a numerical study of riemann problems for the two-dimensional unsteady transonic small disturbance...

A NUMERICAL STUDY OF RIEMANN PROBLEMS FOR THETWO-DIMENSIONAL UNSTEADY TRANSONIC SMALL

DISTURBANCE EQUATION∗

SUNCICA CANIC† AND DRAGAN MIRKOVIC†

SIAM J. APPL. MATH. c© 1998 Society for Industrial and Applied MathematicsVol. 58, No. 5, pp. 1365–1393, October 1998 001

Abstract. We study a two-parameter family of Riemann problems for the unsteady transonicsmall disturbance (UTSD) equation, also called the two-dimensional Burgers equation. The twoparameters, a and b, which define oblique shock initial data, correspond to the slopes of the initialshock waves in the upper half-plane. For each a and b, the three constant states in the upper half-plane satisfy the Rankine–Hugoniot conditions across the shocks. This leads to a two-parameterfamily of oblique shock interaction problems.

In this paper we present a numerical study of global solution behavior for the values of a and bin a previously obtained bifurcation diagram. Our study supplements the related theoretical resultsand conjectures recently obtained by S. Canic and B. L. Keyfitz. We employ a high resolutionnumerical method which reveals fine solution structures. Our findings confirm theoretical resultsand conjectures about the solution patterns and deepen the understanding of the structure of severalintricate wave interactions arising in this model.

Key words. two-dimensional Riemann problems, unsteady transonic small disturbance equa-tion, essentially nonoscillatory numerical scheme

AMS subject classifications. Primary, 35L65, 35L67; Secondary, 35-04

PII. S003613999730884X

1. Introduction. The results of this paper shed light on the structure of solu-tions of the unsteady transonic small disturbance equation (or the two-dimensionalBurgers equation):

ut + uux + vy = 0,−vx + uy = 0,

(1.1)

which was used by several authors to model weak shock reflection phenomena; see, forexample, [2], [8], [11], [16]. We study a two-parameter family of oblique shock initialvalue problems for the UTSD equation with initial data given in the upper half-plane,y ≥ 0, consisting of two shock waves, S1 and S2, defined by

S1 ≡ x = a y and S2 ≡ x = −b y,where a, b > 0 are the two parameters in the problem. The initial discontinuitiesseparate regions of constant states

U0 = (0, 0),U1 = (1,−a),U2 = (1 + a/b, 0).

(1.2)

∗Received by the editors March 21, 1997; accepted for publication (in revised form) August 27,1997; published electronically June 25, 1998. The U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to doso, for U.S. Government purposes. Copyright is owned by SIAM to the extent not limited by theserights.

http://www.siam.org/journals/siap/58-5/30884.html†Department of Mathematics, Iowa State University, Ames, IA 50011 ([email protected],

[email protected]). The research of the first author was supported by the National ScienceFoundation grant DMS-9625831 and in part by Department of Energy grant DE-FG02-94ER25220.The research of the second author was supported by the Office of Naval Research grant ONR N00014-96-1-0279.

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1366 SUNCICA CANIC AND DRAGAN MIRKOVIC

- x

6

y

S1S2

U1U0U2

v = 0

¡¡

¡¡

¡¡

¡¡

¡¡

AA

AA

AA

AA

AA

Fig. 1.1. Riemann data.

The states U0, U1, and U2 satisfy Rankine–Hugoniot conditions across the shock wavesS1 and S2. The Rankine–Hugoniot conditions are given by

xt [u]− 12

[u2]+ xy [v] = 0,

xy [u] + [v] = 0,(1.3)

where x = x(y, t) denotes the shock position. In addition, “no flow through the wall”boundary data are imposed along the x-axis:

v(x, 0) = 0;(1.4)

see Figure 1.1. Thus, initial discontinuities will propagate as shocks. Asymptotically,for t > 0 and y →∞, the positions of S1 and S2 will approach the lines

x = ay + s1t, where s1 = 12 + a2,

x = −by + s2t, where s2 = 1 + a2b + b2.

(1.5)

Following the terminology of [5], we call this the oblique shock interaction problem.Notice that because the two initial shocks are given independently, this initial problemis not that of oblique shock reflection.

