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OURNAL OF COMPUTATIONAL PHYSICS123, 114 (1996)

ARTICLE NO.0001

Extension of the Piecewise Parabolic Method to One-DimensioRelativistic Hydrodynamics

JOSEMa MARTI* ANDEWALDMU LLER

Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild-Str. 1, 85740 Garching, Germany

Received July 25, 1994; revised April 28, 1995

form of the RHD equations. One advantage of this ap

is the possibility of using numerical techniques speciAn extension to 1D relativistic hydrodynamics of the piecewiseparabolic method (PPM) of Colella and Woodward using an exact signed to solve nonlinear hyperbolic systems of cons

elativistic Riemann solver is presented. Results of several tests laws [9]. In fact, the codes described in Refs. [9, 12, nvolving ultrarelativistic flows, strong shocks and interacting dis-

the code HLLE of [14] are based on Godunov-type methcontinuities are shown. A comparison with Godunovs method dem-

an implementation of different approximate Riemannonstrates that the main features of PPM are retained in our relativis-(readers lacking knowledge of Godunov-type methods aic version. 1996 Academic Press, Inc.mann solvers should consult, e.g., the recent book of L

[16]). Judging from the results of several test calculation1. INTRODUCTION in these references, it can be concluded that an accurate

tion of ultrarelativistic flows with strong shock wavesRelativistic hydrodynamics (RHD here after) plays an im- accomplished by writing the system of RHD in conse

portant role in different fields of physics, e.g., in astrophysics, form and using Riemann solvers.cosmology, and nuclear physics. In extragalactic jets emanating In a recent paper [17] we have derived the exact solufrom core dominated radio sources associated with active galac- the Riemann problem for ideal gases in relativistic hydroic nuclei [1] or in current laboratory heavy-ion reactions [2] ics. Similarly to the classical (Newtonian) case, the s

even ultrarelativistic flows are encountered. The necessity of can be obtained by solving an implicit algebraic equatiomodeling such relativistic flows which also involve strong

determines the pressure in the intermediate states that shocks is triggering the development of relativistic hydro-codes.after breakup of the initial discontinuity. Our solution

The first code to solve the RHD equations on an Eulerianpreviously known particular solutions (i.e., [18, 19]

grid was developed by Wilson [3, 4] and collaborators [5, 6].general case of two arbitrary initial states. It can be

The code is based on explicit finite differencing techniques andconstruct an exact Riemann solver and allows one to fo

uses a monotonic transport algorithm to discretize the advectiona relativistic version of Godunovs method.

erms of the RHD equations. The stabilization of the codeOn the other hand, the piecewise parabolic method

across shocks is accomplished by means of a von NeumannPPM hereafter), is a well-known higher-order extension

and Richtmyer [7] artificial viscosity. This code has been widelydunovs method being used extensively in classical hy

used in numerical cosmology, axisymmetric relativistic stellarnamics. Besides the use of an exact Riemann solver,

collapse, accretion onto compact objects, and, more recently,ingredients responsible for the accuracy of PPM are a pa

n collisions of heavy ions. The codes acccuracy decreases as

interpolation of variables inside numerical cells andhe flow becomes strongly relativistic (flow Lorentz factor,monotonicity constraints and discontinuity steepeners

W 2; see [5]).discontinuities sharp and free of numerical oscillations.

Norman and Winkler [8] proposed a fully implicit treatmentsolving Riemann problems for states averaged over the of the equations in order to overcome the numerical problemsof dependence of the interfaces makes PPM second-ordn the ultrarelativistic limit (W 1). Recently, several newrate in time. In this paper, we present an extension of methods for numerical RHD have appeared [915] which withits direct Eulerian version to one-dimensional RHD, inhe exception of [11] (smoothed particle hydrodynamics) andall the key ingredients of PPM have properly been gene10] (flux-corrected transport method for the equations of rela-

The paper is organized as follows. In Section 2 weivistic magnetohydrodynamics) are based on theconservationthe RHD equations in an Eulerian reference frame in co

tion form and describe the transformation of quantiti* Present address: Departamento de Astronom a y Astrofsica, Universidad

de Valencia, 46100 Burjassot, Valencia, Spain. the Eulerian frame to the rest (proper) frame of a fluid e1


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2 MARTIAND MULLER

This transformation is a key ingredient in our algorithm. Section equations and throughout the whole paper the speed of

set equal to unity.3 is devoted to a description of the reconstruction procedure,

he determination of effective second-order-accurate (in time) From Eqs. (4)(6) the following relation between p

the conserved quantities can easily be derived:eft and right states defining the Riemann problems and the

calculation of the numerical fluxes using an exact relativistic

Riemann solver. Several numerical test calculations includingv

S

D p.

he RHD generalization of the interaction of two blast waves

22] are presented in Section 4. In Appendix I, the explicit

formulae used in the interpolation step are given. Finally, the In the non-relativistic limit (v 0, h 1) D, Sanalytical solution corresponding to the problem of the interac- approach their Newtonian counterparts,,v, andEion of two relativistic blast waves introduced in Section 4 is v2/2, and Eqs. (1)(3) reduce to the classical ones,

described in Appendix II.

t

v

x 0

2. THE EQUATIONS OF RELATIVISTIC

HYDRODYNAMICSv

t

(v2 p)

x 0

In an Eulerian reference frame, the equations of RHD of a

perfect fluid in one spatial (Cartesian) coordinate x can beE

t

v(Ep)

x 0. written in conservation form as

The system of Eqs. (1)(3) with definitions (4)(8) iD

t

Dv

x 0 (1)

by means of an equation of state (EOS), which we shall

as given in the formS

t

(Sv p)

x 0 (2)

p p(, ).

t

(SDv)

x 0, (3)

In order to use the exact solution of the relativistic R

problem derived in [17] we restrict ourselves to an id

EOS, i.e.,where D, S, and are the rest-mass density, the momentum

density, and the energy density in a fixed frame, respectively.

p ( 1),These variables are related to quantities in the local rest frameof the fluid through

where is the adiabatic index. A very important quan

rived from the EOS is the local sound velocity c, whicD W (4)case is defined through

