(a(u, x, t)

∂u

∂x), (x, t) ∈ R× (0, T ), (7.3.1)

7.4. MULTI SPACE DIMENSIONS 117

where a(u, x, t) ≥ 0. Following the work of Kurganov and Tadmor ([KuTa00]), wecan discretize equation (7.3.1) in the new setting as follows

Un+1i = θ( 1

∆x

∫ xi+1/2

xi−1/2νn(x)dx) + (1 − θ)Un

i − ∆tn∆x

[f(νn(xi+1/2)) − f(νn(xi−1/2))]

+∆tn∆x

[a(Uni+1/2, xi+1/2, tn)

Uni+1−Un

i

∆x− a(Un

i−1/2, xi−1/2, tn)Un

i −Uni−1

∆x],

Un+1i+1/2 = θ( 1

∆x

∫ xi+1

xiµn(x)dx) + (1 − θ)Un

i+1/2 − ∆tn∆x

[f(µn(xi+1)) − f(µn(xi))]

+∆tn∆x

[a(Uni+1, xi+1, tn)

Uni+3/2

−Uni+1/2

∆x− a(Un

i , xi, tn)Un

i+1/2−Un

i−1/2

∆x],

(7.3.2)where θ = ∆tn/∆τn, ∆τn is maximum time step size determined by the CFL re-

striction for the hyperbolic part of the equation (7.3.1), ∂u∂t

+ ∂f(u)∂x

= 0. We have thefollowing stability theorem.

Theorem 2 Let the schemes (7.2.2) and (7.3.2) start from the same time tn withthe same initial values Un

i , Uni+1/2. If the scheme (7.2.2) is TVD from time step tn

to tn + ∆τn, then the scheme (7.3.2) is also TVD from time tn to tn + ∆tn, for any∆tn ≤ ∆τn∆x2

∆x2+2an∆τn, with

an = supa(Uni+1, xi+1, tn), a(Un

i+1/2, xi+1/2, tn) : i = 0,±1,±2, · · · .

We can also obtain a semi-discrete form of (7.3.2) similar to (7.2.4). Explicit Runge-Kutta methods with larger time step size have been developed in Medovikov ([Me98]), Verwer ([Ve96]) etc for semi-discrete equations. Implicit-explicit Runge-Kutta time discretizations, e.g. Ascher et. al. ([AsRuSp97]), Kennedy and Carpen-ter ([KeCa03]), etc, may also be applied to the semi-discrete form of (7.3.2).

7.4 Multi Space Dimensions

For rectangular grid, dimension by dimension line methods are the most convenienthigh order (≥ 3) methods for multi-dimensional problems. For example, in Shuand Osher ([ShOs88; ShOs89]), the ENO scheme is formulated in a dimension bydimension approach; in Kurganov and Tadmor ([KuTa00]), the semi-discrete centralscheme is also formulated in a dimension by dimension approach for multi dimen-sional problems. A 2D diagonal line method (DLM) is introduced in Liu ([Li04]) forcentral scheme on overlapping cells using 1D quadratic ENO reconstruction alongdiagonal lines on combined two classes of overlapping cells, and using dimensionby dimension (diagonal) approximation to the flux derivatives (without evaluationat quadrature points). The most common (staggered) overlapping cell averages aredefined as in Fig. 7.2.2(g). Each of the two diagonal axises passes through exactly5 overlapping cells. We may view them as 1D overlapping cell averages as in Fig.


∆x, ∆y 1/20 1/40 1/80 1/160 1/320 1/640l1 error 0.231 0.0359 0.00464 0.000583 7.30e-05 9.22e-06order - 2.69 2.95 2.99 3.00 2.99

Table 7.4.2: Convergence test for DLM for ut + (2u)x + uy = 0, (x, y) ∈ [0, 1]2 withperiodic boundary condition. u(x, y, 0) = sin(2πx+ 4πy) + 1

3cos(2πy), T = 1.

7.2.1(B) following the strategy of PPM (Colella and Woodward, [CoWo84]), thusthe 1D quadratic ENO reconstruction can be adapted to the 1D overlapping cells.The convergence tests on 2D linear translations show that DLM has 3rd order ac-curacy even though the 1D quadratic ENO reconstruction along diagonal lines onlyhas 2nd order accuracy (since the cell averages are 2D), see Table 7.4.2. The other2D tests also confirm the high resolution of DLM, see Fig. 7.2.2(h). Note thatthere is no overlapping within the cells with solid boundary lines (Fig. 7.2.2(g)) orwithin the cells with dash boundary lines, therefore conventional line reconstructioncan also be applied to each class of the cells separately. The reason for combiningthe overlapping cells in reconstruction is to achieve better resolution ([Li04]) takingadvantage of the “effectively refined grid” (in space, not time! The time step sizedoesn’t change).

Finite volume ENO type high order reconstruction for combined overlappingcells is also important because it can be applied to unstructured meshes such as theVoronoi mesh. Second order reconstruction (e.g., MUSCL) for combined overlappingcells is straight forward as described in [Li04] and has been tested to have goodrobustness. Higher order finite volume ENO reconstruction separately for each classof the cells (no overlapping within each class, see Fig. 7.2.2(g)) should be standard.

7.5 Central Discontinuous Galerkin Methods on

Overlapping Cells for Convection Diffusion Equa-

tions

Following the general strategy of discontinuous Galerkin methods (see e.g. Lesaintand Raviart ([LeRa74]), Cockburn ([Co98]) etc) and the series works of Cockburnand Shu ([CoSh89] etc), the central type discontinuous Galerkin method can bederived on overlapping cells. Consider the system of convection diffusion equations

∂ui

∂t+ · fi(u) = · (Ai(u,x, t) ui), (x, t) ∈ Rd × (0, T ), i = 1, · · · ,m,

(7.5.1)

7.5. CENTRAL DISCONTINUOUS GALERKIN METHODS 119

where ui is the ith component of u, Ai is a matrix. For simplicity, assume a uniformstaggered rectangular mesh (see Fig. 7.2.2(g) for the 2D case). The formulation forirregular staggered mesh, e.g., the Voronoi mesh which is a triangular mesh plus itsdual, is similar. Let CI , I = (i1, i2, · · · , id) be a (open) cell of a uniform rectangularmesh in Rd (the cells bounded by solid lines in Fig. 7.2.2(g))and xI be the cellcentroid. Let ∆x be the cell size. Let M be the set of piece-wise polynomials ofdegree r over the cells CI with no continuity assumed across the cell boundary.Let DJ , J = (i1 + 1/2, i2 + 1/2, · · · , id + 1/2) be a (open) cell of the dual meshwhich is the shift of the original mesh along the vector (1

2∆x, 1

2∆x, · · · , 1

2∆x) (the

cells bounded by dash lines in Fig. 7.2.2(g)). Let xJ be the cell centroid of the cellDJ . Let N be the set of piece-wise polynomials of degree r over the cells DJ withno continuity assumed across the cell boundary. The smooth solution of (7.5.1) willsatisfy

ddt

∫CIuiΦidx =

∫CI

fi · Φidx −∫

∂CI(fi · n)Φids+

∫CI

· (ATi Φi)uidx −

∫∂CI

ui(ATi Φi) · nds+

∫∂CI

Φi(Ai ui) · nds, ∀Φi ∈ M, i = 1, · · · ,m, ∀I,

(7.5.2)

ddt

∫DJuiΨidx =

∫DJ

fi · Ψidx −∫

∂DJ(fi · n)Ψids+

∫DJ

· (ATi Ψi)uidx −

∫∂DJ

ui(ATi Ψi) · nds+

∫∂DJ

Ψi(Ai ui) · nds, ∀Ψi ∈ N , i = 1, · · · ,m, ∀J,

(7.5.3)

where n is the unit outer normal of the corresponding cell,∫

∂CIF (s)ds denotes∫

∂CIF (s)|CI

ds. Let Uni ∈ M and V n

i ∈ N both be the numerical approximations of

the solution ui(·, tn). On the right hand side of the equation (7.5.2), replace the ui

(u) by V ni (Vn) and use the integration by part formula (similarly for (7.5.3)). The

central discontinuous Galerkin formulation on overlapping cells with forward Eulertime discretization from tn to tn+1 = tn +∆tn (with dissipation control as in (7.2.3))


is to find Un+1i ∈ M and V n+1

i ∈ N such that∫CIUn+1

i Φidx = θ∫

CIV n

i Φidx + (1 − θ)∫

CIUn

i Φidx + ∆tn∫

CIfi(V

n) · Φidx−∫∂CI

(fi(Vn) · n)Φids+

∑J [∫

CI∩DJ · (Ai V n

i )Φidx+∫CI∩∂DJ

V ni (AT

i Φi) · nds−∫

CI∩∂DJΦi(Ai V n

i ) · nds],∀Φi ∈ M, i = 1, · · · ,m, ∀I,

∫DJV n+1

i Ψidx = θ∫

DJUn

i Ψidx + (1 − θ)∫

DJV n

i Ψidx + ∆tn∫

DJfi(U

n) · Ψidx−∫∂DJ

(fi(Un) · n)Ψids+

∑I [∫

DJ∩CI · (Ai Un

i )Ψidx+∫DJ∩∂CI

Uni (AT

i Ψi) · nds−∫

DJ∩∂CIΨi(Ai Un

i ) · nds],∀Ψi ∈ N , i = 1, · · · ,m, ∀J,

(7.5.4)where θ = ∆tn/∆τn ≤ 1, ∆τn is the maximum time step size determined by theCFL restriction for the hyperbolic part of the equation (7.5.1) (i.e., assuming theright hand side is 0), ∆tn = tn+1 − tn is the current time step size. Note that thelast two boundary integral terms of (7.5.2) and (7.5.3) are canceled out in (7.5.4)due to the continuity, which is different from usual discontinuous Galerkin methods.Semi-discretized version of the (7.5.4) can be easily obtained similar to (7.2.4), towhich various higher order Runge-Kutta time discretization methods can be applied.If we make the corresponding diffusive fluxes in the above formula implicit (or usean implicit-explicit Runge-Kutta method to the corresponding semi-discretized ver-sion), we obtain an implicit central discontinuous Galerkin method which enableslarge time step size. It is the use of overlapping cells that makes it possible becauseotherwise the fluxes may not be able to be represented in the implicit form. Eventhough the two equations in (7.5.4) are coupled, when written down in the matrixform, one can easily eliminate one class of unknowns, say Un+1

i , which is similar tothe procedure of mixed finite element method for the incompressible Navier-Stokesequation (see. e.g. Fortin, [Fo93]). Therefore we are actually solving two N × Nsparse systems instead of a 2N × 2N sparse system.

7.6 Application to Elliptic Equations with Con-

vection

Consider the equation (7.5.1) without the time dependence term

− · (Ai(u,x) ui) + · fi(u) = 0, (x, t) ∈ Rd × (0, T ), i = 1, · · · ,m, (7.6.1)

The corresponding central discontinuous Galerkin approach on overlapping cellscan be obtained as follows: (a) write equation (7.5.4) in a semi-discrete form bymoving the corresponding

∫CIUn

i Φidx or∫

DJV n

i Ψidx terms to the left hand side

7.6. APPLICATION TO ELLIPTIC EQUATIONS WITH CONVECTION 121

and multiplying both side by 1/∆tn, then passing to the limit as ∆tn → 0; (b) dropthe time derivative term. (7.5.4) becomes

1∆τ

∫CI

(Vi − Ui)Φidx + ∫

CIfi(V) · Φidx−∫

∂CI(fi(V) · n)Φids+

∑J [∫

CI∩DJ · (Ai Vi)Φidx+∫

CI∩∂DJVi(A

Ti Φi) · nds−

∫CI∩∂DJ

Φi(Ai Vi) · nds] = 0,

∀Φi ∈ M, i = 1, · · · ,m, ∀I,

1∆τ

∫DJ

(Ui − Vi)Ψidx + ∫

DJfi(U) · Ψidx−∫

∂DJ(fi(U) · n)Ψids+

∑I [∫

DJ∩CI · (Ai Ui)Ψidx+∫

DJ∩∂CIUi(A

Ti Ψi) · nds−

∫DJ∩∂CI

Ψi(Ai Ui) · nds] = 0,

∀Ψi ∈ N , i = 1, · · · ,m, ∀J.

(7.6.2)

Here ∆τ is inherited from ∆τn in (7.5.4) and can be determined similarly. Thiscentral discontinuous Galerkin formulation is locally conservative. Existing locallyconservative finite element methods to this type of equation are control volumemethods (see Baliga and Patankar ([BaPa80]), Liu and McCormick ([LiMc88]), Caiand McCormick ([CaMc90]), etc), skew-symmetric discontinuous Galerkin methods(see e.g. Oden et. al., [OdBaBa98], Baumann and Oden, [BaOd99], Riviere et. al.,[RiWhGi01]) and mixed methods (see e.g. [BrFo91]). The distinctive feature of thecentral discontinuous Galerkin formulation on overlapping cells would be its centraltype treatment to the convection without using any upwind information. The useof overlapping cells allows a convex combination of the overlapping cell elementsto control the dissipation from staggering and the resulting semi-discrete form of(7.5.4) makes (7.6.2) possible. A benefit is that this formulation doesn’t require thediffusive matrix A to be symmetric.

References

P. Arminjon and A. St-Cyr, Nessyahu-Tadmor-type central finite volume methodswithout predictor for 3D Cartesian and unstructured tetrahedral grids, Appl. Nu-mer. Math. 46 (2003), no. 2, 135–155.

U. Ascher, S. Ruuth and R. J. Spiteri, Implicit-explicit Runge-Kutta methods fortime dependent partial differential equations, Appl. Numer. Math. 25 (1997),151–167.

B. R. Baliga and S. V. Patankar, A new finite-element formulation for convection-diffusion problems, Numer. Heat Transfer 3 (1980), 393–409.


C. E. Baumann and J. T. Oden, A discontinuous hp finite element method forconvection-diffusion problems, Comput. Methods Appl. Mech. Engrg. 175 (1999),no. 3-4, 311–341.

B. Rivire, M. F. Wheeler and V. Girault, A priori error estimates for finite elementmethods based on discontinuous approximation spaces for elliptic problems, SIAMJ. Numer. Anal. 39 (2001), no. 3, 902–931.

F. Bianco, G. Puppo and G. Russo, High-order central schemes for hyperbolic sys-tems of conservation laws, SIAM J. Sci. Comput. 21 (1999), no. 1, 294–322.

F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Seriesin Computational Mathematics 15 (1991), Springer-Verlag, New York.

S. Bryson and D. Levy, High-order central WENO schemes for multidimensionalHamilton-Jacobi equations, SIAM J. Numer. Anal., 41 (2003), 1339–1369.

Z. Q. Cai and S. McCormick, On the accuracy of the finite volume element methodfor diffusion equations on composite grids, SIAM J. Numer. Anal. 27 (1990), no.3, 636–655.

P. Colella and P. Woodward, The piecewise parabolic method (PPM) for gas-dynamical simulation, J. Comput. Phys. 54 (1984), 174–201.

B. Cockburn, An introduction to the discontinuous Galerkin method for convection-dominated problems, Advanced numerical approximation of nonlinear hyperbolicequations (Cetraro, 1997), Lecture Notes in Math. 1697 (1998), 151–268, Springer,Berlin.

B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuousGalerkin finite element method for conservation laws II, General framework, Math.Comp. 52 (1989), no. 186, 411–435.

B. Cockburn and C.-W. Shu, The Runge-Kutta local projection P 1-discontinuous-Galerkin finite element method for scalar conservation laws, RAIRO Modl. Math.Anal. Numr. 25 (1991), no. 3, 337–361.

B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (1998), no. 6,2440–2463.

A. Esmaeeli and G. Tryggvason, Direct numerical simulations of bubbly flows, part1. low Reynolds number arrays, J. Fluid Mech. 377 (1998), 313–345.

M. Fortin, Finite element solution of the Navier-Stokes equations, Acta Numer.(1993), 239–284.

J. Glimm, Solution in the large for nonlinear hyperbolic systems of equations, Comm.Pure. Appl. Math. 18 (1965), 697–715.

7.6. APPLICATION TO ELLIPTIC EQUATIONS WITH CONVECTION 123

J. Glimm, J. W. Grove, X.-L. Li, K.-M. Shyue, Q. Zhang and Y. Zeng, Threedimensional front tracking, SIAM J. Sci. Comp. 19 (1998), 703–727.

J. Glimm, X.-L. Li, Y. Liu, Z. Xu and N. Zhao, Conservative front tracking withimproved accuracy, SIAM J. Numer. Anal., 41 (2003), 1926-1947.

A. Harten, B. Engquist, S. Osher and S. R. Chakravarthy, Uniformly high orderaccuracy essentially non-oscillatory schemes III, J. Comput. Phys. 71 (1987), no.2, 231–303.

G.-S. Jiang, D. Levy, C.-T. Lin, S. Osher and E. Tadmor, High-resolution non-oscillatory central schemes with non-staggered grids for hyperbolic conservationlaws, SIAM J. Numer. Anal. 35 (1998), 2147.

G.-S. Jiang and E. Tadmor, Non-oscillatory central schemes for multidimensionalhyperbolic conservation laws, SIAM J. Sci. Comput. 19 (1998), no. 6, 1892–1917.

S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbi-trary space dimensions, Comm. Pure Appl. Math. 48 (1995), no. 3, 235–276.

C. A. Kennedy and M. H. Carpenter, Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math. 44 (2003), 139–181.

A. Kurganov and D. Levy, A third-order semi-discrete central scheme for conserva-tion laws and convection-diffusion equations, SIAM J. Sci. Comput. 22 (2000),no. 4, 1461–1488.

A. Kurganov, S. Noelle and G. Petrova, Semidiscrete central-upwind schemes for hy-perbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput.23 (2001), no. 3, 707–740.

A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear con-servation laws and convection-diffusion equations, J. Comput. Phys. 160 (2000),no. 1, 241–282.

P. Lax and X.-D. Liu, Solution of two dimensional Riemann problem of gas dynamicsby positive schemes, SIAM J. Sci. Comput. 19 (1998), No 2, pp.319-340.

P. Lesaint and P. A. Raviart, On a finite element method for solving the neutrontransport equation, in Mathematical Aspects of Finite Elements in Partial Differ-ential Equations, C. de Boor. ed., Academic Press, NY 1974.

D. Levy, G. Puppo and G. Russo, A fourth-order central WENO scheme for multi-dimensional hyperbolic systems of conservation laws, SIAM J. Sci. Comput. 24(2002), no. 2, 480–506.