The main theme of this paper is a numerical study of fine solution structureswhich occur in the zone of interaction of shock waves S1 and S2 for values of a and bin a bifurcation diagram introduced in [5] and [6]. The bifurcation diagram was basedon theoretical results about hyperbolic wave interactions which occur in this modeland on conjectures about interactions of shocks with the subsonic region. We employa high resolution numerical method to resolve the intricate wave interactions.

The numerical method we implemented is described in section 2. We used amodification of methods studied by Osher [12] and by Osher and Sethian [13] toapproximate (1.1) in potential form.

In section 3 we summarize the basic concepts used to obtain the bifurcationdiagram in [5] and [6]. In sections 4 through 6 we study solution patterns thatcomprise the bifurcation diagram.

Our findings have three important consequences. The first consequence is theresolution of a solution pattern arising for the parameter values in the region wecall VN. For the parameter values in question, shock polar analysis for the UTSDequation does not provide a solution, and a theoretical study of the flow in the zoneof interaction of waves indicates the following structure. Shock S1 bends smoothly,

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TWO-DIMENSIONAL BURGERS EQUATION 1367

S 1

S 2

square root singularity

sonic flow

SUPERSONIC FLOW

SUBSONIC FLOW

sonic parabola

U0

U1

U2

SUPERSONIC FLOW

Fig. 1.2. A sketch of the VN wave interaction pattern.

forming a “stem,” while the shock S2 “meets” S1 at a point above the wall wherethe strength of S2 becomes zero and the flow continuously changes from supersonic(ahead of S2) to subsonic (behind S2); see Figure 1.2. Theoretical results of [3] and [4]indicate that the horizontal component of the velocity behind the shock S2, where itsstrength becomes zero, exhibits a square-root–type singularity. Indeed, our numericalstudy of the UTSD equation enabled us to zoom in on the solution and obtain a set ofdata for which the nonlinear least squares fit indicates the square-root–type behaviorpredicted in [4]. Details are given in section 5.

The second consequence of the study presented in this paper is a confirmation ofthe theoretical results and conjectures about the solution structures obtained in [5]for the parameter values corresponding to the regions RR and MR of the bifurcationdiagram shown in Figure 3.1. These are presented in section 4.

Finally, the third consequence is related to a resolution of the intricate waveinteractions arising in the KMR and TMR regions of the bifurcation diagram. As weshall see in section 6, for these parameter regimes numerical simulations indicate atriple shock configuration. It can be proved (see [2]) that the triple shock configurationis not possible in this model. This is the case for the full set of Euler equations andweak enough shocks as well. To resolve the triple shock paradox arising in the UTSDequation, the presence of a logarithmic singularity in the subsonic solution at thetriple point was suspected. However, the work of Gamba, Rosales and Tabak in [7]shows that such a solution cannot be matched with the rest of the flow, and thereforeis inadmissible. Our findings show that the state behind the shock S2 at the triplepoint is, in fact, sonic; see section 6. Thus, a quasi-one–dimensional Riemann problemtakes place at the triple point. We show that this quasi-one–dimensional problem hasa solution which contains a small hyperbolic region behind the point of interaction ofwaves. More precisely, shock polar analysis of this quasi-one–dimensional Riemannproblem implies the existence of a unique local solution consisting of a shock anda thin rarefaction wave. These hyperbolic states soon become sonic and the flowchanges to subsonic (see Figures 6.2 and 6.3). Our numerical simulations indicatethat this solution indeed occurs. Details are given in section 6.

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2. The numerical method. We begin by describing the numerical method weused to study the UTSD equation. We note that our choice of a particular numericalmethod depended mostly on numerical evidence and the present experience in nu-merical approximation of conservation laws that change type. We followed the ideasgiven in [2], [9], [12], [13], [14], [15] and general guidance given in [10].

We solve the oblique shock interaction problem (1.1), (1.2), described in sec-tion 1, written in potential form. Namely, since the flow is irrotational, there exists apotential, φ, such that (φx, φy) = (u, v). The potential satisfies

φxt + φxφxx + φyy = 0.(2.1)

Since φ is continuous across a shock, by fixing the value of φ at one point, e.g.,φ(∞, 0) = 0, one recovers the initial data for (2.1) from (1.2):

φ0(x, y) = 0, x ≥ a y,φ1(x, y) = x− ay, −b y ≤ x < a y,φ2(x, y) =

(1 + a

b

)x, x < −b y.