S hW2v (5)

hW2 p D, (6) c2 p

h.

where , p, v, W, and h are the proper rest-mass density, theIn any RHD code evolving the conserved quantitiepressure, the fluid velocity, the fluid Lorentz factor, and the

in time, the variablesp,, , vhave to be computspecific enthalpy, respectively. The latter two quantities are

the conserved quantities at least once per time stepgiven by approach, like in Refs. [9, 12], this is achieved using r

(4)(8) and (14) to construct the function

W 1

1 v2(7)

f(p) ( 1)

p

and with

and

given by

h 1 p

, (8)

D

W

andwhere

is the specific internal energy. Note that in the previous


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PPM METHOD FOR 1D RELATIVISTIC HYDRODYNAMICS

ues. Then, we concentrate on the construction of left a

D(1 W

) p(1 W2

)

DW

, (18) interface states averaged over the domain of depend

each interface, which are used as initial states for the R

problem to be solved at each interface. Finally, we discrespective, wheresolution of the relativistic Riemann problem and the co

tion of the numerical fluxes.W

1

1 (S/(D p))2. (19)

3.1. Interpolation Procedure

The zero off(p) in the physically allowed domain p ]pmin, In our relativistic version of PPM the interpolation al[ determines the pressure, the monotonicity of f(p) in that described in the original paper by Colella and Woodwdomain ensuring the uniqueness of the solution. The lower is applied to zone-averaged values ofV (p,, v), whbound of the physically allowed domain, pmin, defined by obtained from zone averaged values of the conserved

ties uj.(20)pmin S D, Interpolating in V instead of, for example, in u has

advantages. First, in this case the solution of Eq. (16)

involves an iteration, has to be computed only once ps obtained from (9) by taking into account that (in our units)

per time step, and second, one can easily avoid the occv 1. Knowing p, Eq. (9) then directly gives v, while theof unphysical values (i.e., larger than the speed of ligremaining state quantities are straightforwardly obtained from

the interpolated flow velocity. Third, interpolating inEqs. (4)(8).

simplifies the implementation of the contact discodetector (see Appendix I).3. A RELATIVISTIC VERSION OF PPM

Like in the Newtonian version of PPM, we determEquations (1)(3) can be written in the form each zone j the quartic polynomial which has zone-av

values aj2, aj1, aj, aj1, aj2 to interpolate the structur

the zone, wherea is one of the quantities p,, or v. Usu

t

F(u)

x 0 (21)

quartic polynomial values ofa at the left and right int

of the zones, aL,j and aR,j, are obtained. These recons

values are then modified such that the parabolic profilewithis uniquely determined by aL,j,aR,j, andaj, is monotoni

the zone (monotonization). Finally, the interpolation pr(22)u (D, S, )T

is slightly modified near discontinuities to produce najumps (see Appendix I).and

3.2. Construction of Effective Left and Right States(23)F (Dv, Sv p, S Dv)T.

To obtain time-averaged fluxes at an interfacej sepIn order to solve system (21) we consider the conservative zones j and j 1, in PPM two spatially averageddifference scheme Vj1/2,S(S L,R, whereL and R denote the left and rig

of the interface, respectively), are constructed, which t

account the characteristic information reaching the iun1j u

nj

t

xFj1/2 Fj1/2, (24)

from both sides during the time step.

In the Lagrangian version of PPM this implies to ca

the average ofV over the domain of dependence of eacwhere unj and un

1j are the zone-averaged values of the statevector u of zone j at times t tn and t tn1 tn t, face. In its direct Eulerian version, however, the cons

of the effective states is more complicated, because the respectively. Fj1/2 are the time-averaged numerical fluxes at the

right and left interfaces of zonej. In a Godunov-type difference of characteristics reaching the interface of a zone from

side can vary from zero to three. In a first step, one coscheme appropriate left and right states are constructed from

he zone-averaged values, which are then used to calculate the the average over that part of the domain of dependence

interface, which lies to the left and right of the interfanumerical fluxes Fj1/2 by solving the corresponding Riemann

problem. initial guess is then corrected by subtracting that am

each characteristic which will not reach the interfaceIn the remainder of this section we discuss in detail how the

numerical fluxes are calculated in our relativistic version of the time step (see Fig. 5 in [20]).

In our relativistic version of PPM the correction of thPPM. First, we describe the interpolation procedure used to

reconstruct the variables inside zones from zone-averaged val- guess is obtained by closely following the procedure in [


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4 MARTIAND MULLER

TABLE I

j1/2,S 1

2Cj1/2,Svj1/2,S vj1/2,S pj1/2,S pj1/2,S

Cj1/2,SSpectral Decomposition of Matrix A

Eigenvalues Left eigenvectors Right eigenvectors

0j1/2,S j1/2,S

0j1/2,S

Rj1/2,S

pj1/2,S p0j1/2,S

C 2j1/2,S

v c

1 vc l 0,12 , 12hW2c r W2c , 1, hW2cotherwise.

0 v r0 (1, 0, 0)l0 1, 0, 1hc2

The quantitiesa#j1/2,

S (SL,R;a p, v) are the aof a over that part of the domain of dependence

v c

1 vc l 0,

1

2,

1

2hW2c r W2

c , 1,hW2c #-characteristic, which lies to the left (right) of the res

interface. They are calculated using the (monotonized) Note. Left and right eigenvectors have been chosen such that l# r# ##;lae determined for a in the interpolation step; i.e.,#, # , 0, .

a#j1/2,L aR,j x

2aR,j aL,j 6 1 2

3x

considering the characteristic speeds and Riemann invariants of

he equations of relativistic instead of Newtonian hydrodynam- aj aL,j aR,j2 ,cs. We rewrite system (21) in terms of V in characteristic

form as

whereV

t A

V

x 0, (25)

x max(0, t#(Vj))

xj1/2 xj1/2where matrix A is defined as

and

a#j1/2,R aL,j1 x

2aR,j1 aL,j1 6 1 2

3x

A

v

1 v2c2v

hW2(1 v2c2)

0 v(1 c2)

1 v2c21

hW4(1 v2c2)

0 hc2

1 v2c2v(1 c2)

1 v2c2

. (26)

aj1

aL,j1 aR,j1

2 ,

where

Note that the local sound speed,c, defined in (15) has explicitly

been introduced in matrix A, whose eigenvalues # (# , x max(0, t#(Vj1))

xj3/2 xj1/2.