C. Liu and S. F. McCormick, The finite volume element method (FVE) for planarcavity flow, in Proc. 11th Intern. Conf. CFD (1988), Williamsburg, VA.


X. D. Liu and S. Osher, Convex ENO high order multi-dimensional schemes withoutfield by field decomposition or staggered grids, J. Comput. Phys. 142 (1998), no.2, 304–330.

X.-D. Liu and E. Tadmor, Third order nonoscillatory central scheme for hyperbolicconservation laws, Numer. Math. 79 (1998), no. 3, 397–425.

Y.-J. Liu, Central schemes on overlapping cells, J. Comput. Phys. (2004),submitted.

A. A. Medovikov, High order explicit methods for parabolic equations, BIT 38(1998), 372-390.

H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic Con-servation Laws, J. Comput. Phys. 87 (1990), 408–463.

J. T. Oden, I. Babuska and C. E. Baumann, A discontinuous hp finite elementmethod for diffusion problems, J. Comput. Phys. 146 (1998), no. 2, 491–519.

J. Qiu and C.-W. Shu, On the construction, comparison, and local characteristic de-composition for high-order central WENO schemes, J. Comput. Phys. 183 (2002),no. 1, 187–209.

R. Sanders and A. Weiser, High resolution staggered mesh approach for nonlinearhyperbolic systems of conservation laws, J. Comput. Phys. 101 (1992), no. 2,314–329.

C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemesfor hyperbolic conservation laws, in Advanced Numerical Approximation of Non-linear Hyperbolic Equations, A. Quarteroni, ed., Lecture Notes in Math. 1697(1998), Springer, Berlin.

C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatoryshock-capturing schemes, J. Comput. Phys. 77 (1988), no. 2, 439–471.

C.-W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes, II, J. Comput. Phys. 83 (1989), no. 1, 32–78.

B. van Leer, Towards the ultimate conservative difference scheme I, Lecture Notesin Phys. 18 (1973), 163–168.

J. G. Verwer, Explicit Runge-Kutta methods for parabolic partial differential equa-tions, Special issue celebrating the centenary of Runge-Kutta methods, Appl. Nu-mer. Math. 22 (1996), no. 1-3, 359–379.

Chapter 8

Dual-Family Viscous Shock Wavesin Systems of Conservation Laws:A Surprising Example ∗

Alexei A. MailybaevInstitute of Mechanics, Moscow State Lomonosov University,Michurinsky pr. 1, 119192 Moscow, [email protected]

Dan MarchesinInstituto Nacional de Matematica Pura e Aplicada – IMPA,Estrada Dona Castorina, 110, 22460-320 Rio de Janeiro RJ, [email protected]

Abstract: We consider shock waves satisfying the viscous profile criterion in ageneral system of conservation laws. We introduce a concept of Si,j dual-family shockwave, which is associated with a pair of characteristic families i and j. We developa constructive method for sensitivity analysis of Si,j shocks. Generic solutions ofthe Riemann problem with Si,j shocks are described. As an example we present asystem of three conservation laws. Remarkably, despite being coupled only throughthe viscous terms, it has an S3,1 shock.

Keywords: Shock, viscous profile, conservation laws, sensitivity analysis, Rie-mann problem

∗This work was supported in part by: CNPq grant 301532/2003-6, FINEP under CTPETROgrant 21.01.0248.00, IM-AGIMB/ IMPA, CAPES grant 0722/2003 (PAEP no. 0143/03-00),FAPERJ E-26/152.163/02 and E-26/171.220/03, and President of RF grant MK-3317.2004.1

125

126 CHAPTER 8. DUAL-FAMILY VISCOUS SHOCK WAVES

Introduction

In this paper, shock waves in a general system of n conservation laws in one spacedimension x are considered. When shock waves are required to possess viscousprofiles rather than to satisfy Lax’s inequalities, new types of shocks arise. Ingeneral, these shocks may be associated with the i-th characteristic family on theleft and the j-th characteristic family on the right. We call such waves Si,j dual-family shocks. It was shown by [Kulikovskii, 1968] (see also [Kulikovskii et al.,2001]) that for i > j the viscous profile requirement provides exactly the number ofadditional equations (i − j equations) that is necessary to ensure that the numberof characteristics emanating from the shock in positive time direction equals thenumber of independent conditions at the shock interface.

For systems of two equations, transitional shock waves (i = j+1) were studiedby [Isaacson et al., 1990], [Schecter et al., 1996], and [Shearer et al., 1987], andnovel structures of Riemann solutions resulting from such shocks were described.Shock waves with one or several additional equations for the viscous profile werefound in problems of wave propagation in ferromagnetics, elastic media, and MHD,see [Kulikovskii et al., 2001], and in three phase flow in porous media they wereanalyzed for the case S2,1 by [Plohr and Marchesin, 2001]. A program for studyingthe Hadamard stability of Si,j shocks was initiated in [Liu and Zumbrun, 1995],where an example of S3,1 shock was presented.

In this paper, we provide a constructive method for perturbation analysis ofgeneral dual-family shocks, in which relationships between states at opposite sidesof the shock and shock speed resulting from perturbations of problem parametersare derived. The role of Si,j shocks in generic solutions of the Riemann problem isdescribed. As an example, we exhibit S3,1 shocks in a particularly simple system ofthree conservation laws that are coupled only through viscous terms; five separatedwaves appear in the Riemann solution containing this S3,1 shock.

8.1 Dual-family shock waves

We consider a system of partial differential equations of the form

∂G(U)

∂t+∂F (U)

∂x= ε

∂

∂x

(D(U)

∂U

∂x

), t ≥ 0, x ∈ R (8.1.1)

in the vanishing viscosity limit ε0. The function representing conserved quantitiesG(U) ∈ R

n, the flux function F (U) ∈ Rn, and the n × n viscosity matrix D(U)

depend smoothly on the state vector U ∈ Rn. Taking ε = 0 in (8.1.1) yields a

system of n first-order conservation laws. Real eigenvalues λ(U) of the characteristic

8.1. DUAL-FAMILY SHOCK WAVES 127

equation det(∂F/∂U − λ ∂G/∂U) = 0 are the characteristic speeds. When they arereal and distinct in a region of state space U (the strictly hyperbolic region), we listthem in increasing order λ1 < λ2 < · · · < λn.

A shock wave is a discontinuous (weak) solution of system (8.1.1) with ε = 0consisting of a left state U− = limx/ts U(x, t) and a right state U+ = limx/ts U(x, t),where s is the shock speed. A shock wave is considered admissible if it agrees withthe traveling wave solution (or viscous profile) U(x, t) = U(ζ), ζ = (x − st)/ε ofsystem (8.1.1) in the vanishing viscosity limit ε 0. The admissibility conditionimplies that U(ζ) is a solution of the system of ordinary differential equations

D(U)U = F (U) − F (U−) − s(G(U) −G(U−)), (8.1.2)

“connecting” the left and right equilibria U(−∞) = U−, U(+∞) = U+.

By linearizing (8.1.2) about U− and U+ we obtain

∆U = B(U±, s)∆U, ∆U(ζ) = U(ζ) − U±, (8.1.3)

where B(U, s) is the n× n matrix

B(U, s) =∂

∂U

[D−1(U)

(F (U) − F (U−) − s(G(U) −G(U−))

)]. (8.1.4)

Let µi(U, s), i = 1, . . . , n, be the eigenvalues of B(U, s) ordered with increasing realparts Reµ1 ≤ Reµ2 ≤ · · · ≤ Reµn. Let us assume that

Reµi−1(U−, s) < 0 < Reµi(U−, s),

Reµj(U+, s) < 0 < Reµj+1(U+, s)(8.1.5)

(if i− 1 = 0 or j + 1 = n+ 1, the corresponding inequality is disregarded). One cansee that µi(U−, s) = 0 and µj(U+, s) = 0 if s = λi(U−) and s = λj(U+), respectively.Under rather general conditions (e.g. [Kulikovskii et al., 2001] and [Majda and Pego,1985]), inequalities (8.1.5) reduce to

λi−1(U−) < s < λi(U−), (8.1.6)

λj(U+) < s < λj+1(U+). (8.1.7)

Now we define an Si,j shock as an admissible shock satisfying inequalities(8.1.5). Shocks with µi(U−, s) = 0, µj(U+, s) = 0, and µi(U−, s) = µj(U+, s) = 0 aredenoted by S−

i,j, S+i,j, and S±

i,j, respectively.

For i = j, inequalities (8.1.6) and (8.1.7) are the Lax conditions. Thus, anSi,i shock is a classical i-shock. Shocks with i < j are called overcompressive. For


i = j + 1 such a shock is termed transitional or undercompressive. We will focus onthe case i ≥ j, as according to [Isaacson et al., 1990], [Kulikovskii et al., 2001], and[Schecter et al., 1996], generically only for shocks with i ≥ j there may be a uniquesolution of the linearized problem for interaction with small perturbations.

Inequalities (8.1.6) coincide with the Lax conditions for the left state U− of ani-shock. Analogously, inequalities (8.1.7) coincide with the Lax conditions for theright state U+ of a j-shock. Therefore, the Si,j shock can be seen as a dual-familyshock wave associated with the i-th characteristic family on the left and with thej-th characteristic family on the right. For the dual-family shocks S−

i,j, S+i,j, and S±

i,j,the shock speeds coincide with the characteristic speeds at one or both sides.

8.2 Sensitivity analysis of an Si,j shock

Let us consider Si,j as a point in the space (U−, U+, s) of the left and right statesand speed of shocks. The set of all Si,j shocks generically defines a smooth surfaceSi,j of dimension n − i + j + 1 in the space (U−, U+, s), which can be given locallyby n+ i− j equations. Basic n equations relating U−, U+, and s are the Rankine–Hugoniot conditions

H(U−, U+, s) ≡ F (U+) − F (U−) − s(G(U+) −G(U−)) = 0 ∈ Rn, (8.2.1)

which follow from requiring that U+ is an equilibrium of (8.1.2). For i > j, thereare i− j additional requirements

Hadd(U−, U+, s) = 0 ∈ Ri−j (8.2.2)

determined by the existence of the viscous profile. Boundaries of the manifold Si,j

are typically related to characteristic shocks S−i,j, S

+i,j, and S±

i,j, or to bifurcations ofthe viscous profile, see [Schecter et al., 2001].

Let us consider a specific point (U0−, U

0+, s

0) ∈ Si,j. A corresponding viscousprofile U0(ζ) satisfies equation (8.1.2) and the boundary conditions U0(−∞) = U0

−,U0(+∞) = U0

+. Linearizing equation (8.1.2) near the solution U0(ζ), we obtain thesystem of ordinary differential equations

V = B(U0(ζ), s0)V, V (−∞) = V (+∞) = 0, V ∈ Rn, (8.2.3)

where the matrix B(U, s) is given in (8.1.4). The corresponding adjoint linear systemtakes the form

W = −BT (U0(ζ), s0)W, W (−∞) = W (+∞) = 0, W ∈ Rn. (8.2.4)

8.2. SENSITIVITY ANALYSIS OF AN SI,J SHOCK 129

For any bounded function X(ζ) ∈ Rn, solutions W (ζ) of (8.2.4) have the property∫ +∞

−∞ W T (X−B(U0(ζ), s0)X)dζ = 0. Because of inequalities (8.1.5), the linear spaceW of solutions of (8.2.4) has dimension dimW = i− j; so we choose i− j functionsW1(ζ), . . . ,Wi−j(ζ) as a basis of W .

In a neighborhood of the point (U0−, U

0+, s

0), the manifold Si,j is given byequations (8.2.1) and (8.2.2). The local form of Si,j is determined by the linearizationof these equations presented in the following theorem.

Theorem 1 The tangent plane (dU−, dU+, ds) of the manifold Si,j at the point(U0

−, U0+, s

0) ∈ Si,j is given by dH = 0, dHadd = 0 with

dH =

(∂F

∂U− s0∂G

∂U

)U=U0

+

dU+ −(∂F

∂U− s0∂G

∂U

)U=U0

−

dU−

− (G0+ −G0

−)ds,

(8.2.5)

dHadd =

(∫ +∞

−∞W TD−1

0

(∂F

∂U− s0∂G

∂U

)U=U0

−

dζ

)dU−

+

(∫ +∞

−∞W TD−1

0 (G0 −G0−)dζ

)ds,

(8.2.6)

where W (ζ) = [W1(ζ), . . . ,Wi−j(ζ)] is an n × (i − j) matrix. The short notationsD0(ζ) = D(U0(ζ)), F0(ζ) = F (U0(ζ)), G0(ζ) = G(U0(ζ)), F 0

± = F (U0±), and

G0± = G(U0

±) are used in (8.2.5), (8.2.6), where U0(ζ) is the viscous profile ofthe Si,j shock at the initial point (U0

−, U0+, s

0).

Equation (8.2.5) represents the differential of the Rankine–Hugoniot conditions(8.2.1), and (8.2.6) is the differential of the specially chosen equations (8.2.2). Weomit the proof of the theorem, which is based on perturbation analysis of equation(8.1.2) using the adjoint linear system (8.2.4).

As a model of a physical system, equation (8.1.1) typically depends on oneor several problem parameters. Under variations of these parameters, the functionsG(U), F (U), and D(U) undergo perturbations δG(U), δF (U), and δD(U). If theseperturbations are small, the manifold Si,j undergoes a small perturbation. The firstorder approximation of the perturbed manifold can be determined as follows.

Theorem 2 Let (U0−, U

0+, s

0) ∈ Si,j and consider perturbations δG(U), δF (U), δD(U)of the system functions. Then the first order approximation of the perturbed manifoldSi,j near the point (U0

−, U0+, s

0) is given by the equations

dH = −δF 0+ + δF 0

− + s0(δG0+ − δG0

−), (8.2.7)


dHadd = −∫ +∞

−∞W TD−1

0 δD0D−10

(F0 − F 0

− − s0(G0 −G0−)

)dζ

+

∫ +∞

−∞W TD−1

0

(δF0 − δF 0

− − s0(δG0 − δG0−)

)dζ,

(8.2.8)

where the differentials dH and dHadd are given by (8.2.5), (8.2.6).

Theorems 1 and 2 determine all nearby Si,j shock waves, even when problemparameters are changed, using the information on a particular shock and its viscousprofile. This method is useful for constructing Riemann solutions possessing Si,j

shocks, continuation procedures, and parametric analysis.

The characteristic shock waves S−i,j, S

+i,j, and S±

i,j can be studied in the sameway. In addition to equations (8.2.1) and (8.2.2), one should use conditions ensuringthat the shock speed is equal to the corresponding characteristic speed at one orboth sides of the shock.

8.3 Dual-family shocks in Riemann solutions

The basic initial-value problem for a system of conservation laws (equations (8.1.1)with ε = 0) is the Riemann problem, given by piecewise constant initial data witha single jump at x = 0: U(x, 0) = UL for x < 0 and U(x, 0) = UR for x > 0.The solution is found in the form U(x, t) = U(ξ), ξ = x/t, consisting of continu-ously changing waves (rarefaction waves), jump discontinuities (shock waves), andseparating constant states. Classically, there are n rarefactions, one for each charac-teristic speed, which we denote by R1, . . . , Rn. We require all shocks to have viscousprofiles, i.e., there can be Si,j, S

−i,j, S

+i,j, and S±

i,j shocks.

The structure of a Riemann solution is given by a sequence of waves wk

w1, w2, . . . , wm, (8.3.1)

appearing with increasing value of ξ. Here each wave wk ∈ Ri, Si,j, S−i,j, S

+i,j, S

±i,j

is a rarefaction or shock. The wave wk has left and right states U(k)− and U(k)+

and speeds ξ(k)− < ξ(k)+ for a rarefaction wave and s(k) = ξ(k)− = ξ(k)+ for a shockwave. The left state of the first wave w1 and the right state of the last wave wm

are the initial conditions of Riemann problem: U(1)− = UL and U(m)+ = UR. Thenatural requirements in sequence (8.3.1) are U(k)+ = U(k+1)− and ξ(k)+ ≤ ξ(k+1)−. Ifξ(k)+ < ξ(k+1)− then there is a separating constant state between wk and wk+1. Inthis case we will use the notation wk — wk+1. If ξ(k)+ = ξ(k+1)− then the waves donot possess a separating constant state. This situation will be denoted by wk |wk+1.

8.3. DUAL-FAMILY SHOCKS IN RIEMANN SOLUTIONS 131

The most important structures of a Riemann solution are the generic ones:they do not change under perturbations of initial conditions UL, UR and of systemfunctions. Only shocks with i ≥ j may appear in generic structures; recall thatthe shock Si,i (or simply Si) is a classical shock of i-th family. Overcompressiveshocks (i < j) bifurcate to a set of waves under perturbations with arbitrarily smallamplitudes. The following theorem describes all generic structures of a Riemannsolution (see also [Schecter et al., 1996] for the case of two conservation laws).

Theorem 3 If (8.3.1) is a generic structure of a Riemann solution, then w1 ∈R1, S1, S

+1 , wm ∈ Rn, Sn, S

−n , and each pair wk, wk+1 has one of the types

Rj or Si,j — Ri′ or Si′,j′, i′ = j + 1, i ≥ j, i′ ≥ j′,

Ri | S−i,j or S±

i,j, i ≥ j,

S+i,j or S±

i,j | Rj, i ≥ j.

(8.3.2)

A classical Riemann solution consists of n separated waves Ri or Si correspond-ing to different characteristic families. The classical structure R1 — R2 — S3 of aRiemann solution in a system of three conservation laws is shown in Fig. 8.3.1(a)using characteristic lines in the space-time plane (shocks are shown as bold linesand rarefactions as thin line fans).

As an example, we list two nonclassical Riemann solution structures:

R1 — S2 — R3 |S−3,2 — R3, (8.3.3)

R1 — S2 — S3,1 — S2 — R3. (8.3.4)

The Riemann solution with structure (8.3.3) contains a transitional characteristicshock S−

3,2. A distinctive feature of structures (8.3.3) and (8.3.4) is that the classicalwaves R3 and S2 appears twice. Riemann solutions with these structures are shownin Fig. 8.3.1(b,c).

Riemann solutions with dual-family shock waves violate the classical structureof sequences of only n classical waves with increasing family number from left toright. The shocks with i > j+1 introduce a “jump back” capability in this sequenceallowing classical waves of (j+1), . . . , (i− 1)-th characteristic families to appear re-peatedly. Moreover, from the theoretical point of view, there is no general bound onthe number of separated classical waves or of nonclassical shock waves in a Riemannsolution for systems of n > 2 conservation laws. The existence of several separatedwaves with the same characteristic family is a property of Riemann solutions thathas not appeared in previous works.