(2.2)

Finally, the boundary data at the wall read

φy = 0.(2.3)

To solve this problem we first integrate (2.1) with respect to x from ∞ to x (sincethe state ahead of the incident shock is unperturbed) to obtain

φt +φ2x

2+

∫ x

∞φyydx = 0.

Our discretization procedure is effectively divided into two steps. First, we use asemidiscrete approximation to discretize

φt = −φ2x

2−∫ x

∞φyydx(2.4)

with respect to the space variables. Next, a full discretization is obtained using aRunge–Kutta method for the system of ordinary differential equations obtained inthe first step. The spatial domain of (2.4) is the upper half-plane. Our numericalsimulations are restricted to a rectangle in the upper half-plane (symmetric aboutx = 0), denoted by [x0, xNx ]× [y0, yNy ], with transmission boundary conditions alongthe “infinite” boundary. Our computational domain is padded with two extra meshblocks, and along the “infinite” boundary the potential φ is extrapolated so thatthe normal to the surface of φ remains constant. This means that the flow at theinfinite boundaries will be “transmitted” through the boundary. This can be achievedeverywhere where the potential is smooth. At the points where the gradient of φis discontinuous (shock points) there is some reflection present; see, for example,Figure 5.2. However, the reflected wave is very weak and it does not contaminate theflow in the zone of interaction of waves.

The approach proposed here is similar to that of Brio and Hunter [2]. To obtainbetter resolution and fewer oscillations for the parameter regimes that are of interestto our study, we introduce different spatial discretization of the nonlinear term u2/2and employ transmission boundary conditions which minimize reflection from the topboundary.

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Before we describe the discretization, we introduce some standard notation. Formesh sizes ∆x, ∆y and ∆t, φni,j will denote a numerical approximation to the solutionφ(xi, yj , t

n), where

xi = x0 + i∆x, i = 0, 1, . . . , Nx,yj = y0 + j∆y, j = 0, 1, . . . , Ny,tn = n∆t.

Also, let φi,j(t) denote an approximation of a true solution φ at (xi, yj) for t ≥ 0. Wedenote the forward and backward difference operators in the usual way:

∆x± = ±(φi±1,j − φi,j), ∆y

± = ±(φi,j±1 − φi,j),

Dx± = ∆x

±/∆x, Dy± = ∆y

±/∆x.

We first discretize (2.4) with respect to the spatial variables to obtain a systemof ordinary differential equations of the form

dφi,jdt

= [F (φ)]i,j ,(2.5)

where [F (φ)]i,j is a particular spatial discretization, i.e., a discrete equivalent, of thenonlinear operator

F (φ) = −1

2φ2x −

∫ x

∞φyydx.

Let [N(φ)]i,j and [L(φ)]i,j denote an approximation of the nonlinear (convection) termand the linear (diffraction) term, respectively, so that

[F (φ)]i,j = −1

2[N(φ)]i,j − [L(φ)]i,j .

We describe the approximations in the following subsections.

2.1. Discretization of the nonlinear term. To discretize the nonlinear term,we used the results from [9] and [13]. In [13] Osher and Sethian constructed ENO(essentially nonoscillatory) type schemes and applied them to a class of Hamilton–Jacobi equations

φt +H(φx) = 0

arising in front propagation problems. High order ENO schemes are usually con-structed in two steps. One begins with a first-order monotone scheme and then buildsup a higher order approximation using an ENO interpolation procedure. We followthe same steps for the spatial discretization of the nonlinear convection term

N(φ) = H(φx) ≡ φ2x.

We begin with the following upwind, first order, monotone numerical flux which ap-proximates H:

h(unj , unj+1) :=

[min(unj+1, 0)

]2+[max(unj , 0)

]2.

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This flux satisfies the numerical entropy condition

h(w, z) ≤ minz≤a≤w

H(a) if w ≥ z,

h(w, z) ≥ maxw≤a≤z

H(a) if z ≥ w.(2.6)

Next, a second-order ENO approximation of u = φx can be written as

[φx]i,j = Dx−φ

ni,j +

∆x

2m[Dx−D

x−φ

ni,j , D

x−D

x+φ

ni,j

],(2.7)

where m is the min-mod function

m(x, y) =1

2(sgn(x) + sgn(y)) min (|x|, |y|) .