0, ) and corresponding left and right eigenvectors l# and r#are given in Table I. In analogy with [20] the effective left and

Note that we have defined x in Eqs. (35) and (37), anright states, Vj1/2,S(S L, R), are then found to bea#j1/2,S, slightly different from Colella and Woodward ([

their Eq. (3.5)). However, the resulting #j1/2,S, the onlypj1/2,S pj1/2,S C 2j1/2,S(j1/2,S j1/2,S) (27)

ties depending ona#

j1/2,S, are identical to those of [20], they have to fulfill Eqs. (30) and (31). With our mvj1/2,S vj1/2,S Cj1/2,S(j1/2,S j1/2,S) (28)definition ofa#j1/2,S, the quantitiesaj1/2,Sappearing in E

j1/2,S j1/2,S Rj1/2,S(j1/2,S 0j1/2,S j1/2,S) (29) to (33) are then simply given by

aj1/2,L aj1/2,L with

aj1/2,R aj1/2,R. #j1/2,L 0 if#(Vj) 0 (30)

#j1/2,R 0 if#(Vj1) 0 (31) Finally, the quantitiesCj1/2,S andRj1/2,S are defined acco

Cj

1/2,S

j

1/2,S

hj

1/2,S

W 2j

1/2,S

cj

1/2,S and


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where S(p) (S(p)) denotes the family of all states whRj1/2,S 2j1/2,Shj1/2,SW 4j1/2,S (41) be connected through a rarefaction (shock) with a giv

Sahead of the wave (for more details see [17]). Becahj1/2,S, Wj1/2,S, and cj1/2,S being given as functions of pj1/2,S, Riemann invariantsj1/2,S, and vj1/2,S(see Eqs. (8), (7), and (15), respectively).

It is worthwhile to note that the Newtonian limits of

j1/2,Sand

j1/2,Scoincide with the corresponding coefficients J 1

2ln 1 v

1 v c

d

n Eq. (3.7) of [20]. However, the limit of0j1/2,Sdiffers from

ts Newtonian counterpart, because contrary to [20] we have are constant through rarefaction waves propagating toused the density instead of the specific volume as a variable(J) or right (J), one can derivefor the characteristic equations.

3.3. Solution of the Riemann Problem and Computation of the expressionthe Numerical Fluxes

In Godunovs approach the numerical fluxesFj1/2 are calcu- S(p) (1 vS)A(p) (1 vS)

(1 vS)A(p) (1 vS)

ated according to

Fj1/2 F(uj1/2), (42) with

whereuj1/2 is an approximation to (1/

t) u(xj1/2, t) dt, i.e., A(p)

1 c(p) 1 c(p)

1 cS 1 cs

2/

he time-averaged value of the solution at xj1/2 , which is ob-

ained solving the Riemann problem atxj1/2 with left and right

states Vj1/2,L and Vj1/2,R, respectively.the () sign ofAcorresponding to S L ( S R)In a recent paper [17] we have shown that an exact solution(47),cSis the sound speed of the state S, and c (p) is gof a Riemann problem for a polytropic gas in RHD can be

obtained, in complete analogy to the Newtonian case, by solving

a non-linear algebraic equation.c(p) ( 1)p

( 1)S(p/pS)1/ p

1/2. Both in relativistic and Newtonian hydrodynamics the dis-continuity between the two constant initial states VL and VRdecays into two elementary nonlinear waves (shocks or rarefac- The family of all states S(p), which can be coions), one moving towards the initial left state and the other

through a shock with a given state S ahead of the wowards the initial right state. Between the waves two new determined by the shock jump conditions. One obtaistatesVLand VRappear, which are separated from each other [17])hrough a contact discontinuity moving along with the fluid.

Across the contact discontinuity, pressure and velocity are con-

inuous, while the density exhibits a jump. As in classical S(p) hSWSvS p pSj (p)1 V(p)2

hydrodynamics [24] the self-similar character of the flow

hrough rarefaction waves and the RankineHugoniot condi-

ions across shocks provide the conditions to link the intermedi- hSWS (p pS) 1SWS

vS

j (p)1 V(pate states VS (S L, R) with their corresponding initial stateVS. In particular, one can express the velocity of the intermedi-

ate states vS as a function of the pressure pS of these states.

where the () sign corresponds to S R (S L)The smoothness of the velocity across the contact discontinuityand j (p) denote the shock velocity and the modulushen givesmass flux across the shock front, respectively. They are g

vL(p) vR(p), (43)

V(p) 2SW

2SvS j (p)

21 (S/j (p))2

2SW2S j (p)

2

wherep

pL pR. This is the above mentioned non-linear

algebraic equation to be solved at each interface in each time

step. The functions vS(p) are defined by and

vS(p)

S(p) ifp pS

S(p) ifp pS,(44) j (p)

(pS p)

h2S h(p)2

pS

p

2hS

S

,


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6 MARTIAND MULLER

S vS cS

1 vScSifp

pS

V(p

) ifp

pS,

whereV(p

) is the velocity of the shock (see Eq. (50))

ing states VSand VS. Note that S S. Hence, for

or S

0, i.e., outside rarefaction waves, the solutionRiemann problem at a given interface is given by

V VS ifS 0VS ifS 0.