Figure 8.3.1: Riemann solutions: (a) classical, (b) with S−3,2 shock, (c) with S3,1

shock.

8.4 Nonclassical shocks in three conservation laws

coupled through the viscosity term

Let us consider system (8.1.1) with state vector U = (u, v, w)T and

G(U) = U, F (U) =

⎛⎝ u2

v2

w2

⎞⎠ , D(U) =

⎛⎝ 9 8 2

8 9 22 2 1

⎞⎠ . (8.4.1)

Equations in this system are coupled only through the viscous term D∂2U/∂x2 withpositive definite matrix D. Putting ε = 0 yields three uncoupled inviscid Burgersequations. Thus, classically this system has Riemann solutions containing threewaves of different families.

Consider a shock with states U0− = (−3, 7,−1)T , U0

+ = (5,−5, 3)T and speeds0 = 2. The choice of D in (8.4.1) facilitates the verification that this shock is oftype S3,1 with straight line viscous profile

U0(ζ) = (1, 1, 1)T + ρ(ζ)(4,−6, 2)T , ρ(ζ) = tanh(2ζ). (8.4.2)

Two independent solutions of adjoint system (8.2.4) are

W1(ζ) = (−1, 0, 2)T exp(−8

∫ ζ

0ρ(ζ ′)dζ ′

),

W2(ζ) = (12, 9, 3)T exp(−4

3

∫ ζ

0ρ(ζ ′)dζ ′

).

(8.4.3)

Using Theorem 1, we find the approximation of the manifold S3,1 consisting of allpossible states and speeds (U−, U+, s) in the neighborhood of the point (U0

−, U0+, s

0),in the form (dU−, dU+, ds) satisfying

dU− + dU+ = (1, 1, 1)Tds,

(−2 0 44 3 1

)dU− =

(14

)ds. (8.4.4)

8.4. NONCLASSICAL SHOCKS 133

Table 8.4.1: Riemann solution with the S3,1 dual-family shock wave for the initialconditions U0

L = (−6, 7, 1)T and U0R = (−1,−5, 0)T .

wave types states speeds

R1 U = (ξ/2, 7, 1)T −12 ≤ ξ ≤ −6

S2 U− = (−3, 7, 1)T , U+ = (−3, 7,−1)T s = 0

S3,1 U− = (−3, 7,−1)T , U+ = (5,−5, 3)T s = 2

S2 U− = (5,−5, 3)T , U+ = (5,−5, 0)T s = 3

S3 U− = (5,−5, 0)T , U+ = (−1,−5, 0)T s = 4

Figure 8.4.2: Riemann solutions with S3,1 shock wave.

It turns out that the approximation (8.4.4) coincides locally with the manifold S3,1,which is a part of plane (8.4.4) in the space (U−, U+, s).

The flux function variation δF (U) = δp (v, w, u)T (where p is a parameter)couples the conservation laws. Theorem 2 gives the perturbed manifold S3,1 near(U0

−, U0+, s

0) for small values of δp as

dU− + dU+ = ds (1, 1, 1)T + δp (3/2, 1/3,−2)T ,(−2 0 44 3 1

)dU− =

(ds− 11δp/2

4ds+ 5δp/2

).

(8.4.5)

In Table 8.4.1 we give an example of a Riemann solution with the genericstructure R1 — S2 — S3,1 — S2 — S3 containing an S3,1 shock. Fig. 8.4.2 shows


Figure 8.4.3: Riemann solution for a system of three equations with S3,1 and S3,2

dual-family shocks.

the space distribution for waves in this solution at t = 1. For any initial conditions(UL, UR) near (U0

L, U0R) the solutions are similar.

Numerical experiments using a linearized Crank-Nicolson scheme were carriedout for system (8.1.1), (8.4.1) with different initial conditions. It turned out thatthe solution of Table 8.4.1 is the only stable asymptotic solution for the given U0

L

and U0R. The classical solution, obtained by solving each conservation law sepa-

rately, consists of a rarefaction for the first state coordinate u and two shocks forv and w separated by constant states. Numerical calculations showed that thissolution is unstable and that the shock S3 corresponding to the change of v inthe classical solution does not possess a viscous profile. If the right initial condi-tion is changed to U0

R = (−1,−5, 6)T , then the stable Riemann solution is evenmore complex: it contains six waves separated by constant states with structureR1 — S2 — S3,1 — S2 — S3,2 — R3; see Fig. 8.4.3.

These examples highlight the importance of dual-family shocks for the globalstructure of Riemann solutions of systems of conservation laws.

References

Isaacson, E.L., Marchesin, D. and Plohr, B.J. (1990). Transitional waves for conser-vation laws, SIAM J. Math. Anal. 21, 837–866.

Kulikovskij, A.G. (1968). Surfaces of discontinuity separating two perfect media ofdifferent properties. Recombination waves in magnetohydrodynamics, Prikl. Mat.Mekh. 32, 1125–1131 (in Russian).

Kulikovskii, A.G., Pogorelov, N.V. and Semenov, A.Yu. (2001). Mathematical aspects

8.4. NONCLASSICAL SHOCKS 135

of numerical solution of hyperbolic systems. Boca Raton: Chapman & Hall/CRC.

Liu, T.-P. and Zumbrun, K. (1995). On nonlinear stability of general undercompres-sive viscous shock waves, Comm. Math. Phys. 174, 319–345.

Majda, A. and Pego, R.L. (1985). Stable viscosity matrices for systems of conserva-tion laws, J. Differential Equations 56, 229–262.

Plohr, B. and Marchesin, D. (2001). Wave structure in WAG recovery, SPE 71314,Society of Petroleum Engineers Journal 6, 209–219.

Schecter, S., Marchesin, D. and Plohr, B.J. (1996). Structurally stable Riemannsolutions, J. Differential Equations 126, 303–354.

Schecter, S., Plohr, B.J. and Marchesin, D. (2001). Classification of codimension-oneRiemann solutions, J. Dynam. Differential Equations 13, 523–588.

Shearer, M., Schaeffer, D.G., Marchesin, D. and Paes-Leme, P.L. (1987). Solutionof the Riemann problem for a prototype 2 × 2 system of nonstrictly hyperbolicconservation laws, Arch. Rational Mech. Anal. 97, 299–320.

Chapter 9

Inviscid and Viscous CFDModeling of Plume Dynamics inLaser Ablation

Kedar A. PathakGraduate Ph.D. StudentDepartment of Mechanical EngineeringUniversity of Akron,Akron, OH [email protected]

Alex PovitskyAssociate ProfessorDepartment of Mechanical EngineeringUniversity of Akron,Akron, OH [email protected]

Abstract: We model the dynamics of laser ablated plume in production ofcarbon nanotubes using computational fluid dynamics techniques. A higher orderENO scheme is used for modeling the plume. The exact physical phenomenon oflaser ablation plume dynamics is comparatively complex. For simplicity we modelthe plume and surrounding laser furnace gas as single-species ideal gas. We areaiming at capturing the basic features of laser ablated plume that help in futureto control the plume evolution using simplified models. We solve Euler equations,Navier-Stokes equations, and Navier-Stokes equations with an appropriate model ofturbulent flow field. Euler equations-based model appears to be adequate for firstdozens of microseconds. For modeling of several hundreds of microseconds after the

137

138 CHAPTER 9. CFD MODELING OF PLUME DYNAMICS

ablation, the viscous terms are needed to model the plume diffusion. For millisecond-range modeling, the influence of unsteady turbulent flow in the laser furnace maybecome significant. We propose to develop a simplified model based on baroclinicand viscous deposition of vorticity of the plume. Interaction of plume with reflectedshock waves appear to be the major source of the deposited vorticity. This modelwill be based on evaluation of vorticity on the plume and formation of the vortexcouple.

keywords: laser ablation; plume gas dynamics; vorticity deposition; ENOschemes

Introduction

In application of high-intensity lasers to material processing, the formation of ab-lated plume and its dynamics are of high importance. In general, the ablated plumeis a multi-phase flow and may contain solid particles and liquid clusters of molecules.Nevertheless, the plume dynamics shows well-defined gas dynamic effects (see Kellyet al. [1]). As the ablation plume expands and interacts with the background gas,local distributions of particles velocities eventually evolve towards Maxwellian ones,allowing for application of gas-dynamic continuous computational models. Long-term plume evolution in a near-atmospheric pressure background gas can be welldescribed by continuous gas dynamics. In most of processes for synthesis of nan-otubes, the nanosecond laser pulses are applied to the target and the injectionvelocity of plume is supersonic. Experiments by Puretzky et al. [2] have shown thatthe initial gas pressure in the ablated plume can exceed 100 atm that in turn, pro-duces shock in the ambient gas. In turn, the propagation of incident and reflectedshock waves in a confined space of laser furnace affect the plume behind the incidentshock wave. Before the shock wave hit the side wall of the furnace, its propagationis adequately described by the self-similar analytical solution (see Anisimov et al.[3], Mao et al. [4]). At later time moments, the furnace chamber confines the plumeand produces reflected shock waves that are difficult or impossible to account forin existing analytical models. These reflected shock waves are crucial for plumedynamics since the interaction of plume with reflected shock waves contributes tobaroclinically generated vorticity. In turn, this deposited vorticity determines theformation of vortex couple that controls the plume evolution in later stages of plumedevelopment. Temperature and dynamics of the emitted with the plume catalystparticles are crucial for modeling of growth of nanotubes deposited on these catalystparticles. The previous developed plume dynamics model by Lobao and Povitsky[5; 6] has been based on compressible multi-species Euler equations written in gen-eral curvilinear coordinates. The goal of their model was to obtain the temperature

139

of catalyst particles as a function of time, show the influence of chamber pressure,injection velocity, and periodicity of the plume on the temperature of particles, anddiscuss physical reasons for non-monotonic temperature behavior of catalyst parti-cles. This temperature as a function of time is needed to model chemical reactionsleading to nucleation of nanotubes.

The effective viscous length-scale can be defined as visc =√ντ , where ν is

effective viscosity and τ is the time-scale of the modeled phenomena. This methodto evaluate the influence of viscous forces has been used by Quirk and Karni [7]and other papers dealing with the shock wave interactions with isolated flow inho-mogeneities. This criterion is based on analytical solution for the penetration ofboundary layer into the fluid in rest due to the impulsive motion. For example, inthe laser ablation model in synthesis of carbon nanotubes [5] the considered timeinterval was 200µs and the viscous length-scale is visc∼100µm. This length-scaleis two orders of magnitude smaller than the cross-section of the plume and, there-fore, the viscous effects seem to be negligible for such a time interval. On the otherhand, the model of plume dynamics by Greendyke et al.[8] is based on the Reynolds-stresses model of turbulence and includes adjustment of plume initial conditions tolaser ablation experiments by Puretzky et al. [2] . The choice between inviscid,viscous, and turbulent models for plume dynamics is not straightforward since thehighly compressible ablated plume undergoes several interactions with shock waves,and the above-listed simplified criterion may not be adequate.

Furthermore, modeling of multiple plume injection as a result of laser ablationis important for large-scale production of carbon nanotubes where the laser hits thetarget many times within a short period [5]. In the limit case of very short timebetween two consequent pulses, the under-expanded jet analogy is used by Bulgakov[9] to predict the effect of plume focusing and vortex formation. The choice betweeninviscid and viscous models will become more sophisticated than that for singleplume.

In the current paper we compare our computational results obtained by inviscidand viscous models for plume dynamics. We propose to study the shock-plumeinteraction numerically and will provide with the simplified analytical model forplume evolution based on evolving of deposited vortex pair.


9.1 Governing Equations and Numerical Method

The governing equations are the two-dimensional unsteady compressible Navier-Stokes equations. The conservative equations are given by

∂U

∂t+∂F

∂x+∂G

∂y=∂R

∂x+∂S

∂y(9.1.1)

where,

U =

⎡⎢⎢⎣

ρρuρvρE

⎤⎥⎥⎦, F =

⎡⎢⎢⎣

ρup+ ρu2

ρuv(p+ ρE)u

⎤⎥⎥⎦, G =

⎡⎢⎢⎣

ρvρuvp+ ρv2

(p+ ρE)v

⎤⎥⎥⎦,

R =

⎡⎢⎢⎣

0τxx

τxy

uτxx + vτxy − qx

⎤⎥⎥⎦, S =

⎡⎢⎢⎣

0τyx

τyy

uτyx + vτyy − qy

⎤⎥⎥⎦,

the state equation for an ideal gas in thermodynamic equilibrium can be written as

p

ρ= (γ − 1)(E − u2 + v2

2) (9.1.2)

Since the flow field involves shock waves and smooth areas of flow field, we usethe Essentially non-oscillating (ENO) scheme as given by Shu [10] to discretize thenon-linear terms in above equations. The ENO scheme not only discards the stencilgiving the oscillations in the flux, but also chooses the smoothest stencil. The choiceis made using the Newton divided difference. After choosing the smoothest stencil,reconstruction is done to find the numerical flux function at the cell interface. Thenumerical flux function

fi+ 12

= f(fi−r, ..., fi+s), i = 0, 1, 2, ..., N. (9.1.3)

is calculated using the stencil chosen by ENO scheme with the following k points:

xi−r, ..., xi+s, (9.1.4)

where r + s = k - 1. The numerical flux function fi+ 12

is expressed as

fi+ 12

=k−1∑j=0

crjfi−r+j, (9.1.5)

9.1. GOVERNING EQUATIONS AND NUMERICAL METHOD 141

where the constants crj are given by Shu [10].

We study the plume evolution during the time period of order of a singlemillisecond, that is comparable to the life time of a few turbulent eddies therefore,the turbulence in flow is modeled in a following way:First, the turbulent kinetic energy and the rate of dissipation are calculated usingthe 3 % - level of turbulence in the laser furnace,

u′ = .03u, K =3

2u′2, ε =

cµK32

, (9.1.6)

where = 0.1D, and D is the diameter of the laser furnace. Next, using the discreterandom model employed by Povitsky and Salas [11], the average life time of turbulenteddies is calculated as,

t = cK

ε, (9.1.7)

this provides with the semi period in space of the eddies which is,

L = ut. (9.1.8)

As a first guess, the initial turbulent gust can be set as,

u′ = u′max sin(π

Lx), (9.1.9)

and adding a proper phase shift along the vertical direction, we get gust-type tur-bulent flow-field as shown in Figure (9.1.1).

(a) Horizontal Fluctuating VelocityComponent

(b) Vertical Fluctuating Velocity Com-ponent

Figure 9.1.1: Fluctuating Velocity Component


9.2 Results and Discussion

Solutions for ablation plume are presented at post-ablation time of 300 µs (see Figure(9.2.2). The plume rapidly disperses into base flow and evolves into a vortex pairproceeding down.

(a) Euler Plume concentration. (b) NS Plume concentration.

(c) NS+Turbulence Plume concentra-tion.

Figure 9.2.2: Plume concentration

When viscous effects are included, the plume shape agrees satisfactory withthe experimental shadow photos [12]. Some difference between experimental andsimulation results can be attributed to fact that the real plume in actual carbonnanotube synthesis involves multiple species of carbon, as against the single-speciesideal gas in the present study. The effect of turbulent gust is minor at the consideredtime interval of 300µs.

It is possible to develop an analytical model that determines the plume shapefrom the amount of deposited vorticity. Such a model could be viewed as an ex-

9.2. RESULTS AND DISCUSSION 143

tension of shock-bubble interaction by Yang et al. [13]. This model can be used topredict the behavior of plume through vorticity dynamics-baroclinic and viscous vor-ticity deposition on the plume. The current problem however, is significantly moreinvolved than the single shock wave-bubble interaction since the plume interactswith several oblique shock waves and possibly with expansion waves.

Following Yang et al. [13], extensive numerical simulations for various initialconditions are needed to validate such a simplified model. The behavior of the vortexpair that in turn, controls the plume evolution can be analyzed using the vorticitytransport equation:

Dω

Dt= ω · ∇q + ν∇2ω +

∇ρ×∇pρ2

(9.2.1)

In the above equation the cross product of the density and pressure gradient isthe baroclinic vorticity source. The density gradient exists inside the plume andthe strong pressure gradient is provided by reflected shock waves of incident shockwave. This gives rise to vorticity generation inside the plume.

Both terms, the vorticity source and viscous diffusion of vorticity, depend onthe injection velocity and on the base flow conditions. The parameters of the flowand injection velocity determine the distance between the plume and strong shockwaves. At higher injection velocity the plume penetrates the reflected shock wavegiving rise to baroclinic vorticity source and as a result plume immediately turnsinto the pair of vortex proceeding down as seen in Figure (9.2.3).

(a) Density Contours for High InjectionVelocity

(b) Density Contours for Low InjectionVelocity.

Figure 9.2.3: Density Contours

Extensive numerical experiments can provide some insight into the behavior ofboth these terms with different flow conditions. Thus the results of such experiments


can be used to guide the development of relatively simple analytical model that willallow extrapolation of the numerical results to various practical applications.

9.3 Conclusion

The numerical simulations have captured the basic features of ablation plume. Vis-cosity is a prominent parameter that controls the plume evolution. During the initialtime of plume evolution the effect of turbulence on plume evolution is less promi-nent than viscous and baroclinic effects. However, turbulence may affect the plumeevolution in its later stages when its speed of propagation slows down. We need toevaluate individually the vorticity source term and viscous source term for plume toget better understanding of its behavior.

References

[1] R. Kelly, A. Miotello, A. Mele, and A. G. Guidoni, Plume formation and char-acterization in laser-surface interactions, Experimental Methods in the PhysicalSciences, 30, pp. 225-289, 1998.

[2] A. Puretzky, H. Schittenhelm, X. Fan, M. Lance, L. Allard, and D. Geohegan,Investigation of single-wall carbon nanotube growth by time-restricted laser va-porization, Physical Review B, 65 (245425) 2002.

[3] S. I. Anisimov, D. Bauerle and B. B. Lukyanchuk, Gas dynamics and film profilesin pulsed-laser deposition of materials, Phys. Rev. B 48, pp. 12076-12081 1993.

[4] S.S. Mao, X. Mao, R. Grief, and R.E. Russo, Influence of preformed shock waveon the development of picosecond laser ablation plasma, J. Applied Physics, 89,pp. 4096-4099 2001.

[5] D. Lobao and A. Povitsky, Furnace geometry effects on plume dynamics in laserablation, Mathematics and Computers in Simulation (Special Issue on Wave Prop-agation), 65, pp. 365-383, 2004; short version in Proceedings of ICCSA-2003, Lec-ture Notes in Computer Science 2668, pp. 871-880 2003.