By a result in [9], if φ(x) is piecewise C∞0 and if φx has at most a finite number ofisolated jump discontinuities and is smooth at some point x = x∗, then for ∆x smallenough, {

φx − [φx]i,j

}x=x∗

= O((∆x)2

),

where [φx]i,j is defined by (2.7). Moreover, the following global estimate holds:

total variation [φx]i,j ≤ total variation (φx) +O((∆x)2

).

Therefore, our second-order approximation of the nonlinear flux is given by

[N(φ)]i,j =

[max

(Dx−φ

ni,j +

∆x

2m(Dx−D

x−φ

ni,j , D

x−D

x+φ

ni,j

), 0

)]2

+

[min

(Dx

+φni,j −

∆x

2m(Dx−D

x+φ

ni,j , D

x+D

x+φ

ni,j

), 0

)]2

.

2.2. Discretization of the linear term. To approximate the nonlocal lineardiffraction term, L(φ), we proceed as follows:

L(φ)(x, y, t) =

∫ x

∞φyy(ξ, y, t)dξ

=

∫ x

∞

[Dy

+Dy−φi,j +O(∆y2)

]dξ

=∆x

2Dy

+Dy−φi,j + ∆x

Nx−1∑k=i+1

Dy+D

y−φk,j +O(∆x2) +O(∆y2).

The approximation

[Lφ]i,j =∆x

2Dy

+Dy−φi,j + ∆x

Nx−1∑k=i+1

Dy+D

y−φk,j(2.8)

is therefore formally of second order because of the use of a central difference ap-proximation for φyy and the trapezoidal rule for the integration. We calculate thisapproximation recursively by using

[Lφ]i,j = [Lφ]i+1,j +∆x

2

(Dy

+Dy−φi,j +Dy

+Dy−φi+1,j

).

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2.3. Time discretization. To obtain a fully discrete scheme, second order ac-curate in time, we employ a second-order TVD Runge–Kutta type discretization in-troduced by Shu and Osher [15]:

φn+ 1

2i,j = φni,j + ∆t [F (φn)]i,j ,

φn+1i,j =

1

2

(φni,j + φ

n+ 12

i,j

)+

∆t

2

[F(φn+ 1

2i,j

)]i,j,

which has the same CFL condition as the first-order forward Euler scheme; see [15].We performed several tests to study convergence of the method. We ran all our

examples on several levels of mesh refinement and compared our results with the exactentropy solutions in the parameter regimes for which the exact entropy solutions canbe obtained. This is, for example, the case with supersonic (hyperbolic) solutionsin RR and MR parameter regimes, described in detail in section 4. Our numericalsimulations indicate that this method converges to the entropy solution and thatshocks, which travel at the correct speeds, exhibit little or no spurious oscillations;see Figures 4.3 and 5.3.

Figures 4.3 and 5.3 also indicate the typical numerical shock thickness. Figure 5.3provides more detail. It shows a magnified view of the mesh surface plot of the firstvelocity component u for a solution occurring in the MR parameter regime. Thenumerical results were obtained on the rectangular domain [−23, 23] × [0, 10], withthe spatial discretization of 880× 120 points, which implies ∆x = 0.052,∆y = 0.083.The solution is shown at time t = 1 after 1520 times steps at ∆t = 0.00066. Figure 5.3depicts a mesh surface plot of the solution on the subdomain [−0.2, 1.4]× [0.2, 2] andshows that the typical shock thickness is 4 points.

In addition to the tests mentioned above, we studied the behavior of our numericalmethod on a test problem of regular reflection. We started with initial data consistingof only one shock (incident shock) at an angle at which regular reflection should occur.In spite of a severe singularity in the initial data at y = 0, our method producedexcellent results. The solution is in good agreement with the known regular reflectionsolution; the shocks are well resolved and they travel with the correct speeds. Moredetails are given in the appendix.

3. The bifurcation diagram. In this section we present the basic conceptsused to obtain the bifurcation diagram introduced in [5] and [6], shown in Figure 3.1.They rely on the properties of self-similar solutions of (1.1).

In self-similar coordinates, ξ = x/t, η = y/t, (1.1) reads

(u− ξ)uξ − ηuη + vη = 0,−vξ + uη = 0.