IfS 0 Sholds at an interface, the solution VhFIG. 1. Graphical solution in the p-v plane of the Riemann problemsevaluated inside a rarefaction wave (see [17]). In [22] tdefined by the initial states VL (pL 103, L 1, vL 0.5) and ViR

piR,R 1, vR 0) (i 1, ..., 4) with p1

R 102,p2R 10, p

3R 1,p

4R 10

1. tion inside the rarefaction wave is obtained by interpThe adiabatic index of the equation of state isin all cases. Note the asymptotic linearly between the states on both sides of the wavebehaviour of the functions when they approach v 1.

is sufficiently accurate for Newtonian problems. Howour relativistic variant of PPM we compute the exact s

Finally, the numerical fluxes at each interface are owhere the enthalpy h(p) of the state behind the shock is theaccording to Eq. (42). This concludes the descriptionunique) positive root of the quadractic equationmethod, which is of second-order accuracy in space an

It can be degraded to a first-order Godunov method byh21 ( 1)(pS p)

p ( 1)(pS p)

p(52)

setting aL,j aR,j aj ( a p, , v) in the interpolatio

4. NUMERICAL TESTSh hS(pS p)

S h2S 0,

Traditionally, numerical methods for RHD have beewhich is obtained from the Taub adiabat (the relativistic version

against two kinds of problems, namely wall shocks anof the Hugoniot adiabat) for an ideal gas equation of state.tubes, giving rise to flows with large Lorentz factors and

In Fig. 1, the functions vL

(p) and vR

(p) are displayed in ashock waves. Thus, we have simulated these problem

p-v diagram for a particular set of Riemann problems. In theour relativistic version of PPM, too. In addition, we hav

calculations presented here, Eq. (43) is solved for the pressurelated the more challenging problem of the interaction of

p

of the intermediate states by a combination of interval bi-tinuities, i.e., the relativistic version of the collision

section and quadratic interpolation following the procedureblast waves proposed by Woodward and Colella [22].

described in [25]. Once p

has been obtained, the remaining

state quantities can easily be derived [17].4.1. Shock Heating of a Cold FluidNow we are ready to determine the Riemann solution at a

given interface following a procedure analogous to [26]. Let The initial setup consists of an inflowing cold (i.e., sign(v

), where v

vL(p) vR(p), and set gas with coordinate velocity v1 and Lorentz factor W1

fills the computational grid and hits a wall placed at the oedge of the grid. As the gas hits the wall, it is compres

SL if 1R otherwise. (53) heated up, converting its momentum into internal enegiving rise to a shock, which starts to propagate off th

Behind the shock, the gas is at rest (v2 0) and has a Then, we defineinternal energy

2 W1 1. S vs cS

1 vScSifp

pS

V(p

ifp

pS

(54)

The compression ratio between shocked and unshock

2/1, follows fromand


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FIG. 2. Exact (solid line) and numerical profiles of pressure, density, and FIG. 3. Exact (solid line) and numerical profiles of pressure, de

flowvelocity of a relativistic shock tube(Problem 1; seetext). Thecomflow velocity for the shock heating problem with an inflow velocity v1

0.99999, when the shock has propagated 50 zones off the wall (at x 1). The were performed on an equidistant grid of 400 zones.

computations were performed on an equidistant grid of 100 zones.

wall was placed at x 1. For numerical reasons, the

internal energy of the inflowing gas was set to a small fini

1

1

12, (58)

(1 107 W1). Figure 2 shows the profiles of pressure, re

density and flow velocity for a run with a gas inflow vwhere is the adiabatic index of the equation of state. v1 0.99999aftertheshockhaspropagated50zonesoff

This test problem, which sometimes is also formulated as The profiles obtained for other inflow velocities are qualihe collision of two identical gases moving at equal speed similar.Themeanandmaximumerrorsobtainedforthecon opposite directions in order to avoid reflecting boundary sion ratioare displayed in Table II for a set of inflow ve

conditions, has widely been used to check the accuracy of RHD It shows that with our relativistic PPM the mean relaticodes [5, 8, 9, 12, 14, 15, 27]. Concerning explicit schemes, r() never exceeds a value of 10

3 and that, in accordanhe numerical results improved significantly for this test prob- other codes based on a Riemann solver, the accuracy oem, when numerical methods based on Riemann solvers sults does not exhibit any significant dependence on the

were introduced. factor of the inflowing gas.In our test calculations we have used a gas with an adiabatic

4.2. Relativistic Shock Tubesndex and inflow velocities ranging from nearly Newton-

an to ultrarelativistic values. The computational grid consisted Shock tubes represent a special class of Riemann prof 100 equidistant zones covering the interval x [0, 1]. The in which the initial state on both sides of the disconti

at rest. They have become a useful tool in testing nu

codes, because their evolution involves shock waves aTABLE II

efactions. We have simulated two particular shock tubShock Heating of a Cold Gas Moving with a Velocity v1 and a lems characterized by the following initial states:

Lorentz Factor W1

PROBLEM1.v1 W1

maxr () r()

0.5 1.15 7.6188 4.3E 02 1.3E 04

L 10.0, R 1.0

pL 13.3, pR 0

vL 0, vR 0.0.9 2.29 12.1766 5.6E 02 6.4E 04

0.99 7.09 31.3552 6.9E 02 3.9E 04

0.999 22.4 92.4651 7.5E 02 8.6E 04

0.9999 70.7 285.8498 7.6E 02 5.9E 04 PROBLEM2.0.99999 223.6 897.4294 7.7E 02 3.4E 04

L 1.0, R 1.0Note. Maximum and mean relative errors of the compression ratio ,maxr () and r(), are given after the shock has propagated 50 zones off the pL 10

3, pR 102

wall. The zone next to the wall, which always dominates the maximum error,

has not been considered when calculating the mean error. v

L

0, v

R

0.


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8 MARTIAND MULLER

TABLE III

L 1 Norm Errors of Conserved Quantities and Convergence Rates

Corresponding to Problem 1 at t 0.36 for Relativistic PPM and

Godunovs Method (Superscript G)

x E(D)1 E(S)1 E()1 r E(D)G1 rG

2.53E 01 3.97E 01 2.72E 01 4.64E 01

1.43E 01 2.54E 01 1.67E 01 0.82 2.89E 01 0.68

7.51E 02 1.19E 01 7.80E 02 0.93 1.84E 01 0.65 3.98E 02 5.77E 02 3.83E 02 0.92 1.14E 01 0.69

1.94E 02 2.96E 02 1.93E 02 1.04 7.03E 02 0.70

1.03E 02 1.96E 02 1.28E 02 0.91 4.39E 02 0.68

Note.E(a)1 j xj aj Aj, where Aj is the exact solution at x xj .