[6] D. Lobao and A. Povitsky, Single and Multiple Plume Dynamics in Laser Abla-tion for Nanotube Synthesis, AIAA Journal, accepted for publication, 2004; shortversion -AIAA Paper, 2003-3923 2003.

[7] J. J. Quirk and S. Karni, On the dynamics of a shock-bubble interaction, J. FluidMech., 318, pp. 129-163 1996.

9.3. CONCLUSION 145

[8] R. Greendyke, C. Scott, and J. Swain, CFD simulation of laser ablation in carbonnanotube production, 8th AIAA/ASME joint Thermodynamics and Heat TransferConference, 2002.

[9] A. V. Bulgakov and N. M. Bulgakova, Gas-dynamic effects of the interactionbetween a pulsed laser ablation plume and the ambient gas: analogy with anunder-expanded jet, J. Phys. D: Applied Physics, 31, pp. 693-703 1998.

[10] C. W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatoryschemes for hyperbolic conservation laws, NASA/CR-97-206253, 1997.

[11] A. Povitsky, and M. Salas, Thermal regime of catalyst particles in reactor forproduction of carbon nanotubes, AIAA Journal, 41(11), pp. 2130-2142, 2003.

[12] A. Puretzky, D. Geohegan, X. Fan, and S. Pennycook, Dynamics of single-wallcarbon nanotube synthesis by laser vaporization, Applied Physics, A 70, pp. 153-160, 2000.

[13] J. Yang, T. Kubota, and E. Zukoski, A model for characterization of a vortexpair formed by shock passage over a light-gas inhomogenity, J. Fluid Mech., 258,pp. 217-244, 1994.

Chapter 10

Linearized Richtmyer-MeshkovFlow for Elastic Materials

JeeYeon N. PlohrLos Alamos National LaboratoryTheoretical Division, Equations of State and Mechanics of Materials [email protected]

Bradley J. PlohrLos Alamos National LaboratoryTheoretical Division, Complex Systems [email protected]

Abstract: We present a study of Richtmyer-Meshkov flow for elastic materials.This flow, in which a material interface is struck by a shock wave, was originallyinvestigated for gases, where growth of perturbations of the interface is observed.Here we consider two elastic materials in frictionless contact. If the interface is flat,the flow defines a Riemann problem that is solved numerically; its solution containsa transmitted shock wave and either a reflected shock or rarefaction wave in thelongitudinal modes. We linearize the governing equations around this backgroundsolution under the assumption that the perturbation is small. The resulting linearsystem of partial differential equations is solved numerically using a finite differencemethod supplemented by front tracking. For the elastic case, in contrast to the caseof gases, we find that the material interface remains bounded: the nonzero shearstiffness stabilizes the flow. In particular, the linear theory remains valid at latetime. Moreover, we identify the principal mechanism for the stability of Richtmyer-Meshkov flow for elastic materials: the vorticity deposited on the material interface

147

148 CHAPTER 10. RICHTMYER-MESHKOV FLOW

during shock passage is convected away by the shear waves, whereas for gas dynamicsit stays on the interface.

Keywords: Richtmyer-Meshkov instability, conservation laws, equation of state,shear stiffness, Riemann problem, shock wave, rarefaction wave, front tracking

Introduction

The instability, caused by the passage of a shock wave, of an interface between twogases was first studied by R. D. Richtmyer in a 1954 Los Alamos report ([Richt-myer, 1960]). His theoretical predictions and numerical calculations were confirmedexperimentally by E. Meshkov fifteen years afterward ([Meshkov, 1970]), and thistype of instability is therefore named the Richtmyer-Meshkov instability. Since thisinitial work, extensive theoretical, numerical, and experimental research has beenconducted on the Richtmyer-Meshkov instability. The reader may wish to consult areview article ([Rupert, 1992]) as well as more recent references (e.g., [Grove et al.,1993]; [Yang et al., 1994]; [Holmes, 1994]; [Zhang and Graham, 1998]).

A schematic illustration of the Richtmyer-Meshkov flow configuration is shownin figure 10.0.1. An incident (left-facing) shock wave impinges on a corrugated mate-rial interface, generating transmitted and reflected waves. The type of the reflectedwave (shock or rarefaction) is determined by the material parameters; figure 10.0.1shows the case of a reflected shock wave. During the interaction, the material in-terface is accelerated by the incident shock wave. This acceleration can cause thecorrugations in the interface to grow in amplitude in an unstable fashion.

In this paper, which is based on the Ph.D. dissertation ([Nam, 2001]) of one ofus (JNP, nee Nam), we investigate the behavior of elastic materials, such as metals,in Richtmyer-Meshkov flow. (As we will see, the term “instability” is not appropriatefor elastic materials, so we use the word “flow”.) In contrast to a gas, an elasticmaterial resists shear strain. Emerging from the interaction of the shock wave withthe material interface are transmitted and reflected waves in shear modes, as well aslongitudinal modes, which leads to a significant change in the subsequent evolutionof the material interface. The tensor nature of strain requires that the principles ofsolid mechanics be carried over to the analysis of the Richtmyer-Meshkov flow.

Just as Richtmyer did in his pioneering research on the Richtmyer-Meshkovinstability for gas dynamics, we take the first step of studying the small amplitudelimit. Our approach is based on Richtmyer’s analysis for the case of a reflectedshock wave, but it also draws from the refinements of this analysis ([Yang et al.,1994]). In making the transition from gas dynamics to elasticity, we have organizedthe analysis in a systematic way that facilitates application to general systems of

10.1. GOVERNING EQUATIONS 149

x

y

materialinterface

incidentshockwave

materialinterface

(a) Before interaction. (b) During interaction. (c) After interaction.

transmittedshock wave(s)

reflectedwave(s)

Figure 10.0.1: An illustration of the Richtmyer-Meshkov flow configuration: (a) theincident shock wave and material interface before interaction; (b) the interactionbetween the incident shock and the interface; (c) waves emerging after interaction.

conservation laws.

Shear stiffness has a stabilizing effect in Richtmyer-Meshkov flow: according toour simulations, the amplitude remains bounded, oscillating around an asymptoticvalue, rather than growing linearly with time, as it does for gases; i.e., there isno instability. Moreover, the frequency of the oscillation increases with the shearstiffness of the materials. This result is shown both for reflected shock and reflectedrarefaction cases.

10.1 Governing Equations

The dynamical equations for an elastic material consist of a system of conserva-tion laws along with corresponding jump conditions for discontinuous solutions andconstitutive equations specifying the elastic material response.

The partial differential equations governing an elastic material in the Eulerianframe can be written in first-order conservative form ([Plohr and Sharp, 1988;Trangenstein and Colella, 1991; Plohr and Sharp, 1992; Wagner, 1996]). Theseconservation laws involve, as field variables, the deformation gradient tensor, the ve-locity vector, and a thermodynamic variable, which together characterize the stateof the material; and they represent the physical principles of material continuity,conservation of momentum, and conservation of energy.


We characterize the state of the flow by the vector

U =[g1

1 g12 g2

1 g22 v1 v2 p

]T, (10.1.1)

where gαi for α = 1, 2, i = 1, 2 denotes the inverse deformation gradient, vi for

i = 1, 2 denotes the particle velocity, and p is the (hydrostatic part of the) pressure.The conserved quantity vector is

W = H(U) :=[g1

1 g12 g2

1 g22 ρv1 ρv2 ρe

]T, (10.1.2)

where ρ = ρ0 det g is the mass density (ρ0 being the reference mass density), e =12viv

i + ε is the total specific energy, and ε is the specific internal energy, which isexpressed in terms of ρ and p through an equation of state. The conservation lawstake the form

H(U);t + F(U);x + G(U);y = 0, (10.1.3)

with

F(U) :=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

g11v

1 + g12v

2

0g2

1v1 + g2

2v2

0ρ(v1)2 − σ11

ρv2v1 − σ21

ρev1 − v1σ11 − v2σ

21

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦, G(U) :=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

0g1

1v1 + g2

2v2

0g1

1v1 + g2

2v2

ρv1v2 − σ12

ρ(v2)2 − σ22

ρev2 − v1σ12 − v2σ

22

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10.1.4)

being the fluxes in the x- and y-directions, respectively, and σij, i, j = 1, 2, beinggiven in terms of gα

i and p. In these terms, the Rankine-Hugoniot jump conditions,which are satisfied across shock waves, are

− s∆H(U) + ∆F(U)n1 + ∆G(U)n2 = 0. (10.1.5)

The system of ordinary differential equations satisfied by rarefaction waves can bederived from the jump conditions by taking the limit of infinitesimal jumps.

To complete the governing system of conservation laws, we must specify howthe stress tensor σij relates to the conserved quantities. We adopt a thermoelasticconstitutive equation satisfying the axioms of locality, entropy production, and ma-terial frame indifference (see, e.g., [Marsden and Hughes, 1983]). Such a constitutiveequation amounts to an equation of state

ε = ε(C, η) (10.1.6)

10.2. BACKGROUND FLOW 151

relating the specific internal energy ε to the right Cauchy-Green strain tensor C =F TF (where F is the deformation gradient) and the specific entropy η, which de-termines the Cauchy stress σij and the temperature θ through the formulas

σij = 2ρF iα∂ε

∂Cαβ

F jβ, (10.1.7)

θ =∂ε

∂η. (10.1.8)

For modeling elastic solids, we assume that ε is the sum of two terms, thevolumetric (or “hydrodynamic”) energy and the shear energy. The volumetric energyεh accounts for the response of the material to changes in volume; it depends on Csolely through the specific volume τ = 1/ρ = J/ρ0, where J2 = detC. The shearenergy, on the other hand, accounts for the response to shear strain. To measurethe shear strain, we define the volume-preserving part of the right Cauchy-Greentensor ([Simo and Hughes, 1998]), C = J−2/3C, and the elastic shear distortion, ε,given by

ε2 = 12

(tr C − 3

). (10.1.9)

Because det C = 1, the tensor C and therefore ε are unaffected by volume change;moreover, ε reduces to the usual measure of shear strain (the norm of the deviatorof the symmetric part of the displacement gradient) in the small-strain limit.

Specifically, we adopt the equation of state

ε = εh(τ, η) + τGε2, (10.1.10)

where εh(τ, η) is a hydrodynamic energy and G is the (constant) shear modulus.

On the left and right sides of the material interface, materials are assumedto be homogeneous and behaves according to the constitutive relation given above.The material interface, which separates two different materials, needs to be modeledproperly. The appropriate model for the behavior of a material interface depends onthe problem of interest (welded materials, a lubricated interface, etc.). In the presentpaper, we assume that: (1) no separation and no penetration of the materials occursat the material interface; and (2) no friction resists the relative tangential motionof the materials.

10.2 Background flow

The background problem of the Richtmyer-Meshkov flow is such that the incidentshock wave impinges on the material interface head-on. By symmetry, there is no


shear wave, and therefore the problem becomes uniaxial. We developed a Riemannsolver specialized for the equation of state and the interface conditions for the currentmodel ([Plohr and Plohr, 2004a]).

The solution U of the background problem is independent of y and satisfies

H(U);t + F(U);x = 0. (10.2.1)

It contains of five waves: left-moving longitudinal and shear waves, three contactmodes, and right-moving shear and longitudinal waves. (Although the shear waveshave zero strength in the background solution, they are present in the perturbedsolution.) The right-moving transmitted wave is always a shock wave, whereas theleft-moving reflected wave is either a shock wave or a rarefaction wave, depending onthe character of the materials. (Roughly speaking, when the incident shock impingesfrom a lighter material onto a heavier material, the reflected wave is a shock wave,and vice versa.)

10.3 Linearization

Given the background solution, we linearize the governing equations around it toobtain the linear system of PDE for the first order perturbation. The linearizationfor the reflected shock wave case is straightforward, as the background state is sector-wise constant. However, when the reflected wave is a rarefaction, the linearizationrequires more careful treatment: the nonzero derivative of the background solutionand the perturbation of the interface give rise to extra terms.

In Richtmyer-Meshkov flow, the rarefaction wave is a right-facing longitudinalwave with a perturbed trailing edge but an unperturbed leading edge. If x = φ(t)denotes the trajectory of the trailing edge for the background U , then the trajectoryof the trailing edge for the perturbed solution U ε takes the form x = φε(y, t) :=φ(t) + εφε(y, t). We define the perturbation in the state state U ε by

U ε(x+ εφε(y, t), y, t) = U(x, t) + εU ε(x, y, t). (10.3.1)

The change of variables from x to x = x − εφε(y, t) is crucial for U ε to remainbounded as ε→ 0.

The linearized system of PDE governing U := limε→0 Uε is[

H′(U)U]

;t−H(U);xφ;t +

[F ′(U)U

];x

+[G ′(U)U

];y− G(U);xφ;y = 0, (10.3.2)

where φ := limε→0 φε. A simplification occurs if we introduce the alternate state

variable˜U(x, y, t) = U(x, y, t) − U;x(x, t)φ(y, t), (10.3.3)

10.4. INITIALIZATION 153

which satisfies [H′(U) ˜U

];t

+[F ′(U) ˜U

];x

+[G ′(U) ˜U

];y

= 0 (10.3.4)

([Plohr and Plohr, 2004b]).

The internal boundary conditions, which hold at the background shock waves,the material interface, and the edges of the rarefaction fan, are also linearized.

10.4 Initialization

Unlike the background problem, the linearized system needs initialization becauseduring the interaction of the incident shock and the material interface, the problemis intrinsically nonlinear and we cannot apply the linear theory. We can solve thelinear system of PDE starting from the time when the incident shock exits theperturbed interface.

We approximate the interaction between the shock and interface by modelinga sinusoidal interface as a zigzag, which is a reasonable approximation when theperturbation is small. Then the problem becomes 2-D Riemann problem or obliqueshock polar problem. Using the fact that the incident angle is shallow, we linearizethe 2-D equations around 1-D solution where this angle is zero, and the resultingequations are solved.

10.5 Numerics

We apply the method of finite differences to solve the linear system of PDE. Thenumerical scheme consists of three parts: updating the states in the interior regionsbetween the background waves, updating the states on the front using the front-tracking method, and updating the states near the front (those with stencil thatstraddles the front). Specifically, we used second-order Lax-Wendroff method forthe interior region and linear extrapolation (second-order) for the near-front stateupdate.

In order to verify our numerical code, we reproduced the known results forthe gas dynamics ([Yang et al., 1994]): the growth rate for gas rises quickly in thebeginning and then approaches a nonzero asymptotic value. We also performedsome numerical experiments. First, we added a small artificial viscosity term tothe Lax-Wendroff scheme; we found that the solution deviates unacceptably fromthe correct one at late time, as shown in the plot of the growth rate of the interfaceperturbation (Fig. 10.5.2). Second, we tried constant, instead of linear, extrapolation


0 20 40 60 80 100time

0

0.05

0.1

0.15

0.2

0.25

grow

th r

ate

linear; b = 0linear; b = 0.001linear; b = 0.01constant; b = 0

Figure 10.5.2: Growth rate vs. time for the gas dynamics test problem, as calcu-lated with various artificial viscosity coefficients and with constant, instead of linear,extrapolation at the fronts.

in updating the state near fronts; this modification also had an undesirable effect(see Fig. 10.5.2). In conclusion, it is important to keep the numerical scheme trulysecond-order, not only in the interior but also at the front. As a simulation requiresmany time steps (typically tens of thousand), the first-order errors accumulate overtime.

10.6 Results

Our main goal in this study is to determine and understand the effect of shearstiffness on the growth of the interface perturbation. To this end, we simulatedRichtmyer-Meshkov flow for various choices of the shear moduli while the othermaterial parameters were kept fixed at the values for Aluminum and Tantalum. Wefound that even with very small shear moduli (0.1% of the physical values), the flowremains stable: the growth rate oscillates around zero; moreover, the amplitude(the time integral of the growth rate) stays bounded. Also, there is a relationshipbetween the shear stiffness and the frequency of the amplitude oscillation, shown inFig. 10.6.3: the larger shear modulus, the larger frequency. This can be understoodin terms of the shear energy τGε2, which generates the restoring force, in that theshear modulus G is like the spring constant in Hooke’s law.

Next, we simulated Richtmyer-Meshokov flow with various incident shockstrengths (see Fig. 10.6.4). The qualitative behavior does not change and the fre-quency of the amplitude oscillation remains the same for different incident Mach

10.6. RESULTS 155

0 2 4 6

κ−1/2

0

5

10

15

20

peri

od

Figure 10.6.3: Period of oscillations vs. 1/√κ for various values of the interpolation

parameter κ. Here κ ∈ [0, 1] is the ratio of the shear moduli to their physical values.

0 10 20 30 40 50time

0.75

0.8

0.85

0.9

0.95

1

ampl

itude

M = 1.022M = 1.057M = 1.076M = 1.116

Figure 10.6.4: Amplitude vs. time for Tantalum/Aluminum and various Machnumbers of the incident shock wave.


numbers. For stronger shocks, the mean amplitude becomes smaller because of thelarger compression at the initial time.

Finally, when the reflected wave is a rarefaction, the plot of the amplitude isalmost identical with that for the reflected shock case. The type of the reflectedwave does not change the behavior of the interface.

The stabilizing mechanism can be explained in terms of vorticity. The inter-facial instability is caused by the vorticity deposited on the interface, and in case ofelastic materials, the shear waves carry away the vorticity, stabilizing the flow.

Acknowledgments

The authors thank Professor James Glimm for his continual support and encour-agement on completion of this work.

References

Grove, J., Holmes, R., Sharp, D., Yang, Y., and Zhang, Q. (1993). Quantitativetheory of Richtmyer–Meshkov instability. Phys. Rev. Lett., 71(21):3473–3476.

Holmes, R. L. (1994). A Numerical Investigation of the Richtmyer–Meshkov Insta-bility Using Front Tracking. PhD thesis, State Univ. of New York at Stony Brook.

Marsden, J. and Hughes, T. (1983). Mathematical Foundations of Elasticity. Prentice-Hall.

Meshkov, E. (1970). Instability of a shock wave accelerated interface between twogases. NASA Tech. Trans., F-13:074.

Nam, J. (2001). Linearized Analysis of the Richtmyer–Meshkov Instability for ElasticMaterials. PhD thesis, State Univ. of New York at Stony Brook.

Plohr, B. and Sharp, D. (1988). A conservative Eulerian formulation of the equationsfor elastic flow. Adv. Appl. Math., 9:481–499.

Plohr, B. and Sharp, D. (1992). A conservative formulation for plasticity. Adv. Appl.Math., 13:462–493.