(3.1)

Equation (3.1) changes type across a curve

Pu : ξ +η2

4= u.

The problem is elliptic inside Pu, where the flow is subsonic, and hyperbolic outside, inthe supersonic region. Linearized around a constant state U = (u, v), characteristicsare straight lines tangent to the parabola Pu. The domain of influence of a pointis the union of the forward wave cone through the point and the parabola Pu; seeFigure 3.2.

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0 1 2 30

0.5

1

1.5

2

2.5

3

a

b

Bifurcation Diagram

(a*,b*)

I

L

M

C

D J

K(a*,b*)

I

L

M

C

D J

K(a*,b*)

I

L

M

C

D J

K(a*,b*)

I

L

M

C

D J

K(a*,b*)

I

L

M

C

D J

K(a*,b*)

I

L

M

C

D J

K(a*,b*)

I

L

M

C

D J

K(a*,b*)

I

L

M

C

D J

K

0 1 2 30

0.5

1

1.5

2

2.5

3

a

b

IM

Solution Regions

J

MR

KMR

TMR

RR

vN

Fig. 3.1. The bifurcation diagram and the solution regions.

ξ

η

τ +

τ -

Ξ 0

Subsonic (elliptic) region

Supersonic (hyperbolic) region

Domain of influence

Fig. 3.2. Domain of influence of Ξ0.

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The elliptic regions corresponding to U0, U1, and U2 are conveniently nested. Ifξ = ξS(η) denotes the shock position, the states across the shock satisfy the followingRankine–Hugoniot conditions written in self-similar coordinates:

1

2

[u2]− ξ [u] + ξ′η [u]− ξ′ [v] = 0,

[v] + ξ′ [u] = 0,(3.2)

where [·] denotes the jump in a state variable across a shock. The initial value problem(1.1) and (1.2) in self-similar coordinates becomes a Cauchy problem with data givenalong a space-like curve at infinity. Unlike the full set of compressible Euler equations,or the wave equation, a space-like curve at infinity for the UTSD equation is not acircle. Because the x-axis is time-like, we are not allowed to impose data at x→ −∞.We can think of the Cauchy data as given along a parabola PR, with R→∞. (Moredetails are given in [5].) This is the reason why, using the numerical method describedin the previous section, we sweep the domain from right to left, with data given inthe computational domain (computational rectangle) along the upper and the rightedge of the rectangle, with v = 0 along the wall (bottom edge).

Depending on the values of a and b, shock S2 either overtakes S1, the shock S1

travels faster than S2, or S1 and S2 travel with the same speed, i.e., s1 = s2; see(1.5). The last case corresponds to the curve J , defined below, lying in the RR regionof the parameter space. The corresponding wave pattern looks like the initial shockconfiguration sliding along the wall with the speed s1 = s2. For other parametervalues one of the following three things can happen. The two shocks can interact inthe strictly hyperbolic region, giving rise to new hyperbolic waves; the shock S1 canhit the wall and generate a new wave which “joins” S2; or one or both shocks can firstinteract with the subsonic region leading to a free-boundary problem for the positionof a transonic shock. Based on the type of hyperbolic wave interaction and the type offree-boundary problem that arise for different values of a and b, a bifurcation diagram,shown in Figure 3.1, was conjectured in [5] and [6].

A geometric description of the curves that comprise the bifurcation diagram canbe summarized as follows.

Curve J: The curve J corresponds to the loci of points for which the two shocks,S1 and S2, meet at the wall leading to the “uniform RR-wave pattern.” A simplecalculation, requiring s1 = s2, shows the following.

Proposition 3.1. The shocks S1 and S2 meet at the wall if a = b + 1/(2b) ≡J(b).

The curves I and K: The curve I ∪ K is defined as the locus of points forwhich S1 and S2 intersect at the boundary of the elliptic region P2; See Figure 3.3a.A calculation gives the following.

Proposition 3.2. I ∪K ≡ {(a, b)|a2 +(2b− 3

b

)a+

(12b − b

)2= 0}.

The intersection of S1 and S2, ΞI , is outside P2 if I ∪K > 0. This happens if ais not between the roots

a1,2 ≡ 3

2b− b±

√2

√1

b2− 1(3.3)

of the quadratic defined in Proposition 3.2. Notice that if b > 1, ΞI is always outsideP2.