In both cases the adiabatic index is 5/3 and the initial FIG. 4. Exact (solid line) and numerical profiles of pressure, dediscontinuity is placed at x 0.5. flow velocity of a relativistic shock tube(Problem 2; seetext). Thecom

were performed on an equidistant grid of 400 zones.Problem 1 was chosen because it has been considered by

several authors [5, 6, 14, 15], whose results can directly be

compared with ours. For numerical reasons the pressure of theright state has been set to a small finite value (pR 0.66 the L-1 norm error for different grid resolutions, togeth106). The decay of the initial discontinuity gives rise to an the convergence rate for both the relativistic PPM ntermediate state located between a shock wave and a rarefac- relativistic Godunov method.

It is worthwhile to note that in both shock tube proion propagating to the right (i.e., positive x-direction) and left,

the accuracy obtained on the finest grid with the relarespectively. The fluid in the intermediate state moves to theGodunov variant of our method is already achievedright with a velocity vshell 0.72. Figure 3 shows the resultsrelativistic PPM with a four times coarser grid.for a grid of 400 equidistant zones. One recognizes that the

shock is smeared across 45 zones and that the largest errors

occur for the postshock density. In Table III the errors ofD, 4.3. Collision of Two Relativistic Blast WavesS, and are displayed for different grid resolutions using the

The collision of two strong blast waves [23] was udiscrete L-1 norm. Refining the grid the convergence rate of the Woodward and Colella [22] to compare the performsolution (column 5 of Table III) indicates an order of accuracyseveral numerical methods (including PPM) in classicaof code of roughly 1, which is expected for problems withdynamics. In their test calculations the initial condition cdiscontinuities and which is in good agreement with the New-of three constant states of an ideal gas with 1onian version of PPM [22]. This behaviour indicates that thedensity is unity and the velocity vanishes everywhermain features of the method are retained in our relativisticinterval [0, 1] covered by the grid. Reflecting wall conversion. For comparison, we have also listed in Table III theare used at x 0 and x 1. In the left state (x errors of the first-order accurate relativistic Godunov variantpressure p 103, while in the right state (x 0.9) pof our method. The corresponding convergence rate is only

0.66 (see Table III).

Problem 2, usually referred to as the propagation of a relativ-TABLE IV

stic blast wave, was first considered by Norman and Winkler8]. The flow pattern is similar to that of Problem 1 but some- L 1 Norm Errors of Conserved Quantities and Convergen

Corresponding to Problem 2 at t 0.36 for Relativistic Pwhat more extreme. Relativistic effects reduce the postshockGodunovs Method (Superscript G)state to a thin dense shell. The fluid in the shell moves with

vshell 0.960, while the shock front ahead of it (the blast wave)x E(D)1 E(S)1 E()1 r E(D)G1

propagates with a velocity vS 0.986. Norman and Winkler

8] obtained very good results with an adaptive grid of 400 6.18E 01 1.09E 01 1.10E 01 7.06E 0 4.94E 01 6.61E 00 6.43E 00 0.32 6.38E zones using an implicit hydro-code with artificial viscosity. 3.21E 01 4.25E 00 4.10E 00 0.62 5.45E Figure 4 shows the results obtained with our relativistic PPM 1.78E 01 2.71E 00 2.67E 00 0.85 4.63E

on a fixed grid of 400 equidistant zones. As in Problem 1, the 1.00E 01 1.83E 00 1.89E 00 0.83 3.66E

argest errors arise in the postshock state. To achieve a con-

Note.E(a)1

j

xj aj

Aj , where Aj is the exact solution atverged solution a grid of 2000 zones is required. Table IV gives


9/14


FIG. 5. (1) Exact (solid line) and numerical density profile of the colliding relativistic blast wave problem before the interaction of the wa

computations were performed on an equidistant grid of 4000 zones. (2) Same as (1) but showing the flow velocity.

holds. In the central state (0.1 x 0.9) the initial pressure interaction occurs later than in the Newtonian problem,

att 0.420. For the same reason, the dense shells are p 102.

The early evolution is characterized by the development of having a width at the time of the collision of x

and x 0.019 for the left and right shells, respewo strong blast waves, which propagate into the cold central

gas. These waves are followed by thin shells of dense material. Consequently, the interaction is limited to an extremely

region of size x 0.015. Because of the narrownesAt t 0.028, the shells collide near x 0.69, resulting in a

multiple interaction of strong shocks and rarefactions with each structures one has to use of a very fine grid to reso

structures properly. In the calculations presented here wother and with contact discontinuities. Much of the wave inter-

action takes place in a narrow region of size x 0.2. This used a grid of 4000 equidistant zones.

Figure 5 shows the density and velocity profiles of test is considered as a very severe one, in the sense that it

contains the most challenging ingredients that can appear in prior to the shock collision at timet

0.20. The relatiin the density of the left (right) shell never exceeds 2.0%one-dimensional hydrodynamics, i.e., strong shocks, narrow

structures, and interaction of discontinuities. and has a value of about 1.0% (0.5%) at the moment o

collision. The quality of the numerical solution is draWe have considered the same initial conditions to test our

relativistic code. While in [22] a special version of PPM is degraded when the simulations are performed with the G

variant of our method. At t 0.20 the relative errorused to produce the most accurate solution for the interaction,

we have relied on the exact solution of the relativistic Riemann density of the left (right) shell are about 50% (16%) a

only slightly to a value of about 40% (5%) at the problem [17] to construct the analytical solution of the inter-

acting blast wave problem. Note that this exact solution is also collision (t 0.420).