Plohr, J. and Plohr, B. (2004a). Linearised analysis of Richtmyer-Meshkov flow forelastic materials. Technical Report LA-UR-91-403, Los Alamos National Labora-tory.

10.6. RESULTS 157

Plohr, J. and Plohr, B. (2004b). Linearized analysis of Richtmyer-Meshkov flowfor elastic materials II: Reflected rarefaction case. Technical report, Los AlamosNational Laboratory. in preparation.

Richtmyer, R. (1960). Taylor instability in shock acceleration of compressible fluids.Comm. Pure Appl. Math., 13:297–319.

Rupert, V. (1992). Shock-interface interaction: Current research on the Richtmer–Meshkov problem. In Takayama, K., editor, Shock Waves, proceedings of the 18thinternational symposium on shocks waves. Springer-Verlag, New York.

Simo, J. and Hughes, T. (1998). Computational Inelasticity. Springer-Verlag, NewYork–Heidelberg–Berlin.

Trangenstein, J. and Colella, P. (1991). A higher-order Godunov method for model-ing finite deformation in elastic-plastic solids. Comm. Pure Appl. Math., XLIV:41–100.

Wagner, D. (1996). Conservation laws, coordinate transformations, and differentialforms. In Glimm, J., Graham, M. J., Grove, J. W., and Plohr, B. J., editors, Pro-ceedings of the Fifth International Conference on Hyperbolic Problems Theory,Numerics, and Applications, pages 471–477. World Scientific Publishers, Singa-pore.

Yang, Y., Zhang, Q., and Sharp, D. (1994). Small amplitude theory of Richtmyer–Meshkov instability. Phys. Fluids, 6(5):1856–1873.

Zhang, Q. and Graham, M. J. (1998). A numerical study of Richtmyer–Meshkovinstability driven by cylindrical shocks. Phys. Fluids, 10:974–992.

Chapter 11

Modeling of cavitating and bubblyflows and applications

Roman Samulyak1, Tianshi Lu2, James Glimm1,2, and Yarema Prykarpatskyy1

1Computational Science CenterBrookhaven National Laboratory, Upton, NY 119732Department of Applied Mathematics and StatisticsSUNY at Stony Brook, Stony Brook, NY [email protected], [email protected], [email protected], [email protected]

Abstract: We have studied two approaches to the modeling of bubbly andcavitating fluids. The first approach is based on the direct numerical simulation ofgas/vapor bubbles using the interface tracking technique. The second one uses ahomogeneous description of bubbly fluid properties. Two techniques are complemen-tary and can be applied to resolve different spatial scales in simulations. Numericalsimulations of the dynamics of linear and shock waves in bubbly fluids have beenperformed and compared with experiments and theoretical predictions. Two tech-niques are have been applied to study hydrodynamic processes in liquid mercurytargets for a new generation of accelerators.

Keywords: Cavitation, multiphase flow, front tracking

Introduction

An accurate description of cavitation and wave propagation in cavitating and bubblyfluids is a key problem in modeling and simulation of hydrodynamic processes in a

159

160 CHAPTER 11. MODELING OF CAVITATING AND BUBBLY FLOWS

variety of applications ranging from marine engineering to high energy physics. Themodeling of free surface flows imposes an additional complication on this multiscaleproblem.

The wave propagation in bubbly fluids have been studied using a variety ofmethods. Significant progress has been achieved using various homogeneous descrip-tions of multiphase systems (see for example [[1]; [15]] and references therein). TheRayleigh-Plesset equation for the evolution of the average bubble size distributionhas often been used as a dynamic closure for fluid dynamics equations. This allowsto implicitly include many important physics effects in bubbly systems. Numericalsimulations of such systems require relatively simple and computationally inexpen-sive numerical algorithms. Nevertheless, homogeneous models cannot capture allfeatures of complex flow regimes and exhibit sometimes large discrepancies withexperiments even for systems of non-dissolvable gas bubbles.

The heterogeneous method, or direct numerical simulation, is a powerful methodfor multiphase problems based on techniques developed for free surface flows. Theheterogeneous method is potentially a very accurate technique, limited by only nu-merical errors. It allows to account for drag, surface tension, and viscous forces aswell as the phase transition induced mass transfer. Examples of numerical simula-tions of a single vapor bubble undergoing a phase transition on its surface are givenin [[8]; [16]]. Systems of bubbles in fluids were modeled in [[5]] using the incom-pressible flow approximation for both fluid and vapor and a simplified version of theinterface tracking. In this paper, we describe a direct numerical simulation methodfor systems of compressible bubbles in fluids using the method of front tracking.Our FronTier code is capable of tracking and resolving topological changes of alarge number of fluid interfaces in 2D and 3D spaces. We present the simulation re-sults of the wave dynamics of linear and shock waves in bubbly systems and comparethem with classical experiments.

The direct numerical simulations of wave dynamics in bubbly fluids in large 3Ddomains remain, however, prohibitively expensive even on supercomputers. Homo-geneous models can effectively be used for such systems, especially if the resolving ofspatial scales smaller then the distance between bubbles is not necessary. To modelcavitating and bubbly fluids within the homogeneous approximation, we have re-cently developed and implemented in the FronTier code a two-phase equation ofstate (EOS) model based on the isentropic approximation. Therefore both hetero-and homogeneous approaches have advantages and disadvantages and can be usedto resolve different temporal and spatial scales in numerical simulations. In thispaper, we present results of the validation and comparison of our homogeneous anddirect numerical simulation models.

Two numerical approaches are being used to study hydrodynamic processes in-

11.1. MODELING OF MULTIPHASE FLOWS 161

volving cavitation and bubble dynamics in liquid mercury targets for new generationaccelerators such as the Spallation Neutron Source and the Muon Collider/NeutrinoFactory. Hydrodynamic instabilities and cavitation in the Muon Collider target,which is a mercury jet interacting with high intensity proton pulses in the presenceof a strong magnetic field, will create complications for the machine operation. Thecollapse of cavitation bubbles in the Spallation Neutron Source mercury target, re-sulting in the pitting of steel walls, has been the most critical problem reducing thetarget lifetime. The injection of layers of gas bubbles in mercury has been proposedas a possible pressure mitigation technique. These processes must be studied bymeans of large-scale numerical simulations.

The paper is organized as follows. In Section 1, we describe the homogeneousand heterogeneous EOS models for multiphase flows and their validation. Section 3presents results of the numerical simulation of the liquid mercury jet interacting withhigh intensity proton pulses using two cavitation modeling techniques. Applicationsto the SNS target are discussed in Section 3. We conclude the paper with a summaryof our results and perspectives for future work.

11.1 Modeling of Multiphase Flows

11.1.1 Homogeneous Method

The homogeneous flow approximation provides a simple technique for analyzingtwo-phase (or multiple phase) flows. It is sufficiently accurate to handle a varietyof practically important processes. Suitable averaging is performed over the lengthscale which is large compared to the distance between bubbles and the mixture istreated as a pseudofluid that obeys an equation of state (EOS) of a single componentflow [[14]].

We have recently developed [[11]] a simple isentropic homogeneous equationof state for two-phase liquids and implemented the corresponding software libraryin FronTier, a compressible hydrodynamics code with free interface support. Thebubbly/cavitating liquid is described by the system of equations of compressiblehydrodynamics for a single component fluid. The equation of state which closesthis system describes averaged physics properties of the bubbly liquid at a givenvalues of the void fraction. The isentropic approximation reduces by one the num-ber on independent variables defining the thermodynamic state. As a result, allthermodynamic states in our EOS are functions of density only.

The proposed EOS consists of three branches. The pure vapor branch is de-scribed by the polytropic EOS reduced to an isentrope and the pure liquid phase is


represented by the corresponding reduction of the stiffened equation of state model[[9]]. The two branches are connected by a model for the liquid-vapor mixture

P = Psat,l + Pvllog

[ρsat,vasat,v

2(ρsat,l + α(ρsat,v − ρsat,l))

ρsat,l(ρsat,vasat,v2 − α(ρsat,vasat,v

2 − ρsat,lasat,l2))

], (11.1.1)

where ρsat,v, ρsat,l, asat,v, asat,l are the density and the speed of sound of vapor andliquid in saturation points, respectively, Psat,l is the liquid pressure in the saturationpoint, α is the void fraction

α =ρ− ρsat,l

ρsat,v − ρsat,l

, (11.1.2)

and the parameter Pvl is

Pvl =ρsat,vasat,v

2ρsat,lasat,l2(ρsat,v − ρsat,l)

ρsat,v2asat,v

2 − ρsat,l2asat,l

2. (11.1.3)

The expression (13) was derived by integrating an experimentally validated model forthe sound speed in bubbly mixture [[14]]. A set of the EOS input parameters, most ofwhich are measurable quantities, allows to fit the two-phase EOS to thermodynamicsdata for real fluids. The selection of input parameters and some other details on theEOS model are presented in [[11]].

The most important feature of the homogeneous isentropic EOS model is thecorrect behavior of the sound speed in liquid at void fractions ranging from the pureliquid to pure vapor (gas) phases. The EOS reproduces the well known fact [[14]]that the sound speed in bubbly liquid is lower than the sound speed in not only pureliquid but also in the pure vapor phase.

The homogeneous equation of state has been validated through comparisonwith experimental data [[11]]. The use of two-phase EOS has led to improvementover single phase EOS simulations [[12]] of the Muon Collider mercury target exper-iments [[6]].

11.1.2 Heterogeneous Model

One of the main disadvantages of the homogeneous equation of state model formultiphase flows is its inability to resolve spatial scales comparable to the distancebetween bubbles. In many cases, cavitation in strong rarefaction waves may leadto a rapid growth of a relatively small number of cavitation bubbles. Averaging offluid properties will result in unresolved fine structure of waves which may be crit-ical for understanding the important features of the flow dynamics such as surfaceinstabilities, bubble collapse induced pressure peaks etc. The direct numerical sim-ulation method eliminates this deficiency and improves many other thermodynamicand hydrodynamic aspects of the modeling of cavitating and bubbly flows.

11.2. APPLICATIONS TO THE MUON COLLIDER TARGET 163

In the heterogeneous method, we model a liquid – vapor or liquid – non-dissolvable gas mixture as a system of one phase domains (vapor bubbles in a liq-uid) separated by free interfaces. We have used our code FronTier, a compressiblehydrodynamics code with free interface support (see [[4]] and references therein), tomodel the behavior of bubble/liquid interfaces. Though computationally intensive,such an approach is very accurate in treating important effects in bubbly flows. Themethod makes it possible to model some non-equilibrium thermodynamics featuressuch as finite critical tension in cavitating liquids. The Riemann problem for thephase transition can be solved for the liquid - vapor interface. The direct numericalsimulation is potentially a very accurate technique, limited by only numerical errors.

Numerical simulation of the cavitation presents additional complications, un-certainties and numerical challenges compared to the simulation of wave phenomenain bubbly fluids (fluids containing small non-dissolvable gas bubbles). These prob-lems are associated with the dynamic creation and collapse of bubbles in the com-putational domain. The corresponding software routines were implemented in theFronTier code. Some remarks on the choice of initial bubble size and the distributionof cavitation centers can be found in [[17]].

The direct numerical simulation method has been validated through the studyof the dynamics of linear and nonlinear waves in bubbly liquids [[71]]. Schematicof the numerical experiment is shown in the left part of Figure 11.1.1. The liquidcontains non-dissolvable gas bubbles at normal conditions. The bubble radius is0.12 mm and the void fraction is 1.55 · 10−4. Measuring the dispersion relation andthe attenuation rates from simulations of small amplitude linear (sound) waves inbubbly fluids, we found that results are in good agreement with Fox, Curley andLarson’s experiments [[3]] as well as theoretical predictions. Some numerical andtheoretical results are depicted in Figure 11.1.1. We have also performed numericalsimulations of the interaction of shock waves with bubbly layers and found a goodagreement with experimental data. Some discrepancy in the amplitude of pressureoscillations can be explained by grid related numerical errors [[71]].

11.2 Applications to the Muon Collider Target

In this section, we will apply two approaches for modeling cavitation and bubblyflows to the study of a free mercury jet interacting with high energy proton pulses.Such a jet is a key component of the target for the proposed Muon Collider [[10]].The target will contain a series of mercury jet pulses of about 0.5 cm in radius and60 cm in length. Each pulse will be shot at a velocity of 30-35 m/sec into a 20Tesla magnetic field at a small angle (0.1 rad) to the axis of the field. When the jetreaches the center of the magnet, it is hit with a 3 ns proton pulse depositing about


Figure 11.1.1: Left: Schematic of the numerical experiment. Right: Dispersionrelation in bubbly flows. The bubble radius is 0.12mm and the void fraction is1.55 · 10−4. The grid size is 90 × 10800. δ is the damping coefficient.

100 J/g of energy in the mercury.

In the present studies of cavitation processes, the effect of the magnetic fieldwas not considered. The MHD processes in one-phase liquid mercury jet were studiedin [[12]]. The influence of the magnetic field on cavitation will be discussed in aforthcoming paper. The interaction with the proton pulse was modeled by addingthe proton beam energy density to the internal energy density of mercury at a singletime step. The proton energy deposition in mercury was approximated by a 2DGaussian distribution that accurately reproduces the actual beam energy depositionachieved in experiments.

Our first set of numerical simulations of cavitation in the mercury jet wasbased on the homogeneous EOS model. Numerical results showed the formation ofa large liquid-vapor mixture domain along the jet axis. The average velocity of thejet surface was in agreement with experimentally measured values [[6]]. The secondset of numerical experiments employed the direct simulation of cavitation bubbles.The dynamic interface insertion algorithm was used to create bubbles in rarefactionwaves with tension exceeding the of critical value of -10 bar. The evolution of thetwo phase domain in the mercury jet observed using the direct numerical simulationof cavitation bubbles was very similar to one obtained with the homogeneous modeland not very sensitive to changes in the concentration of nucleation centers. Namely,we did not observe essential changes of the averaged velocity of the mercury jetsurface at different concentrations of nucleation centers. A possible explanation isthat the rapid pressure relaxation in bubbly fluids weakens strong waves. In thiscase, the expansion of the two-phase domain is driven by inertial forces due to theradial momentum gained during short period of time after the interaction with the

11.3. APPLICATIONS TO THE SNS TARGET 165

Figure 11.2.2: Comparison of the homogeneous and direct numerical simulationmethods, and experimental results. Density distribution in the mercury jet obtainedusing the homogeneous EOS model (top left) and direct numerical simulation ofcavitation bubbles (top right). Bottom: Experimental image of the mercury jet.Average velocity of the jet surface

proton pulse. Some results of numerical simulations depicted in Figure 11.2.2.

We believe that one of the most essential differences between the homogeneousmodel and the direct numerical simulation of cavitation is associated with the rangeof numerically resolved spatial scales. Due to the averaging of fluid properties onthe length scale large compared to the distance between bubbles, the homogeneousmodel is not capable of resolving fine structure of waves in the fluid volume and smallscale surface perturbations. The direct numerical simulation approach is free of thisdeficiency. Using the later technique, we were able to obtain complete disintegrationof the jet at a lower concentration of nucleation centers. The modeling of theconcentration of nucleation centers is a subject of our present study.

11.3 Applications to the SNS Target

The Spallation Neutron Source, is a neutron source being built at Oak Ridge Na-tional Lab by the U.S. Department of Energy. The SNS will provide the most intensepulsed neutron beam in the world for scientific research and industrial development.The SNS target will include a flow of liquid mercury in a stainless steel container


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−400

−300

−200

−100

0

100

200

300

400

500

t ( ms )

P (

bar

)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2−600

−400

−200

0

200

400

600

t ( ms )

P (

bar

)

Figure 11.3.3: Pressure profile at the center of the entrance window in pure mercurytarget (left) and in mercury filled with air bubbles (right). Bubble radii are 1.0 mmand the volume fraction is 2.5%.

interacting with proton pulses. The consequent pressure waves induce severe cav-itation [[13]]. The cavitation induced erosion reduces the lifetime of the targetcontainer to only two weeks with 1MW proton pulses at the frequency of 60Hz. Inorder to mitigate the cavitation erosion, broad research is being done on the evalu-ation of cavitation-resistant materials and coatings. It has also been suggested thatthe injection of non-dissolvable gas bubbles into the container can mitigate pressurewaves and cavitation. Using the heterogeneous method described in the paper wehave been working on the estimation of the efficiency of cavitation mitigation frombubble injection. We have shown that the presence of non-dissolvable gas bubblesspeeds up the decay of pressure peaks on the target window, reducing the amount ofcavitation (Figure 11.3.3). Some high frequency transient pressure oscillations in thepresence of gas bubbles shown in the figure do not make a significant contributionto the cavitation.

11.4 Conclusions

We have applied the heterogeneous and homogeneous approaches to the simulationof cavitation in free surface flows. The heterogeneous model of cavitation, or directnumerical simulation, is based on the ability of our hydrodynamics code FronTierto explicitly track complex interfaces and resolve their topological changes. Themethod has numerous potential advantages and applications. It is capable of re-solving small spatial scales and accounting for drag, surface tension, and viscousforces, and the phase transition induced mass transfer. It is, however, prohibitively

11.4. CONCLUSIONS 167

computationally expensive for large 3D domains.

The homogeneous description of bubbly/cavitating fluids is based on the av-eraging of their properties on length scales large compared with the distance be-tween bubbles. Despite being simple and computationally inexpensive, the methodis sufficiently accurate within its domain of validity and applicable to a variety ofpractically important problems. The method has been validated through the studyof the interaction of mercury with intensive proton pulses in the geometry typicalfor neutrino factory liquid mercury target experiments. The simulations are in goodquantitative agreement with experiments. Through the comparison of numericalsimulations with experiments and theoretical predictions on the propagation of lin-ear and shock waves in bubbly fluids, the heterogeneous approach also has beenvalidated. The hetero- and homogeneous approaches are complementary and canbe used to resolve different temporal and spatial scales in numerical simulations.

Two methods have been applied to the study of cavitation in mercury interact-ing with high energy proton pulses in targets for the new generation of acceleratorssuch as the Spallation Neutron Source and the Muon Collider/Neutrino Factory.The heterogeneous method is also being used to study the role of cavitation in theatomization of high speed liquid jets [[17]]. Currently, we have been working on theimprovement of various aspects of the heterogeneous method such as the modelingof the phase transition on the liquid - vapor interface, the distribution of nucleationcenters, and the numerical resolution through the use of adaptive mesh refinement.

Acknowledgments: Roman Samulyak and Yarema Prykarpatskyy would liketo thank Harold Kirk, Kirk McDonald, Robert Palmer, Adrian Fabich, JacquesLettry, and John Haines for useful discussions. Financial support has been providedby the USA Department of Energy, under contract number DE-AC02-98CH10886.