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-6 -4 -2 0 2 4 60

1

2

3

4

5

6

(a)

ξ

η

Ξ0

P2P1P0

S1S2

(a,b) ∈ I

-6 -4 -2 0 2 4 60

1

2

3

4

5

6

(b)

ξ

η

Ξ0

P2P1P0

S1S2

a = 0.5, b = 1.2

Fig. 3.3. Geometry of shocks and the corresponding parabolas.

Definition 3.3. We define K to be the curve

a = a1 for 0 ≤ b ≤ 1,

a = a2 for1

2≤ b ≤ 1.

(3.4)

Notice that not all shock intersections in the hyperbolic region will lead to ahyperbolic wave interaction which is determined by the data at infinity. For example,Figure 3.3b shows that for a = 0.5 and b = 1.2 the intersection, ΞI , lies “beyond”P2. Namely, before intersecting with S1, the shock S2 “passed through” P2 and“emerged” on the other side of P2 moving “backward in time.” Before resolving theinteraction with S1, we need to solve a free-boundary problem for the transonic partof S2. Therefore, the intersection S1 ∩ S2 will most likely not occur at ΞI , but at aperturbed point ΞI .

To distinguish the shock intersections that lead to hyperbolic problems deter-mined by the data at infinity, we have the following.

Definition 3.4 (see [5]). An admissible intersection is one that is completelydetermined by the data at infinity.

Proposition 3.5. All points (a, b) that lie below the curve

I(b) ≡ 3

2b− b−

√2

√1

2b2− 1(3.5)

such that b < 1/√

2 have the property that ΞI ≡ S1∩S2 is an admissible intersection.Curve D: This is the locus of points (a, b) for which shock S1 hits the wall exactly

at a point (ξ, η) = (u1, 0) at which P1 intersects η = 0. A simple calculation showsthat D is defined by the curve a = 1/

√2. As we will see in section 5, this curve serves

as the right boundary of the VN region in the bifurcation diagram of Figure 3.1.

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Curves a =√

2 and a = a∗: These curves play an important role in determin-ing the boundary of the RR region, where the wave pattern consists of a uniform shockS1 and the shock S2 (possibly perturbed), which meet at a point on the wall. For

a <√

2, no RR-wave pattern is possible; see [2] and [5]. For a ≤ a∗ ≡ (1 +√

5/2)1/2

,the state behind the (perturbed) shock S2 cannot be hyperbolic. More details will begiven in section 4.

Curve M: Since the curve M lies in the region above I and below K, the shockS2 intersects the parabola P2 before intersecting S1. If the intersection of S1 and S2

occurs before S2 becomes tangent to some parabola, PT , and starts moving “backwardin time,” the type of interaction between S1 and S2 leads to a perturbed MR-wavepattern described in section 6. Assuming small changes from uniform flow (therefore,assuming small changes in the position of S2 after it enters the subsonic region P2),the upper boundary of the region in the parameter space where this configurationoccurs is determined by calculating the set of points (a, b) for which the shock S2

intersects S1 exactly at a point of tangency to the parabola PT . A simple calculationshows that this happens for

a = M(b) ≡ 1

2b− b.

To study the structure of solutions that occur in different regions of the bifurcationdiagram shown in Figure 3.1, we will have to solve a class of quasi-one–dimensionalRiemann problems. Quasi-one–dimensional Riemann problems for the UTSD equa-tion were studied in detail in [5]. We present here a summary of the techniques anduse it in later sections.

3.1. Quasi-one–dimensional Riemann problems. The study of local solu-tions of quasi-one–dimensional Riemann problems arising in self-similar solutions foroblique shock initial data (1.2) is based on a generalization of wave-curve analysiscoupled with the Lax admissibility criterion. In contrast with one-dimensional Rie-mann problems, the coefficients of the system (3.1) depend not only on the statevariables but on the physical variables as well. This means that the solution of aquasi-one–dimensional Riemann problem will depend on the position where the waveinteraction takes place in the physical space. Therefore, a quasi-one–dimensionalproblem has data consisting of a center ΞI and two states U1 and U2, given along aspace-like line, L, through ΞI . The solution is a function

U(τ) = U

(ξ − ξIη − ηI

)defined downstream from L in a forward half-space below L. To define the downstreamwave locus of a state Ui with respect to a fixed point ΞI , D(Ui,ΞI), we first definethe shock and rarefaction polars. For precise definitions see [5].