The collision of the shells produces a region of veused in our code to calculate the numerical fluxes. Based on the

exact solution of the relativistic Riemann problem, an analytical density bounded by two shocks. The density jump ac

shock propagating to the left (right) is 7.26 (12.06); solution to the blast wave collision problem can be obtainedfor epochs prior to interactions with the rarefaction waves. For value lies well above the classical limit for strong sho

for 1.4). In Fig. 6 a snapshot of the system is dihis reason, we have used outflow boundary conditions at x

0 andx 1 (to avoid the reflection and subsequent interaction after the interaction has occurred. Compared to Fig. 5 a

different scale had to be used in the density plot of Fof the rarefaction waves produced by the initial data) and

stopped our calculations after the interaction of the leading include the narrow dense new states produced by the inte

Obviously, our relativistic PPM code satisfactorily resoshocks.

The evolution of the system is described in detail in Appendix structure of the collision region, the maximum relative

the density distribution being less than 2.0%. Using theII. The (dimensionless) propagation speed of the two blast

waves is slower than in the Newtonian case, but very close to nov variant of our method, the new states are muc

smeared out and the positions of the leading shocks arehe speed of light (0.9776 and 0.9274 for the shock waves

propagating to the right and left, respectively). Hence, the shock (see Fig. 6).


10/14

10 MARTIAND MULLER

FIG. 6. (1) Same as Fig. 5(1) but after the blast wave interaction. Note the change in scale on both axes with respect to Fig. 5(1). (2) Same as

but showing the flow velocity.

5. CONCLUSIONS zone-averaged valuesaj2,aj1,aj,aj1, andaj2. The exp

for aj1/2 then readsWe have presented and tested an extension to one-dimen-

sional relativistic hydrodynamics of the well-known PPMaj1/2 aj

xj

xj xj1(aj1 aj)method of Colella and Woodward [20]. The results obtained

for problems involving ultrarelativistic flows, strong shocks,

and interacting discontinuities and the comparison with Godu-

1

2k1 xjk 2xj1 xj

xj xj1novs method demonstrate the superior accuracy and perfor-

mance of our relativistic PPM hydrodynamics code. In the code,

for the first time, an exact relativistic Riemann solver is used xj1 xj

2xj xj1

xj2 xj1

2xj1 xj(aj1 aj )o compute the numerical fluxes across zone interfaces. The

modular structure of the code very easily allows the incorpora-ion of approximate relativistic Riemann solvers, too.

xjxj1 xj

2xj xj1maj1 xj1

xj1 xj2

xj 2xj1We also provide the exact solution of the relativistic counter-

part of the collision of two blast waves [22] (see Appendix II).

This solution can be used as a challenging 1D test case to withcalibrate relativistic hydrodynamics codes.

Finally, we mention that the method can be extended in a maj min(aj , 2aj aj1,aj1 aj )sign (aj )straightforward manner to treat also multidimensional relativis- if (aj1 aj)(aj aj1) 0, ic flows. A particular multidimensional relativistic PPM code, 0, otherwise,

based on an approximate relativistic Riemann solver has already

been developed and successfully applied to the simulations of where

relativistic jets [28].

aj xj

xj1 xj xj1APPENDIX I: RECONSTRUCTION PROCEDURE

In this apprendix we give the details of the interpolation 2xj1 xj

xj1 xj(aj1 aj )

xj 2xj1xj1 xj

(aj procedure used in our relativistic version of PPM. Although

his procedure is identical to that in the original PPM formula-

ion, we will repeat the formulae here for completeness.Using maj, instead ofaj, in Eq. (60) guarantees thata

in the range of values defined by aj and aj1. This calcStep 1. First, interpolated values ofa (where a stands for

any of the quantitiesp,, v) are calculated at all zone interfaces yields a value ofaj1/2 which is third-order accurate for n

distant grids, even where the zone size changes disc . These interpolated interface values aj1/2 are obtained

using the quartic polynomial uniquely determined by the five ously [20].


11/14


In smooth parts of the flow, away from extrema, the limiting Step 3. Near strong shocks the order of the metho

duced locally to avoid spurious postshock oscillations.values ofa at the left and right interface, aL ,j limxxj1/2

a(x)

and aR ,j limxxj1/2

a(x) are then given by the relation acheived by flattening the distribution inside the corresp

zones. In these zones the quantities aL ,j and aR ,jdefine

are substituted byaL ,j1 aR ,j aj1/2. (62)

aflatL ,j ajfj aL ,j (1 fj ),The values of aL ,j and aR ,j have to be modified later so that

he unique parabola defined by aL ,j, aR ,j, and ajis a monotone aflatR,j ajfj aR ,j (1 fj ).

function in each cell, thereby introducing discontinuities at zonenterfaces (see Step 4). The weight function fj is given by the maximum o

fjsj, whereStep2. The interpolation procedure described in Step 1 has

o be slightly modified to produce narrower profiles in the

vicinity of a contact discontinuity. This process is called contactfj min1,wjmax0, pj1 pj1

pj2 pj2 (1)(2)

steepening. As suggested in [20], we consider that a jump is

predominantly a contact discontinuity, if the condition

The index sjof fjsj

is either 1 or 1 depending on w

the differencepj1 pj1is positive or negative. This wK0j1 j1

min(j1,j1)

pj1 pj1min(pj1,pj1)

(63)j sj is always the next zone upstream of zone j, if th

is in a shock.

In Eq. (71), the quotient (pj1 pj1)/(pj2 pjholds, where K0is a constant. In all zones j satisfying (63) themeasure of the steepness of the pressure jump across tdensity distribution is steepened by modifying the values aL ,j

j, (1) and (2) are constants, and wj is equal to 1, if thand aR ,j according tois inside a shock and zero otherwise. The criterion for

being inside a shock isaL ,j aL ,j(1 j) adL,jj,

(64)aR ,j aR ,j(1 j) ad

R,jj

pj1 pj1min (pj1,pj1)