References

[1] Beylich, A.E. and Gulhan, A. On the structure of nonlinear waves in liquids withgas bubbles. Phys. Fluids A. 2:1412–1428, 1990.

[2] Brennen, C. E. Cavitation and Bubble dynamics. Oxford University Press, 1995.

[3] Fox, F.E., Curley, S.R., and Larson,G.S. J. Acoust. Soc. Am. 27:534, 1955.

[4] Glimm J., Grove J., Li X.-L, and Tan D.C. Robust computational algorithmsfor dynamic interface tracking in three dimensions. SIAM J. Sci. Comp. 21:2240–2256, 2000.

[5] Juric, D. and Tryggvason, G. Computation of boiling flows. Int. J. Multiphase


Flow 24: 387 – 410, 1998.

[6] Kirk, H., et al. Target studies with BNL E951 at the AGS. Particles and Accel-erators 2001. June 18-22, 2001, Chicago IL

[7] Lu, T., Samulyak, R., and Glimm, J. Direct numerical simulation of bubbly flowsand its application. Submitted to J. of Physics of Fluids, 2004.

[8] Matsumoto, Y. and Takemura, F. Influence of internal phenomena on gas bubblemotion. JSME Int. J. 37:288–296, 1994.

[9] Menikoff, R. and Plohr, B. The Riemann problem for fluid flow of real materials.Rev. Mod. Phys. 61:75–130, 1989.

[10] Ozaki, S., Palmer, R., Zisman, M., and Gallardo, J. (editors) Feasibility Study-II of a Muon-Based Neutrino Source. BNL-52623, 2001.

[11] Samulyak, R. and Prykarpatskyy, Y. Richtmyer-Meshkov instability in liquidmetal flows: influence of cavitation and magnetic fields. Mathematics and Com-puters in Simulations. 65:431–446, 2004.

[12] Samulyak, R. Numerical simulation of hydro- and magnetohydrodynamic processesin the Muon Collider target. Lecture Notes in Comp. Sci. 2331:391–400, 2002.

[13] Status report on mercury target related issues, SNS-101060100-TR0006-R00(July 2002).

[14] Wallis, G.B. One-dimensional Tho-phase Flow. McGraw-Hill, New York, 1969.

[15] Watanabe, M. and Prosperetti, A. Shock waves in dilute bubbly liquids. J. FluidMech. 274:349–381, 1994.

[16] Welch, S.W. Local simulation of two-phase flows including interface trackingwith mass transfer. J. Comp. Phys. 121:142–154, 1995.

[17] Xu, Z., Kim, M.N., Oh, W, Glimm, J., Samulyak, R., Li, X., Tzanos, C. Atom-ization of a High Speed Jet. J. Fluid Mech. 2005. Submitted.

Chapter 12

Component-based Adaptive MeshControl Procedures

Mark S. ShephardScientific Computation Research CenterRensselaer Polytechnic [email protected]

Eunyoung SeolScientific Computation Research CenterRensselaer Polytechnic [email protected]

Jie WanScientific Computation Research CenterRensselaer Polytechnic [email protected]

Andrew C. BauerScientific Computation Research CenterRensselaer Polytechnic [email protected]

Abstract: In this paper we demonstrate the use of loosely-coupled adaptiveloops created from already available software components. Using interoperable func-tions, we demonstrate the effectiveness of the adaptive loops through two examples,an electromagnetics simulation and a metal forming simulation. Both are completelyautomated and give accurate results.

169

170 CHAPTER 12. ADAPTIVE MESH CONTROL PROCEDURES

Keywords: adaptive loop, mesh, model, field, mesh adaptation, error estima-tion

12.1 Introduction

A large number of methods and associated analysis codes for the numerical solutionof partial differential equations are in common use today. Although these analysiscodes are capable of providing results to the required levels of accuracy for manyclasses of problems, the ability to provide these predictions is not automaticallycontrolled by the analysis code, but instead is a strong function of the input infor-mation provided. This deficiency is further complicated by the fact that most ofthe effort required to execute an analysis is associated with the generation of theinput, and that substantial expertise and training is required to successfully definethis input to provide reliable results. Efforts to address these problems are focusedon automatically constructing analysis code input (e.g., [[6]; [18]]) and employingthe development of adaptive analysis procedures [[4]; [21]]. The ultimate results ofthese efforts will be the production of automated adaptive analysis software.

Commercial software is available that can accept a general problem definitionand automatically generate the analysis code input. Such tools are dramaticallyreducing the time and effort required to perform analyses. However, the abilityto automatically generate input for and execute an analysis does not ensure thatresults to the level of accuracy needed are obtained. For the current discussion itis assumed that there is sufficient a-priori information to know that the results ofinterest can be obtained by solving sets of partial differential equations. In thiscase, the errors that must be controlled are the discretization errors associated withusing a finite-dimensional basis over a mesh. Although adaptive methods to controldiscretization errors are well known, their application is limited to some researchand specific special purpose codes.

Consideration of the steps of an automated adaptive analysis process and theinteractions of the components used in those steps help to explain why adaptiveanalysis methods are not more common. The steps of a finite element based auto-mated adaptive analysis system are:

i. Create a general geometry-based problem definition in terms of a non-manifold boundary representation of solid models as supported by commercialCAD systems that are well suited to provide the geometric domain definition.When supplemented with an attributing capability, general problem definitionscan be completed [[14]].

ii. Create an initial mesh using an automatic mesh generator, which gener-

12.1. INTRODUCTION 171

ates meshes directly from CAD representation [[6]], including procedures thatprovide flexible spatially-based control of the mesh [[17]].

iii. Perform analysis using a finite element procedure which constructs the ele-ment level contributions, assembles them into a global system and solves theglobal system. A wide variety of codes have been developed to support thisprocess. In most cases these codes operate on a mesh-based problem specifi-cation and employ data structures for a fixed mesh.

iv. Postprocess analysis results using error estimation and correction indi-cation procedures. These procedures are responsible for determining usefulestimates of the discretization error on the current mesh and indicating whereand how to modify the mesh to most effectively reduce the errors to an ac-ceptable level [[2]; [3]; [21]].

v. Improve mesh by altering it to obtain the mesh sizes indicated by the correc-tion indication procedures. These mesh improvements can be carried out byregenerating a new mesh, or by the modification of the existing mesh. The pro-cedures used in the current paper are based on generalized mesh modificationprocedures [[12]; [13]; [20]].

vi. Repeat the steps 3-5 until the desired level of accuracy is obtained.

Steps 1 and 2 interact with the geometry-based problem definition and gen-erate the mesh-based problem definition operated on by the finite element analysisprocedures. Because the only interaction between these components and the analysiscode is the output from the two components, the introduction of the automatic meshgeneration is straightforward. In the common case of performing the analysis on asingle mesh, the analysis procedures (performed in step 3) operate on the mesh-basedproblem definition only. The proper execution of steps 4 and 5 require interactionswith both the mesh-based and geometry-based problem definitions (e.g., improvingthe geometric approximation as the mesh is refined and associating the appropri-ate traction values to newly defined boundary nodes). In addition, as a result ofexecuting step 5 the mesh is modified which must be reflected in the mesh-basedproblem description used in the next analysis. The complexity of dealing with theseinteractions has dramatically slowed the introduction of adaptive analysis methodsinto practice.

One approach to address the mismatch between the needs of fixed mesh andadaptive mesh analysis procedures is to alter the analysis code to directly interactwith the adaptive analysis processes. The advantage of this approach is that theresulting code can minimize the total computation and data manipulation time re-quired (which appears to be important for transient adaptive analyses using explicit


time stepping [[15]]). Although it is possible to construct such adaptive codes thatinteract directly with the geometry-based problem specification [[5]], the modifica-tion of an existing fixed mesh code requires the introduction of entirely new datastructures thus forcing an extensive rewrite of the code. The expense and time re-quired to do this is large and in most cases considered prohibitive, particularly forwell established codes.

The alternative approach for use with existing fixed mesh analysis codes, whichis the focus of this paper, is to leave the analysis code unaltered and to use a set ofinteroperable information communication tools [[23]] to control the flow of informa-tion between the set of components used for creating the problem definition (step 1),mesh generation (step 2), error estimation and correction indication (step 4), andmesh improvement procedure (step 5). Section 12.2 overviews the components andinformation flow between components required in an automated adaptive process.Section 3 demonstrates the effectiveness of this approach for the construction of au-tomated adaptive loops using existing analysis codes, one for electromagnetic fieldsimulations and one for large deformation forming problems. In both these examplesthe finite element systems are solved using implicit methods and it is found thatthe added information transfer cost associated with constructing the adaptive loopwith a set of components that are external to the analysis code is small.

12.2 Integration of Interoperable Components

The steps of the adaptive loop need to be integrated such that software componentscan properly share information to be able to perform the steps. The componentsnecessary for performing the adaptive loop which interact with more than one adap-tive loop steps are:

• Geometry interface is a high level topological model of the domain whichsupports the integration to multiple CAD systems. The interoperable API ofthe modeler enables interactions with mesh generation, mesh modification andanalysis code input construction to obtain all domain geometry informationneeded [[5]; [19]].

• Mesh interface provides the services for storing and modifying mesh data [[5];[7]; [19]] during the adaptive process. The current procedures used the Algorithm-Oriented Mesh Database (AOMD) [[16]] to support these processes.

• Field interface provides complex functions to obtain the solution informationneeded for error estimation and to support the transfer of solution fields asthe mesh is adapted [[5]; [19]].

12.3. EXAMPLES 173

The modules used are the CAD modeler, analysis engine, mesh generator andmesh modification procedures. The geometry interface is used in all steps exceptstep 3. The mesh interface interacts with all steps except for the first one. The fieldinterfaces interacts with steps 3 through 5. The analysis engine is only used duringstep 3, the mesh generator is only used during step 2 and the mesh modification pro-cedures are only used during step 5. The analysis engine, mesh generator and meshmodification procedures are easily replaceable since all interactions with them arethrough the geometry, mesh, and/or field interfaces. With a well-defined interoper-able interface to the geometry, mesh and field modules, various adaptive loops areeasily built through vertical integration of the other components. The interoperablecomponents used here are being designed to be compliant with the TSTT inter-face [[23]] as they evolve so as to provide interchangeability that allows horizontalintegration across a number of different tools that provide similar functionalities.

12.3 Examples

Given a flexible set of adaptive error control components, adaptive loops have beenbuilt around two fixed mesh finite element codes. The first is a frequency domainelectromagnetics simulation code Omega3P [[11]] developed at the Stanford LinearAccelerator Center (SLAC). The second is a commercial metal forming simulationcode, DEFORM-3DTM [[8]] where the adaptive loop tracks the evolving geometry.

12.3.1 Adaptive Loop for Accelerator Design

SLAC’s eigenmode solver Omega3P, which is used in the design of next generationlinear accelerators, has been integrated with adaptive mesh control [[12]; [13]] toimprove the accuracy and convergence of wall loss (or quality factor) calculationsin accelarating cavities. The simulation procedure consists of interfacing Omega3Pto solid models, automatic mesh generator, general mesh modification, and errorestimator components to form an adaptive loop as depicted in Figure 12.3.1.

The accelerator geometries are defined as ACISTM solid models [[1]] and phys-ical parameters required by the simulation are associated with geometric modelentities. Using functional interfaces between geometric model and meshing tech-niques [[6]], the automatic mesh generation tools of SimmetrixTM [[20]] generatesan initial mesh. After Omega3P calculates the solution fields, the error indicationprocedure determines a new mesh size field and the mesh modification proceduresmodify the mesh to generate a new mesh for the next execution of Omega3P. Thisiterative procedure repeats until the desired accuracy is reached.


Initital meshGeometric model

Adapted mesh

Solution field Size field

− Mesh generatorSimmetrix

Mesh modification

SLACMesh database

− Eigensolver

Field

− Omega3P

Error estimator

Figure 12.3.1: Framework of adaptive loop for accelerator design

Figure 12.3.2: Mesh and wall-loss distribution for 3 adaptive steps

The adaptive procedure has been applied to a Trispal 4-petal acceleratingcavity. Figure 12.3.2 shows the mesh and wall loss distribution on the cavity surfacefor three adaptive steps with an increasingly denser mesh in the area of highfieldconcentration (from left to right). The procedure has been shown to reliably produceresults of the desired accuracy for approximately one-third the number of unknownsthe previous user controlled procedure produced [[9]].

12.3.2 Metal Forming Simulation

In 3D metal forming simulations the deformable parts undergo large plastic deforma-tions that result in major changes in the analysis domain geometry. The meshes ofthe deforming parts typically need to be frequently modified to continue the analysisdue to large element distortions, mesh discretization errors and/or geometric approx-

12.3. EXAMPLES 175

Figure 12.3.3: Automated adaptive forming simulation process.

imation errors. In these cases, it is necessary to replace the deformed mesh withan improved mesh that is consistent with the current configuration [[8]; [10]; [24]].History dependent field variables also need to be accurately transferred from theold mesh to the new mesh [[8]; [10]; [24]]. Remeshing generates an entirely newmesh even though there might be only a limited number of elements that need to bemodified. To more effectively address the needed mesh updates and field transfersand to provide higher solution accuracy, a component-based adaptive mesh controlprocedure as depicted in Figure 12.3.3 was developed. Detailed discussions on theinvolved components are presented in [[24]].

A steering link manufacturing problem as shown in Figure 1.6 is investigatedto demonstrate the developed capabilities. A total stroke of 41.7 mm is simulated.The allowed maximum geometric interference is 0.60 mm. The initial workpiecemesh consists of 6,765 mesh vertices and 28,885 mesh regions (Figure 12.3.5a). Thesimulation is completed with 20 mesh modification steps performed by adopting theautomated adaptive mesh control procedure. The final mesh of the achieved work-piece consists of 23,525 mesh vertices and 102,249 mesh regions (Figure 12.3.5c).The workpiece mesh is adapted to control solution error and the geometric approx-imation. The effects of mesh adaptivity are demonstrated in Figure 12.3.5. Thelarge elements as seen in the far left of these pictures are satisfactory because ofthe low strain gradient. In the regions of high strain gradient, smaller elements areneeded to control the discretization error while fine mesh is needed near the contactboundaries to control the geometric approximation errors [[8]; [24]].


Figure 12.3.4: Setup of the steering link problem.

(a) Initial workpiece mesh (28,885 mesh regions)

(b) Stroke =27.6 mm (50,424 mesh regions)

(c) Final workpiece mesh (102,249 mesh regions)

Figure 12.3.5: Mesh adapted consistently with the effective strain profile.

12.4. CLOSING REMARKS 177

Figure 12.3.6: Element quality control through remeshings and mesh modifications.

The element quality of the workpiece mesh measured in terms of the maximumdihedral angles before and after the mesh modification steps is shown in Figure12.3.6. It can be seen that throughout the mesh modification based simulation, theelement quality of the workpiece mesh is effectively controlled through the elementdistortion monitoring and local mesh modification based shape improvement.

12.4 Closing Remarks

This paper has shown two automated adaptive loops which accurately simulate theircorresponding physics. The adaptive loops are created using a set of interoperablecomponents that link analysis codes with geometry-based problem definitions, auto-matic mesh generation, error estimation procedures and generalized mesh modifica-tion procedures. Other adaptive loops can be easily constructed using this approach.

References

[1] Spatial Inc. (2004) http://www.spatial.com/components/acis/.

[2] Ainsworth M, Oden JT (2000) A Posteriori Error Estimation in Finite ElementAnalysis. Wiley-Interscience, John Wiley & Sons.

[3] Babuvska I, Strouboulis T (2001) The Reliability of the FE Method. Oxford Press.

[4] Bangerth W, Rannacher R (2003) Adaptive Finite Element Methods for Differ-ential Equations, Lectures in Mathematics VIII Vol. 207. Birkhauser.

[5] Beall MW, Shephard MS (1999) An Object-Oriented Framework for ReliableNumerical Simulations. Eng. w. Comp. 15:61-72.


[6] Beall MW, Walsh J, Shephard MS (2004) A comparison of techniques for geom-etry access related to mesh generation. Eng. w. Comp. 20(3):210-221.

[7] Beall MW, Shephard MS (1997) A general topology-based mesh data structure.Int. J. Numer. Meth. Eng. 40:1573-1596.

[8] Fluhrer J (2004) DEFORM-3DTM Versoin 5.0 User’s Manual, Scientific FormingTechnologies Corporation.

[9] Ge L, Lee LQ, Li Z, Ng C, Ko K, Luo Y, Shephard (2004) MS Adaptive MeshRefinement for High Accuracy Wall Loss Determination in Accelerating CavityDesign. Eleventh Biennial IEEE Conference on Electromagnetic Field Computa-tion Seoul, Korea, June 6-9.

[10] Kobayashi S, Oh S-I, Altan T (1989) Metal Forming and the Finite ElementMethod. Oxford University Press.

[11] Lee L-Q, et al. (2004) Solving large sparse linear systems in end-to-end acceler-ator structure simulations. SLAC-PUB-10320 January.

[12] Li X, Shephard MS, Beall MW (2002) Accounting for curved domains in meshadaptation. Int. J. Numer. Meth. Eng. 58:247-276.

[13] Li X, Shephard MS, Beall MW (2003) 3-D Anisotropic Mesh Adaptation byMesh Modifications. to appear in Comp. Meth. App. Mech. and Eng..

[14] O’Bara RM, Beall MW, Shephard MS (2002) Attribute Management Systemfor Engineering Analysis. Eng. w. Comp. 18:339-351.

[15] Remacle J-F, Flaherty JE, Shephard MS (2003) An adaptive discontinuousGalerkin technique with an orthogonal basis applied compressible flow problems.SIAM Review 45(1):53-72.

[16] Remacle J-F, Shephard MS (2003) An algorithm oriented mesh database. Int.J. Numer. Meth. Eng. 58:349-374

[17] Shephard MS (2000) Meshing environment for geometry-based analysis. Int. J.Numer. Meth. Eng. 47:169-190.

[18] Shephard MS, Beall MW, O’Bara RM, Webster BE (2004) Toward simulation-based design. Fin. Elem. in An. and Des. 40:1575-1598.

[19] Shephard MS, Fisher P, Chand KK, Flaherty JE (2003) Simulation InformationStrucutures http://www.tstt-scidac.org/.

[20] Simmetrix Inc. (2004) Simulation Modeling Suite http://www.simmetrix.com/.

[21] Stein E, ed. (2002) Error-Controlled Adaptive Finite Elements in Solid Mechan-ics. J. Wiley & Sons.