The shock polar. For a straight line shock ξ = ξI + τ(η − ηI), the Rankine–Hugoniot conditions (3.2) imply, after eliminating τ = ξ′(η), that the equation for theshock polar through Ui is given by

[v]

[u]=ηI2±√η2I

4+ ξI −

(u+ ui

2

).(3.6)

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If Ui is hyperbolic at ΞI , there are two sonic points Us± on the shock polar. At Us±the value of u equals us = η2

I/4 + ξI . If

τ±(U) ≡ −ηI2±√η2I

4+ ξI − u(3.7)

denote the characteristics passing through ΞI , then the standard Lax admissibilitycondition implies that τ = ξ′(η) must be between τ+(U) and τ+(Ui) for an admissible+-shock and between τ−(U) and τ−(Ui) for an admissible −-shock. Transonic shocksare admissible if the state with real characteristics, Ui, is upstream, i.e., if ui ≤ u.More details can be found in [5].

The rarefaction polar. Writing system (3.1) in the form

A(U,Ξ)Uξ +B(U,Ξ)Uη ≡[(

u− ξ 00 −1

)∂ξ +

( −η 11 0

)∂η

](uv

)= 0

implies that a centered simple wave U(τ) satisfies

(A− τB)U ′ = 0,(3.8)

where U ′ = dU/dτ . Therefore, the null vector U ′ satisfies τu′ + v′ = 0, whereτ = τ±(U) or

dv

du=ηI2∓√η2I

4+ ξI − u.

After integrating we obtain the rarefaction polar through Ui, centered at ΞI :

v = vi +η2I

2(u− ui)∓

{(η2I

4+ ξI − u

) 32

−(η2I

4+ ξI − ui

) 32

}.(3.9)

Since both U and Ui must be hyperbolic at ΞI , the curve is defined for u ≤ us =η2I/4 + ξI . We will denote by R− and R+ the branches of the rarefaction polar that

correspond to the − or + sign, i.e., to τ− or τ+ characteristics.The downstream wave locus. The downstream wave locus of a state Ui is the

set of all states U which can be joined to Ui by a shock or a rarefaction wave in whichUi is the upstream state [5]. If Ui is hyperbolic, the downstream wave locus is theunion of the loop part of the shock polar, defined for u > ui, and the two branches ofthe rarefaction polar, defined for u ≤ ui; see Figure 3.4.

We conclude this section with a result on the existence and uniqueness of solu-tions of quasi-one–dimensional Riemann problems for the UTSD equation. In con-trast with one-dimensional Riemann problems, quasi-one–dimensional problems donot have solutions for all choices of initial data. Even when the existence is guaran-teed, uniqueness of a solution may fail. The following theorem, proved in [5], givesa complete description of the existence and uniqueness properties of local solutionsfor quasi-one–dimensional Riemann problems arising in self-similar solutions of theUTSD equation.

Theorem 3.6. Let (U1, U2,ΞI) be a Riemann data triple with hyperbolic statesU1 and U2 at ΞI , with U1 on the lower and U2 on the upper half of a space-like line Lthrough ΞI . Then if U2 lies in any of the regions 1, 2, or 4 in the complement of thedownstream wave locus of U1, D(U1,ΞI), shown in Figure 3.4, the Riemann problemhas a unique admissible solution. If U2 lies in region 3, the Riemann problem mayhave none, one, or two solutions.

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U M

S -

S +

R -

R +

U1

u

v

Region 1

Region 2

Region 3

Region 4

Fig. 3.4. Downstream wave locus and solution regions of quasi-one–dimensional problems.

4. The RR- and MR-wave patterns. We begin the description of the structureof solutions by first considering the cases in which quasi-one–dimensional Riemannproblems arise in the strictly hyperbolic region. We present global solutions andcompare the numerical simulations with the exact local solutions in the neighborhoodof the point where the shock interaction takes place.