(2), vj1 vj1. with

Step 4. Now we are ready to describe the monotoadL,j aj1

maj1

2 , adR,j aj1

maj1

2 , (65) step (see Eq. (1.10) of [20]). In smooth parts of the flow

from extrema, aL ,j1 aR ,j aj1/2, so that the interpis continuous atxj1/2. However, near discontinuities theandofaL ,j and aR ,j obtained in Step 1 to 3 are modified su

in each zonej the interpolation parabola is a monotone fuj max[0, min(

(1)(j (2)), 1)]. (66)which takes on only values between aL,jand aR,j . Acco

[20] the following modifications are necessary:In this last expression, (1) and (2) are free parameters, while

j is defined asaL ,j aj,aR ,j aj if (aR ,j aj )(aj aL ,j ) 0

aL ,j 3aj 2aR ,j if (aR ,j aL ,j )j 2aj1 2aj1

xj1 xj1 (xj xj1)3 (xj1 xj )3

aj1 aj1 ,

aj

aL ,j aR ,j

2

(aR ,j aL ,j )2

6

if 2aj1 2aj1 0,aj1 aj1 (67) (1) min(aj1,aj1) 0

(68)

aR ,j 3aj 2aL ,j if (aR ,j aL ,j )

0, otherwise, aj aL ,j aR ,j

2 (aR ,j aL ,j )2

6 .

where

Note that in RHD the monotonic character of the recons

algorithmm ensures that the interpolated interface ve2aj 1

xj1 xj xj1(69)

are always smaller than the speed of light, if this holds

zone averaged values, too.

aj1 aj

xj

1

xj

aj aj1

xj

xj

1

.

The parameters K0, (1)

, (2)

, and (1)

, introduced in


12/14

12 MARTIAND MULLER

TABLE V

Values of the Reconstruction Parameters Used in the Calculations

K0 (1)

(2) (1) (1) (2) (2)

1.0 5.0 0.05 0.1 0.52 10.0 1.0

and (1), (2), and (2) introduced in Step 3 are, in principle,

problem dependent; i.e., their values have to be fixed for every

calculation. Although the quality of the results depends on the

values of these parameters, a set of values can be found, how-

ever, which is well suited for a wide range of problems. The

parameter values used in all our test calculations are given in

Table V.FIG. 7. Flow pattern of the colliding relativistic blast wave probl

the interaction of the waves. The values of the hydrodynamical qua

the regions R1 to R9, which characterize the flow, are given in TableAPPENDIX II: EXACT SOLUTION FOR THE COLLISIONhave been computed using the formulae of Appendix II.OF TWO RELATIVISTIC BLAST WAVES

In this appendix we describe the exact solution to the problemand x2 are the positions of the head and tail of the rareof the collision of relativistic blast waves for the initial datawave in region R2. They move according togiven in Section 4.3. The problem of the collision of two blast

waves was introduced in classical hydrodynamics by Wood-x1(t) 0.1 0.6324t ward and Colella [22] to test the accuracy and performance

of various finite difference methods in case of a challengingx2(t) 0.1 0.8222t.

numerical problem. The multiple interactions of discontinuities

and rarefactions together with the resulting narrow flow struc-Inside R2, i.e.,x1(t) x x2(t), the distribution of the h

ures make this problem an extremely difficult test case for anynamical quantities can be obtained in two steps. Fi

Eulerian method. In its relativistic version, we have changedsolves the algebraic system of equations for the sound a

he boundary conditions, however (from reflecting to outflow),

velocity given byo avoid the reflection and subsequent interaction of rarefactionwaves. Thus, the flow structure is less complicated than in the

c2(x,t) v2(x,t) (x 0.1)/t

1 v2(x,t)(x 0.1)/tNewtonian case. However, due to relativistic effects narrower

structures and larger jumps occur in the flow.

Early on, the flow consists of two blast waves created byv2(x,t)

(1 v1)A(x,t) (1 v1)

(1 v1)A(x,t) (1 v1), he decay of the initial discontinuities at x 0.1 andx 0.9.

The two waves propagate towards each other and collide at 0.4200. We have used the procedure described in [17] to

solve the Riemann problems at x 0.1 and x 0.9 which

determine the solution before the collision. Up to this stage, TABLE VIhe solution consists of nine different regions (R1 to R9; see

Constant States of the Relativistic Blast Wave Collision PrFig. 7) linked at points x1to x8, where x iis the position of thenterface between regions Ri and R(i 1). Regions R1, R5, Region p v

and R9 correspond to the initial states, whereas regions R3 andR1 1.000E 03 1.000E 00 0.00E 00 6.32R4 as well as regions R6 and R7 are the intermediate states ofR3 1.471E 01 4.910E 02 9.57E 01 6.32he Riemann problem defined by the initial discontinuity atx R4 1.471E 01 1.439E 01 9.57E 01 5.59

0.1 andx 0.9, respectively. The values of the hydrodynamicalR5 1.000E 02 1.000E 00 0.00E 00 1.16

quantities in these constant states are given in Table VI. Finally, R6 4.639E 00 9.720E 00 8.82E 01 5.00R7 4.639E 00 1.120E 01 8.82E 01 6.30regions R2 and R8 are rarefaction waves.R9 1.000E 02 1.000E 00 0.00E 00 6.31Besides the values of the hydrodynamical quantities in the

constant states, the complete analytical solution must also giveC1 3.698E 02 1.044E 02 4.56E 01 6.08

he position of the points xi(i 1, ... , 8) as a function of time, C2 3.698E 02 1.173E 02 4.56E 01 6.05

and the flow quantities inside the rarefaction waves. Points x1


13/14


FIG. 8. (1) Sequence of snapshots showing the evolution of the density profile of the colliding relativistic blast wave problem up to the mom

he interaction of the waves occurs. The profiles have been computed using the formulae of Appendix II. (2) Same as Fig. 8(1) but showing the flow

where x7andx8. The positions of all these points as a function

are given by

A(x,t) 1 c2(x,t) 1 c1 1 c2(x,t) 1 c1

2/1. (80)x5(t) 0.9 0.9274t

x6(t) 0.9 0.8820t Then, the isentropic character of the flow inside the rarefaction

can be used to get the density and pressure profiles through x7(t) 0.9 0.5668t

x8(t) 0.9 0.6315t.