12.4. CLOSING REMARKS 179

[22] Thompson JF, Soni BK, Weatherill NP, eds. (1999) CRC Handbook of GridGeneration. CRC Press, Inc.

[23] http://www.tstt-scidac.org (2004).

[24] Wan J, Kocak S, Shephard MS (2005) Automated Adaptive 3-D Forming Sim-ulation Processes. To appear in Eng. w. Comp.

Chapter 13

The Hubble Length as a CriticalLength Scale in Shock WaveCosmology

Joel SmollerDepartment of MathematicsUniversity of Michigan, Ann Arbor, MI 48109

Blake TempleDepartment of MathematicsUniversity of California, Davis, Davis CA 95616

Abstract: We describe a two parameter family of shock wave refinements ofthe standard Friedmann Universe which reduce the Big Bang to an explosion offinite mass and extent. The Hubble length is a critical length scale in the sensethat whenever the shock wave lies beyond one Hubble length from the center ofthe explosion, the explosion can be interpreted as a White Hole explosion occuringinside the event horizon of an ambient Schwarzschild spacetime.

13.1 Introduction

In the standard model of cosmology, the expanding universe of galaxies is describedby a Friedmann-Robertson-Walker (FRW) metric, which in spherical coordinates

181

182 CHAPTER 13. THE HUBBLE LENGTH

has a line element given by [[2]; [24]],

ds2 = −dt2 +R2(t)

dr2

1 − kr2+ r2[dθ2 + sin2θ dφ2]

. (13.1.1)

In this model, which accounts for things on the largest length scale, the universeis approximated by a space of uniform density and pressure at each fixed time,and the expansion rate is determined by the cosmological scale factor R(t) thatevolves according to the Einstein equations. Astronomical observations show thatthe galaxies are uniform on a scale of about one billion lightyears, and the expansionis critical—that is, k = 0 in (13.1.1)—and so, according to (13.1.1), on the largestscale, the universe is infinite flat Euclidian space R3 at each fixed time. Matchingthe Hubble constant to its observed values, and invoking the Einstein equations, theFRW model implies that the entire infinite universe R3 emerged all at once from asingularity, (R=0), some 14 billion years ago, and this event is referred to as the BigBang.

In this paper, an expanded version of which will appear in [[21]], (see also[[12]; [19]]), we discuss a family of exact solutions of the Einstein equations, (thatdepend on two parameters r∗ and σ), that refine the FRW metric by a sphericalshock wave cut-off. In these exact solutions the expanding FRW metric is reducedto a region of finite extent and finite total mass at each fixed time t > 0, and thisFRW region is bounded by an entropy satisfying shock wave that emerges from theorigin, (the center of the explosion), at the instant of the Big Bang t = 0. The shockwave, which marks the leading edge of the FRW expansion, propagates outward intoa larger ambient spacetime from time t = 0 onward. Thus, in this refinement of theFRW metric, the Big Bang that set the galaxies in motion is an explosion of finitemass that looks more like a classical shock wave explosion than does the Big Bangof the Standard Model3. The Hubble length is a critical length scale in this twoparameter family of solutions in the sense that the explosion occurs inside a BlackHole precisely when the shock wave lies beyond one Hubble Length from the centerof the explosion. This corresponds to the transition from r∗ = 0 to r∗ > 0. Sinceastronomical observations show that on the largest scale the universe is uniformout to a distance beyond one Hubble length, the case r∗ > 0 is most relevant tocosmology.

3The fact that the entire infinite space R3 emerges at the instant of the Big Bang, is, looselyspeaking, a consequence of the Copernican Principle, the principle that the earth is not in a specialplace in the universe on the largest scale of things. With a shock wave present, the CopernicanPrinciple is violated in the sense that the earth then has a special position relative to the shockwave. But of course, in these shock wave refinements of the FRW metric, there is a spacetime onthe other side of the shock wave, beyond the galaxies, and so the scale of uniformity of the FRWmetric, the scale on which the density of the galaxies is uniform, is no longer the largest lengthscale.


In order to construct a mathematically simple family of shock wave refinementsof the FRW metric that meet the Einstein equations exactly, we assume k = 0,(critical expansion), and we restrict to the case that the sound speed in the fluid onthe FRW side of the shock wave is constant. That is, we assume an FRW equationof state p = σρ, where σ, the square of the sound speed, is constant, 0 < σ ≤ c2. Atσ = c2/3, this gives the important equation of state p = c2

3ρ which is correct at the

earliest stage of Big Bang physics, [[24]]. Also, as σ ranges from 0 to c2, we obtainqualitatively correct approximations to general equations of state. Taking c = 1,(we use the convention that c = 1, and Newton’s constant G = 1 when convenient),the family of solutions is then determined by two parameters, 0 < σ ≤ 1 and r∗ ≥ 0.The second parameter r∗ is the FRW radial coordinate r of the shock in the limitt → 0, the instant of the Big Bang4. The FRW radial coordinate r is singular withrespect to radial arclength r = rR at the Big Bang R = 0, so setting r∗ > 0 doesnot place the shock wave away from the origin at time t = 0. The distance from theFRW center to the shock wave tends to zero in the limit t→ 0 even when r∗ > 0. Inthe limit r∗ → ∞ we recover from the family of solutions the usual (infinite) FRWmetric with equation of state p = σρ. That is, we recover the standard FRW metricin the limit that the shock wave is infinitely far out. In this sense our family ofexact solutions of the Einstein equations represents a two parameter refinement ofthe standard Friedmann-Robertson-Walker metric.

The exact solutions for the case r∗ = 0 were first constructed in [[12]], andare qualitatively different from the solutions when r∗ > 0, which were constructedlater in [[19]]. The difference is that when r∗ = 0, the shock wave lies closer thanone Hubble length from the center of the FRW spacetime throughout its motion,but when r∗ > 0, the shock wave emerges at the Big Bang at a distance beyond oneHubble length. (The Hubble length depends on time, and tends to zero as t → 0.)

We show in [[19]] that one Hubble length, equal to cH

where H = RR, is a critical

length scale in a k = 0 FRW metric because the total mass inside one Hubble lengthhas a Schwarzschild radius equal exactly to one Hubble length5. That is, one Hubblelength marks precisely the distance at which the Schwarzschild radius rs ≡ 2M ofthe mass M inside a radial shock wave at distance r from the FRW center, crossesfrom inside (rs < r) to outside (rs > r) the shock wave. If the shock wave is ata distance closer than one Hubble length from the FRW center, then 2M < r andwe say that the solution lies outside the Black Hole, but if the shock wave is at a

4Since when k = 0, the FRW metric is invariant under the rescaling r → αr and R → α−1R,we fix the radial coordinate r by fixing the scale factor α with the condition that R(t0) = 1 forsome time t0, say present time.

5Since c/H is a good estimate for the age of the universe, it follows that the Hubble lengthc/H is approximately the distance of light travel starting at the Big Bang up until present time.In this sense, the Hubble length is a rough estimate for the distance to the further most objectsvisible in the universe.


distance greater than one Hubble length, then 2M > r at the shock, and we say thesolution lies inside the Black Hole. Since M increases like r3, it follows that 2M < rfor r sufficiently small, and 2M > r for r sufficiently large, so there must be a criticalradius at which 2M = r, and in Section 2, (taken from [[19]]), we show that whenk = 0, this critical radius is exactly the Hubble length. When the parameter r∗ = 0,the family of solutions for 0 < σ ≤ 1 starts at the Big Bang, and evolves thereafteroutside the Black Hole, satisfying 2M

r< 1 everywhere from t = 0 onward. But when

r∗ > 0, the shock wave is further out than one Hubble length at the instant of theBig Bang, and the solution begins with 2M

r> 1 at the shock wave. From this time

onward, the spacetime expands until eventually the Hubble length catches up to theshock wave at 2M

r= 1, and then passes the shock wave, making 2M

r< 1 thereafter.

Thus when r∗ > 0, the whole spacetime begins inside the Black Hole, (with 2Mr> 1

for sufficiently large r), but eventually evolves to a solution outside the Black Hole.The time when r = 2M actually marks the event horizon of a White Hole, (the timereversal of a Black Hole), in the ambient spacetime beyond the shock wave. Weshow that when r∗ > 0, the time when the Hubble length catches up to the shockwave comes after the time when the shock wave comes into view at the FRW center,and when 2M = r, (assuming t is so large that we can neglect the pressure fromthis time onward), the whole solution emerges from the White Hole as a finite ballof mass expanding into empty space, satisfying 2M

r< 1 everywhere thereafter. In

fact, when r∗ > 0, the zero pressure Oppenheimer-Snyder solution outside the BlackHole gives the large time asymptotics of the solution, (c.f. [[7]; [20]; [14]] and thecomments after Theorems 6-8 below).

The exact solutions in the case r∗ = 0 give a general relativistic version of anexplosion into a static, singular, isothermal sphere of gas, qualitatively similar to thecorresponding classical explosion outside the Black Hole, [[12]]. The main differencephysically between the cases r∗ > 0 and r∗ = 0 is that when r∗ > 0, (the casewhen the shock wave emerges from the Big Bang at a distance beyond one Hubblelength), a large region of uniform expansion is created behind the shock wave at theinstant of the Big Bang. Thus, when r∗ > 0, lightlike information about the shockwave propagates inward from the wave, rather than outward from the center, as isthe case when r∗ = 0 and the shock lies inside one Hubble length6. It follows thatwhen r∗ > 0, an observer positioned in the FRW spacetime inside the shock wave,will see exactly what the standard model of cosmology predicts, up until the timewhen the shock wave comes into view in the far field. In this sense, the case r∗ > 0gives a Black Hole cosmology that refines the standard FRW model of cosmologyto the case of finite mass. One of the surprising differences between the case r∗ = 0

6One can imagine that when r∗ > 0, the shock wave can get out through a great deal of matterearly on when everything is dense and compressed, and still not violate the speed of light bound.Thus when r∗ > 0, the shock wave “thermalizes”, or more accurately “makes uniform”, a largeregion at the center, early on in the explosion.


and the case r∗ > 0 is that, when r∗ > 0, the important equation of state p = 13ρ

comes out of the analysis as special at the Big Bang. When r∗ > 0, the shock waveemerges at the instant of the Big Bang at a finite non-zero speed, (the speed oflight), only for the special value σ = 1/3. In this case, the equation of state on bothsides of the shock wave tends to the correct relation p = 1

3ρ as t→ 0, and the shock

wave decelerates to subliminous speed for all positive times thereafter, (see [[19]]and Theorem 8 below).

In all cases 0 < σ ≤ 1, r∗ ≥ 0, the spacetime metric that lies beyond the shockwave is taken to be a metric of Tolmann-Oppenheimer-Volkoff (TOV) form,

ds2 = −B(r)dt2 + A−1(r)dr2 + r2[dθ2 + sin2θ dφ2]. (13.1.2)

The metric (13.1.2) is in standard Schwarzschild coordinates, (diagonal with radialcoordinate equal to the area of the spheres of symmetry), and the metric componentsdepend only on the radial coordinate r. Barred coordinates are used to distinguishTOV coordinates from unbarred FRW coordinates for shock matching. The massfunction M(r) enters as a metric component through the relation A = 1 − 2M(r)

r.

The TOV metric (13.1.2) has a very different character depending on whetherA > 0 or A < 0; that is, depending on whether the solution lies outside the BlackHole or inside the Black Hole. In the case A > 0, r is a spacelike coordinate, and theTOV metric describes a static fluid sphere in general relativity.7 When A < 0, r isthe timelike coordinate, and (13.1.2) is a dynamical metric that evolves in time. Theexact shock wave solutions are obtained by taking r = R(t)r to match the spheresof symmetry, and then matching the metrics (13.1.1) and (13.1.2) at an interfacer = r(t) across which the metrics are Lipschitz continuous. This can be done ingeneral. In order for the interface to be a physically meaningful shock surface, weuse the result in [[11]] that a single additional conservation constraint is sufficientto rule out delta function sources at the shock, (the Einstein equations G = κT aresecond order in the metric, and so delta function sources will in general be presentat a Lipschitz continuous matching of metrics), and guarantee that the matchedmetric solves the Einstein equations in the weak sense. The Lipschitz matching ofthe metrics, together with the conservation constraint, leads to a system of ordinarydifferential equations (ODE’s) that determine the shock position, together with theTOV density and pressure at the shock. Since the TOV metric depends only onr, the equations thus determine the TOV spacetime beyond the shock wave. Toobtain a physically meaningful outgoing shock wave, we impose the constriant p ≤ ρto ensure that the equation of state on the TOV side of the shock is qualitativelyreasonable, and as the entropy condition we impose the condition that the shockbe compressive. For an outgoing shock wave, this is the condition ρ > ρ, p > p,

7The metric (13.1.2) is, for example, the starting point for the stability limits of Buchdahl andChandresekhar for stars, [[24]; [15]; 6].


that the pressure and density be larger on the side of the shock the receives themass flux—the FRW side when the shock wave is propagating away from the FRWcenter. This condition breaks the time reversal symmetry of the equations, and issufficient to rule out rarefaction shocks in classical gas dynamics, [[10]; [19]]. TheODE’s, together with the equation of state bound and the conservation and entropyconstraints, determine a unique solution of the ODE’s for every 0 < σ ≤ 1 andr∗ ≥ 0, and this provides the two parameter family of solutions discussed here,[[12]; [19]]. The Lipschitz matching of the metrics implies that the total mass Mis continuous across the interface, and so when r∗ > 0, the total mass of the entiresolution, inside and outside the shock wave, is finite at each time t > 0, and boththe FRW and TOV spacetimes emerge at the Big Bang. The total mass M on theFRW side of the shock has the meaning of total mass inside radius r at fixed time,but on the TOV side of the shock, M does not evolve according to equations thatgive it the interpretation as a total mass because the metric is inside the Black Hole.Nevertheless, after the spacetime emerges from the Black Hole, the total mass takeson its usual meaning outside the Black Hole, and time asymptotically the Big Bangends with an expansion of finite total mass in the usual sense. Thus, when r∗ > 0,our shock wave refinement of the FRW metric leads to a Big Bang of finite totalmass.

A final comment is in order regarding our overall philosophy. The family ofexact shock wave solutions described here are rough models in the sense that theequation of state on the FRW side satisfies σ = const., and the equation of stateon the TOV side is determined by the equations, and therefore cannot be imposed.Nevertheless, the bounds on the equations of state imply that the equations of stateare qualitatively reasonable, and we expect that this family of solutions will capturethe gross dynamics of solutions when more general equations of state are imposed.For more general equations of state, other waves, such as rarefaction waves andentropy waves, would need to be present to meet the conservation constraint, andthereby mediate the transition across the shock wave. Such transitional waves wouldbe pretty much impossible to model in an exact solution. But the fact that we canfind global solutions that meet our physical bounds, and that are qualitatively thesame for all values of σ ∈ (0, 1] and all initial shock positions, strongly suggests thatsuch a shock wave would be the dominant wave in a large class of problems.

In Section 2 we introduce the FRW solution when σ = const. In Section 3 wediscuss the construction of the family of solutions in the case r∗ = 0, and in Section4 we discuss the case r∗ > 0. (See [[12]; [19]; [20]] for details.)

13.2. THE FRW METRIC 187

13.2 The FRW Metric

Our shock wave refinement of the FRW metric in both cases r∗ = 0 and r∗ > 0,assumes the equation of state p = σρ, where σ is assumed to be constant, 0 < σ < 1.In this case, an exact solution of the Einstein equations with FRW ansatz (13.1.1)and k = 0, can be given in closed form, c.f. [[19]]. The solution is given in thefollowing theorem.

Theorem 4 Assume k = 0 and the equation of state p = σρ, where σ is taken to beconstant, 0 ≤ σ ≤ 1. Then, (assuming an expanding universe R > 0), the solutionof the Einstein equations satisfying R = 0 at t = 0 and R = 1, H = H0 at t = t0 isgiven by,

ρ =4

3κ(1 + σ)2

1

t2, R =

(t

t0

) 23(1+σ)

,H

H0

=t0t. (13.2.1)

Moreover, the age of the universe t0 and the infinite red shift limit r∞ are givenexactly in terms of the Hubble length by

t0 =2

3(1 + σ)

1

H0

, r∞ =2

1 + 3σ

1

H0

. (13.2.2)

From (13.2.2) we conclude that a shock wave will be observed at the FRW originbefore present time t = t0 only if its position r at the instant of the Big Bang satisfiesr < 2

1+3σ1

H0. Note that r∞ ranges from one half to two Hubble lengths as σ ranges

from 1 to 0, taking the intermediate value of one Hubble length at σ = 1/3.

13.3 FRW-TOV Shock Matching Outside the Black

Hole—The Case r∗ = 0

To construct the family of shock wave solutions for parameter values 0 < σ ≤ 1 andr∗ = 0, we match the exact solution (13.2.1) of the FRW metric (13.1.1) to the TOVmetric (13.1.2) outside the Black Hole, assuming A > 0. In this case, we can derivethe exact solution of the Einstein equations of TOV form that matches the FRWmetric across a shock wave. This exact solution represents the general relativisticversion of a static, singular isothermal sphere—singular because it has an inversesquare density profile, and isothermal because the relationship between the densityand pressure is p = σρ, σ = const.

Assuming the stress tensor for a perfect fluid, and assuming that the densityand pressure depend only on r, the Einstein equations for the TOV metric (13.1.2)


outside the Black Hole, (that is, when A = 1 − 2Mr

> 0), are equivalent to theOppenheimer-Volkoff system

dM

dr= 4πr2ρ, (13.3.1)

− r2 d

drp = GMρ

1 +

p

ρ

1 +

4πr3p

M

1 − 2GM

r

−1

. (13.3.2)

Integrating (13.3.1) we obtain the usual interpretation of M as the total mass insideradius r,

M(r) =

∫ r

0

4πξ2ρ(ξ)dξ. (13.3.3)

The metric component B ≡ B(r) is determined from ρ and M through the equation

B′(r)B

= −2p′(r)p+ ρ

. (13.3.4)

Assuming

p = σρ, ρ(r) =γ

r2, (13.3.5)

for some constants σ and γ, and substituting into (13.3.3), we obtain

M(r) = 4πγr. (13.3.6)

Putting (13.3.5)-(13.3.6) into (13.3.2) and simplifying yields the identity

γ =1

2πG

(σ

1 + 6σ + σ2

). (13.3.7)

From (13.3.3) we obtainA = 1 − 8πGγ < 1. (13.3.8)

Applying (13.3.4) leads to

B = B0

(ρ

ρ0

)− 2σ1+σ

= B0

(r

r0

) 4σ1+σ

. (13.3.9)

By rescaling the time coordinate, we can take B0 = 1 at r0 = 1, in which case(13.3.9) reduces to

B = r4σ

1+σ . (13.3.10)

We conclude that when (13.3.7) holds, (13.3.5)-(13.3.9) provide an exact solution ofthe Einstein field equations of TOV type10, for each 0 ≤ σ ≤ 1. By (13.3.8), these

10In this case, an exact solution of TOV type was first found by Tolman [[22]], and rediscoveredin the case σ = 1/3 by Misner and Zapolsky, c.f. [[24]], page 320.