2(x,t) c22(x,t)(c21 1)c21(c

22(x,t) 1)

1/( 1) 1 (81)Inside the rarefaction on the right, i.e., for x7(t) x the solution of the following algebraic system of equaand

p2(x,t) 2(x,t)1p1. (82) c8(x,t) v8(x,t) (x 0.9)/t

1 v8(x,t)(x 0.9)/t

Pointx3is the locus of a contact discontinuity. Hence, it moves v8(x,t) (1 v9)A(x,t) (1 v9)

(1 v9)A(x,t) (1 v9)

with the velocity of the fluid in regions R3 and R4, i.e.,

(83)x3(t) 0.1 0.9570t. with

Point x4, finally, gives the position of the shock heading theeft blast wave propagating towards the right; i.e., its motion

A(x,t) 1 c8(x,t) 1 c8(x,t) 1 c9 1 c92/1

s governed by

(84)x4(t) 0.1 0.9776t,

gives the sound and fluid velocity. Then, Eqs. (81) a

allow one to calculate 8(x, t) and p8(x, t), if the indicewhere the shock speed has been calculated from the Rankine

Hugoniot conditions for the jump between states R4 and R5. 2 are substituted by 9 and 8, respectively.

Figure 8 shows four snapshots of the evolution of thThe blast wave on the right has a similar structure, with a

heading shock at x5 propagating into the central initial state, a including the moment of the collision of the blast w

t 0.4200. At this moment, the collision (of regions contact discontinuity at x6 separating regions R6 and R7, and

a rarefaction wave propagating to the right bounded by points R6) occurs at x

0.5106, giving rise to two new st


14/14

14 MARTIAND MULLER

REFERENCES

1. S. Phinney, in Superluminal Radio Sources, edited by J. A. Ze

T. J. Pearson (Cambridge Univ. Press, Cambridge, UK, 1987).

2. D. Strottman, in Relativistic Fluid Dynamics, edited by A. Ani

Choquet-Bruhat (Springer-Verlag, New York/Berlin, 1989).

3. J. R. Wilson,Astrophys. J. 173, 431 (1972).

4. J. R. Wilson, inSources of Gravitational Radiation,edited by L.

(Cambridge Univ. Press, Cambridge, UK, 1979).

5. J. Centrella and J. R. Wilson,Astrophys. J. Suppl. Ser. 54, 229

6. J. F. Hawley, L. L. Smarr, and J. R. Wilson,Astrophys. J. Supp

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7. J. von Neumann and R. D. Richtmyer, J. Appl. Phys. 21, 232 (

8. M. L. Norman and K.-H. A. Winkler, inAstrophysical Radiation

namics, edited by M. L. Norman and K.-H. A. Winkler (Rei

drecht, 1986).

9. J. Ma. Mart, J. Ma. Ibanez, J. A. Miralles, Phys. Rev. D 43, 379FIG. 9. Details of the density profile of the colliding relativistic blast wave 10. M. R. Dubal, Comput. Phys. Commun. 64, 221 (1991).

problem showing the new states (regions C1 and C2; see Table VI) produced11. P. J. Mann, Comput. Phys. Commun. 67 , 245 (1991).

by the interaction of the two waves. Note the change in scale on both axes12. A. Marquina, J. Ma. Mart, J. Ma. Ibanez, J. A. Miralles, and Rwith respect to Fig. 8(1). The profile has been computed using the formulae

Astron. Astrophys.258, 566 (1992).of Appendix II.

13. M. H. P. M. van Putten, J. Comput. Phys. 105, 339 (1993).

14. V. Schneider, U. Katscher, D. H. Rischke, B. Waldhauser, J. A

and C.-D. Munz, J. Comput. Phys. 105, 92 (1993).

and C2 (see Table VI and Fig. 9). The solution of the Riemann 15. F. Eulderink, Ph.D. thesis, Rijksuniversiteit te Leiden, 1993 (unpproblem defined by the states R4 (left) and R6 (right) allows 16. R. J. LeVeque, Numerical Methods for Conservation Laws (Bus to determine these new states, as well as the positions of Basel, 1992).he two shock waves and the contact discontinuity (x41,x42, and 17. J. Ma. Mart and E. Muller, J. Fluid Mech. 258, 317 (1994).

x43, respectively) which separates the newly created regions 18. K. W. Thompson, J. Fluid Mech. 171, 365 (1986).from each other and from the former states R4 and R6, 19. B. J. Plohr and D. H. Sharp, in The VIIIth International Confe

Mathematical Physics, edited by M. Mebkhout and R. Seneo

Scientific, Singapore, 1987).x41(t) 0.5105 0.088(t 0.4200) (92)20. P. Colella and P. R. Woodward,J. Comput. Phys. 54, 174 (198

x42(t) 0.5105 0.456(t 0.4200) (93) 21. S. K. Godunov,Mat. Sb. 47, 271 (1959).

22. P. R. Woodward and P. Colella,J. Comput. Phys. 54, 115 (198x43(t) 0.5105 0.703(t 0.4200). (94)23. P. R. Woodward, inParallel Computation, edited by G. Rodrig

demic Press, New York/London, 1982).The solution described above applies until t 0.4300, when

24. R. Courant and K. O. Friedrichs, Supersonic Flow and Shoche next interaction (between states R4 and C1) takes place.(Springer-Verlag, New York/Berlin, 1976).

25. G. E. Forsythe, M. A. Malcolm, and C. B. Moler,Computer MeACKNOWLEDGMENTS Mathematical Computations(PrenticeHall, Englewood Cliffs, N

26. P. Colella and H. M. Glaz, J. Comput. Phys. 59, 264 (1985).This work has been supported by the Human Capital and Mobility Program

27. T. L. McAbee, J. R. Wilson, J. A. Zingman, and C. T. Alonso,Mof the Commission of the European Communities (Contract ERBCHBICTLett. A 4, 983 (1989).930496). Additional financial support is acknowledged from the Spanish

DGYCIT (Ref. PB91-0648). One of the authors (J.Ma.M.) would like to thank 28. J. Ma. Mart, E. Muller, and J. Ma. Ibanez, Astron. Astrophys

(1994).he Max-Planck-Institut fur Astrophysik for the kind hospitality during his stay.

ppm muller marti

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