13.3. THE CASE R∗ = 0 189

solutions are defined outside the Black Hole, since 2Mr< 1. When σ = 1/3, (13.3.7)

yields γ = 356πG , (c.f., [[24]], equation (11.4.13)).

To match the FRW exact solution (13.2.1) with equation of state p = σρto the TOV exact solution (13.3.5)-(13.3.10) with equation of state p = σρ acrossa shock interface, we first set r = Rr to match the spheres of symmetry, andthen match the timelike and spacelike components of the corresponding metrics instandard Schwarzschild coordinates. The matching of the dr2 coefficient A−1 yieldsthe conservation of mass condition that implicitly gives the shock surface r = r(t),

M(r) =4π

3ρ(t)r3. (13.3.11)

Using this together with (13.3.6) and (13.3.7) gives the following two relations thathold at the shock surface:

r =

√3γ

ρ(t), ρ =

3

4π

M

r(t)3=

3γ

r(t)2= 3ρ.

The dt2 coefficient B on the shock surface can be matched in a neighborhood of theshock surface by solving a linear PDE with initial data coming from (13.3.11), c.f.[[13]]. Finally, the conservation constraint [Tij]n

inj = 0, (which guarantees that thematched metric is a veritable weak solution of the Einstein equations), leads to thesingle condition

0 = (1 −A)(ρ+ p)(p+ ρ)2 +

(1 − 1

A

)(ρ+ p)(ρ+ p)2 + (p− p)(ρ− ρ)2, (13.3.12)

which upon using p = σρ and p = σρ is satisfied assuming the condition

σ =1

2

√9σ2 + 54σ + 49 − 3

2σ − 7

2≡ H(σ). (13.3.13)

Note that H(0) = 0, and to leading order σ = 37σ + O(σ2) as σ → 0. Within the

physical region 0 ≤ σ, σ ≤ 1, H ′(σ) > 0, σ < σ, and H(1/3) =√

17 − 4 ≈ .1231,

H(1) =√

1122

− 5 ≈ .2915.

Using the exact formulas for the FRW metric in (13.2.1), and setting R0 = 1at ρ = ρ0, t = t0, we obtain the following exact formulas for the shock position:

r(t) = αt, (13.3.14)

r(t) = r(t)R(t)−1 = β t1+3σ3+3σ , (13.3.15)

where α = 3(1 + σ)√

σ1+6σ+σ2 and β = α

1+3σ3+3σ

(3γρ0

) 13+3σ

.


It follows from (13.3.6) that A > 0, and from (13.3.15) that r∗ = limt→0 r(t) = 0.The entropy condition stating that the shock wave be compressive follows from thefact that σ = H(σ) < σ. Thus we conclude that for each 0 < σ ≤ 1, r∗ = 0, thesolutions given in (13.3.5)-(13.3.15) define a one parameter family of shock wavesolutions that evolve everywhere outside the Black Hole, which implies that thedistance from the shock wave to the FRW center is less than one Hubble length forall t > 0. The result is the following11, (see [[12]] for details),

Theorem 5 There exist values 0 < σ1 < σ2 < 1, (σ1 ≈ .458, σ2 =√

5/3 ≈ .745),such that, for 0 < σ ≤ 1, the Lax characteristic condition holds at the shock if andonly if 0 < σ < σ1; and the shock speed is less than the speed of light if and only if0 < σ < σ2.

The explicit solution in the case r∗ = 0 can be interpreted as a general rela-tivistic version of a shock wave explosion into a static, singular, isothermal sphere,known in the Newtonian case as a simple model for star formation, (see [[12]]). Asthe scenario goes, a star begins as a diffuse cloud of gas. The cloud slowly contractsunder its own gravitational force by radiating energy out through the gas cloudas gravitational potential energy is converted into kinetic energy. This contractioncontinues until the gas cloud reaches the point where the mean free path for transmis-sion of light is small enough that light is scattered, instead of transmitted, throughthe cloud. The scattering of light within the gas cloud has the effect of equalizingthe temperature within the cloud, and at this point the gas begins to drift towardthe most compact configuration of the density that balances the pressure when theequation of state is isothermal. This configuration is a static, singular, isothermalsphere, the general relativistic version of which is the exact TOV solution beyondthe shock wave when r∗ = 0. This solution in the Newtonian case is also inversesquare in the density and pressure, and so the density tends to infinity at the centerof the sphere. Eventually, the high densities at the center ingnite thermonuclearreactions. The result is a shock-wave explosion emanating from the center of thesphere, and this signifies the birth of the star. The exact solutions when r∗ = 0represent a general relativistic version of such a shock-wave explosion.

11Note that even when the shock speed is larger than c, only the wave, and not the sound speedsor any other physical motion, exceeds the speed of light. See [[9]] for the case when the shockspeed is equal to the speed of light

13.4. THE CASE R∗ > 0. 191

13.4 Shock Wave Solutions Inside the Black Hole—

The case r∗ > 0.

When the shock wave is beyond one Hubble length from the FRW center, we obtaina family of shock wave solutions for each 0 < σ ≤ 1 and r∗ > 0 by shock matchingthe FRW metric (13.1.1) to a TOV metric of form (13.1.2) under the assumptionthat

A(r) = 1 − 2M(r)

r≡ 1 −N(r) < 0. (13.4.1)

In this case, r is the timelike variable. Assuming the stress tensor T is taken tobe that of a perfect fluid co-moving with the TOV metric, the Einstein equationsG = κT, inside the Black Hole, take the form, (see [[20]] for details),

p′ =p+ ρ

2

N ′

N − 1, (13.4.2)

N ′ = −N

r+ κpr

, (13.4.3)

B′

B= − 1

N − 1

N

r− κρr

. (13.4.4)

The system (13.4.2)-(13.4.4) defines the simplest class of gravitational metrics thatcontain matter, evolve inside the Black Hole, and such that the mass functionM(r) < ∞ at each fixed time r. System (13.4.2)-(13.4.4) for A < 0 differs sub-stantially from the TOV equations for A > 0 because, for example, in the Einsteinequations, the energy density T 00 is equated with the timelike component Grr of theEinstein tensor when A < 0, but with Gtt when A > 0. In particular, this impliesthat, inside the Black Hole, the mass function M(r) does not have the interpretationas a total mass inside radius r as it does outside the Black Hole.

The equations (13.4.3), (13.4.4) do not have the same character as (13.3.1),(13.3.2) and the relation p = σρ with σ = const. is inconsistent with (13.4.3),(13.4.4) together with the conservation constraint and the FRW assumption p = σρfor shock matching. Thus, instead of looking for an exact solution of (13.4.3),(13.4.4) ahead of time, as in the case r∗ = 0, we assume the FRW solution (13.2.1),and derive the ODE’s that describe the TOV metrics that match this FRW metricLipschitz continuously across a shock surface, and then impose the conservation,entropy and equation of state constraints at the end. Matching a given k = 0 FRWmetric to a TOV metric inside the Black Hole across a shock interface, leads to thesystem of ODE’s, (see [[20]]) for details),

du

dN= −

(1 + u)

2(1 + 3u)N

(3u− 1)(σ − u)N + 6u(1 + u)

(σ − u)N + (1 + u)

, (13.4.5)


dr

dN= − 1

1 + 3u

r

N, (13.4.6)

with conservation constraint

v =−σ (1 + u) + (σ − u)N

(1 + u) + (σ − u)N, (13.4.7)

where

u =p

ρ, v =

ρ

ρ, σ =

p

ρ. (13.4.8)

Here ρ and p denote the (known) FRW density and pressure, and all variablesare evaluated at the shock. Solutions of (13.4.5)-(13.4.7) determine the (unknown)TOV metrics that match the given FRW metric Lipschitz continuously across ashock interface, such that conservation of energy and momemtum hold across theshock, and such that there are no delta function sources at the shock, [[4]; [13]].Note that the dependence of (13.4.5)-(13.4.7) on the FRW metric is only throughthe variable σ, and so the advantage of taking σ = const. is that the whole solutionis determined by the inhomogeneous scalar equation (13.4.5) when σ = const. Wetake as the entropy constraint the condition that

0 < p < p, 0 < ρ < ρ, (13.4.9)

and to insure a physically reasonable solution, we impose the equation of stateconstriant on the TOV side of the shock

0 < p < ρ. (13.4.10)

Condition (13.4.9) implies that outgoing shock waves are compressive. Inequalities(13.4.9) and (13.4.10) are both implied by the single condition, (see [[20]])),

1

N<

(1 − u

1 + u

)(σ − u

σ + u

). (13.4.11)

Since σ is constant, equation (13.4.5) uncouples from (13.4.6), and thus solutionsof system (13.4.5)-(13.4.7) are determined by the scalar non-autonomous equation(13.4.5). Making the change of variable S = 1/N, which transforms the “Big Bang”N → ∞ over to a rest point at S → 0, we obtain,

du

dS=

(1 + u)

2(1 + 3u)S

(3u− 1)(σ − u) + 6u(1 + u)S

(σ − u) + (1 + u)S

. (13.4.12)

Note that the conditions N > 1 and 0 < p < p restrict the domain of (13.4.12) tothe region 0 < u < σ < 1, 0 < S < 1. The next theorem gives the existence ofsolutions for 0 < σ ≤ 1, r∗ > 0, inside the Black Hole, c.f. [[19]]:

13.4. THE CASE R∗ > 0. 193

Theorem 6 For every σ, 0 < σ < 1, there exists a unique solution uσ(S) of(13.4.12), such that (13.4.11) holds on the solution for all S, 0 < S < 1, andon this solution, 0 < uσ(S) < u, limS→0 uσ(S) = u, where

u = Min 1/3, σ , (13.4.13)

and

limS→1

p = 0 = limS→1

ρ. (13.4.14)

For each of these solutions uσ(S), the shock position is determined by the solutionof (13.4.6), which in turn is determined uniquely by an initial condition which canbe taken to be the FRW radial position of the shock wave at the instant of the BigBang,

r∗ = limS→0

r(S) > 0. (13.4.15)

Concerning the the shock speed, we have:

Theorem 7 Let 0 < σ < 1. Then the shock wave is everywhere subluminous, thatis, the shock speed sσ(S) ≡ s(uσ(S)) < 1 for all 0 < S ≤ 1, if and only if σ ≤ 1/3.

Concerning the shock speed near the Big Bang S = 0, the following is true:

Theorem 8 The shock speed at the Big Bang S = 0 is given by:

limS→0

sσ(S) = 0, σ < 1/3, (13.4.16)

limS→0

sσ(S) = ∞, σ > 1/3, (13.4.17)

limS→0

sσ(S) = 1, σ = 1/3. (13.4.18)

Theorem 8 shows that the equation of state p = 13ρ plays a special role in the analysis

when r∗ > 0, and only for this equation of state does the shock wave emerge at theBig Bang at a finite non-zero speed, the speed of light. Moreover, (13.4.13) impliesthat in this case, the correct relation p

ρ= σ is also achieved in the limit S → 0. The

result (13.4.14) implies that, (neglecting the pressure p at this time onward), thesolution continues to a k = 0 Oppenheimer-Snyder solution outside the Black Holefor S > 1.


It follows that the shock wave will first become visible at the FRW center r = 0at the moment t = t0, (R(t0) = 1), when the Hubble length H−1

0 = H−1(t0) satisfies

1

H0

=1 + 3σ

2r∗, (13.4.19)

where r∗ is the FRW position of the shock at the instant of the Big Bang. At thistime, the number of Hubble lengths

√N0 from the FRW center to the shock wave

at time t = t0 can be estimated by

1 ≤ 2

1 + 3σ≤

√N0 ≤

2

1 + 3σe√

3σ( 1+3σ1+σ ).

Thus, in particular, the shock wave will still lie beyond the Hubble length 1/H0 atthe FRW time t0 when it first becomes visible. Furthermore, the time tcrit > t0at which the shock wave will emerge from the White Hole given that t0 is the firstinstant at which the shock becomes visible at the FRW center, can be estimated by

2

1 + 3σe

14σ ≤ tcrit

t0≤ 2

1 + 3σe

2√

3σ1+σ , (13.4.20)

for 0 < σ ≤ 1/3, and by the better estimate

e√

64 ≤ tcrit

t0≤ e

32 , (13.4.21)

in the case σ = 1/3. Inequalities (13.4.20), (13.4.21) imply, for example, that at theOppenheimer-Snyder limit σ = 0,

√N0 = 2,

tcrit

t0= 2,

and in the limit σ = 1/3,

1.8 ≤ tcrit

t0≤ 4.5, 1 <

√N0 ≤ 4.5.

We can conclude that the moment t0 when the shock wave first becomes visible atthe FRW center, the shock wave must lie within 4.5 Hubble lengths of the FRWcenter. Throughout the expansion up until this time, the expanding universe mustlie entirely within a White Hole—the universe will eventually emerge from this WhiteHole, but not until some later time tcrit, where tcrit does not exceed 4.5t0.

13.5. CONCLUSION 195

13.5 Conclusion

We believe that the existence of a wave at the leading edge of the expansion ofthe galaxies is the most likely possibility. The only alternative possibilites are thateither the universe of expanding galaxies goes on out to infinity, or else the universeis not simply connected. Although the first possibility has been believed for most ofthe history of cosmology based on the Friedmann universe, we find this implausibleand arbitrary in light of the shock wave refinements of the FRW metric discussedhere. The second possibility, that the universe is not simply connected, has receivedconsiderable attention recently12. However, since we have not seen, and cannotcreate, any non-simply connected 3-spaces on any other length scale, and sincethere is no observational evidence to support this, we view this as less likely thanthe existence of a wave at the leading edge of the expansion of the galaxies, leftover from the Big Bang. Recent analysis of the microwave background radiationdata shows a cut-off in the angular frequencies consistent with a length scale ofaround one Hubble length, [[1]]. This certainly makes one wonder whether thiscutoff is evidence of a wave at this length scale, especially given the consistency ofthis possibility with the case r∗ > 0 of the family exact solutions discussed here.

References

[1] Private discussions with cosmologist Andy Albrecht.

[2] S.K. Blau and A.H. Guth, Inflationary cosmology. In: Three Hundred Years ofGravitation, ed. by S.W. Hawking and W. Israel, Cambridge University Press,1987, pp. 524-603.

[3] A. Einstein, Der feldgleichungen der gravitation, Preuss. Akad. Wiss., Berlin,Sitzber. (1915b), pp. 844-847.

[4] W. Israel, Singular hypersurfaces and thin shells in General Relativity , IL NuovoCimento, XLIV B, No. 1, (1966), 1-14.

[5] P.D. Lax, Shock–waves and entropy. In: Contributions to Nonlinear FunctionalAnalysis, ed. by E. Zarantonello, Academic Press, 1971, pp. 603-634.

[6] C. Misner and D. Sharp, Relativistic equations for adiabatic, spherically symmet-ric gravitational collapse, Phys. Rev., 26, (1964), p. 571-576.

[7] J.R. Oppenheimer and J.R. Snyder, On continued gravitational contraction, Phys.Rev., 56, (1939), pp. 455-459.

12See for example, The shape of space by Erica Klarreich, Science News, November 8, 2003 Vol.164


[8] J.R. Oppenheimer and G.M. Volkoff, On massive neutron cores, Phys. Rev.,55(1939), pp. 374-381.

[9] M. Scott, General relativistic shock waves propagating at the speed of light, (doc-toral thesis, UC-Davis, June 2002).

[10] J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer Verlag,1983.

[11] J. Smoller and B. Temple, Shock–wave solutions of the Einstein equations: theOppenheimer-Snyder model of gravitational collapse extended to the case of non–zero pressure, Arch. Rat. Mech. Anal., 128, (1994), pp 249-297.

[12] J. Smoller and B. Temple, Astrophysical shock–wave solutions of the Einsteinequations, Phys. Rev. D, 51, No. 6, (1995), pp. 2733-2743.

[13] J. Smoller and B. Temple, General relativistic shock–waves that extend theOppenheimer–Snyder model, Arch. Rat. Mech. Anal. 138, (1997), pp. 239-277.

[14] J. Smoller and B. Temple, Shock-wave solutions in closed form and the Oppenheimer-Snyder limit in General Relativity, with J. Smoller, SIAM J. Appl. Math, 58, No.1, (1988), pp. 15-33.

[15] J. Smoller and B. Temple, On the Oppenheimer-Volkov equations in GeneralRelativity, with J. Smoller, Arch. Rat. Mech. Anal., 142, (1998), 177-191.

[16] J. Smoller and B. Temple, Solutions of the Oppenheimer-Volkoff equations inside9/8’ths of the Schwarzschild radius, with J. Smoller, Commun. Math. Phys. 184,(1997), 597-617.

[17] J. Smoller and B. Temple, Cosmology with a shock wave, Commun. Math. Phys.,210, (2000), 275-308.

[18] J. Smoller and B. Temple, Shock-wave solutions of the Einstein equations: Ageneral theory with examples, (to appear), Proceedings of European Union Re-search Network’s 3rd Annual Summerschool, Lambrecht (Pfalz) Germany, May16-22, 1999.

[19] J. Smoller and B. Temple, Shock wave cosmology inside a White Hole, Proc.Natl. Acad. Sci., 100, no. 20, (2003), 11216-11218.

[20] J. Smoller and B. Temple, Cosmology, black holes, and shock waves beyond theHubble length, (preprint).

[21] J. Smoller and B. Temple, A Shock Wave Refinement of the Friedmann Robert-son Walker Metric, (To Appear), Encyclopedia of Mathematical Physics, Acad-emic Press, Editor Gregory Nabor.

13.5. CONCLUSION 197

[22] R. Tolman, Static Solutions of Einstein’s Field Equations for Spheres of Fluid ,Physical Review, 55, (1939), pp. 364-374.

[23] R.M. Wald, General Relativity, University of Chicago Press, 1984.

[24] S. Weinberg, Gravitation and Cosmology: Principles and Applications of theGeneral Theory of Relativity, John Wiley & Sons, New York, 1972.

proceedings of the conference on analysis, modeling and

Documents