4.1. The MR-wave pattern. The region denoted by MR in the bifurcationdiagram of Figure 3.1 is bounded by a = 0, b = 0, and by the curve a = I(b),defined in the previous section. For the parameter values in this region the twoshock waves, S1 and S2, intersect in the hyperbolic region before either of theminteracts with the subsonic region. The intersection point, ΞI , is given by ΞI =(1/2+a/2b+ab, 1/2b+b−a). The solution of the corresponding wave interaction canbe obtained locally, near ΞI , by solving a quasi-one–dimensional Riemann problemwith data (U0, U2,ΞI). The following theorem, proved in [5], describes the solution.

Theorem 4.1. If oblique shock data (1.2) given for (1.1) are such that a ≤ I(b),then a hyperbolic quasi-one–dimensional Riemann problem occurs at ΞI . Furthermore,there is a unique admissible solution which consists of a rarefaction wave and a shock.

A sketch of the solution is shown in Figure 4.1; a numerical simulation is presentedin Figure 4.2. Figure 4.2 shows a contour plot of v at t = 1 for the point (a, b) =(0.2, 0.1) ∈ MR. The corresponding values of ∆x and ∆y are ∆x = 0.026, ∆y = 0.046.The solution was obtained after 2653 time steps. The values of v along the cross-sections 1 and 2 are shown in Figure 4.3. Figure 4.2 shows S1 and S2 intersectingat a point ΞI that lies in the region of supersonic (hyperbolic) flow, giving rise tonew hyperbolic waves: a +-shock between U0 and an intermediate state UM and a −-rarefaction wave between U2 and UM . The region of nonuniform flow corresponds tothe subsonic region which is bounded by the sonic parabola PuM . Along cross-section1, moving in the direction from left to right, Figure 4.3 shows the value of v0 = 0.0, ashock from v0 to v1 = −0.2 (S1), a shock from v1 to v2 = 0.0 (S2), a thin rarefactionwave from v2 to vM , a thin region with constant state vM , and a region of subsonicflow. Along cross-section 2, Figure 4.3 shows a shock from v0 to vM followed by theregion of subsonic flow.

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SS12

U

U

U

0

1

2

η

ξ

Elliptic

P

(subsonic)

UM

M

free boundary

ΞI

Fig. 4.1. Sketch of the MR-wave pattern.

-20 -15 -10 -5 0 5 10 15 200

2

4

6

8

10

12

x

y

Contour plot of v at t = 1

Cross-section 1

Cross-section 2

U0

U1

U2

UM

Fig. 4.2. Numerical simulation of the MR-wave pattern with ∆x = 0.026, ∆y = 0.046 after 2653time steps; contour plot of v at t = 1 for a = 0.2, b = 0.1. The values of v along the cross-sections1 and 2 are shown in Figure 4.3.D

ownl

oade

d 11

/18/

14 to

155

.33.

120.

209.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

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php


0 200 400 600 800 1000 1200 1400 1600-0.03

-0.02

-0.01

0

0.01

0.02

0.03

n

v

Cross-section 2

v0

vM

subsonic flow

0 200 400 600 800 1000 1200 1400 1600-0.2

-0.15

-0.1

-0.05

0

0.05

n

v

Cross-section 1

v0

v1

v2subsonic flow

↑rarefaction wave

Fig. 4.3. The figure shows v vs. n (mesh points) at cross-sections 1 and 2 from Figure 4.2.

0 5 10 15-10

0

10

20

30

40

50(a) Downstream Wave Loci Configurations

U0 U2SUBSONIC

REGION

a = 0.2, b=0.1

2.9 2.95 3 3.05-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04(b) Magnified View of the Intersection

(2.995,-0.0129)

U2

R- (U2)

S+(U0)

Fig. 4.4. Downstream wave loci configurations for the MR-wave pattern data.

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To compare the numerical and the exact value of UM we solved the local hyper-bolic wave interaction at ΞI by using the downstream loci through U0 and U2. Theconfigurations of the downstream wave loci through U0 and U2 are given in Figure 4.4.The intermediate state, UM , which lies in the intersection of the R−-branch of therarefaction polar through U2, and the S+-branch of the shock polar through

U0, is given by UM = (2.995,−0.0129). The numerical value at a point which lies inthe middle of the region UM of Figure 4.2 reads UM = (2.997,−0.0125), which is inexcellent agreement with the exact solution.

In order to complete the proof of the existence of a global solution shown inFigure 4.2, what needs to be done is the study of the existence of the subsonic part ofthe solution. This leads to a free-boundary problem for the position of the transonicshock, SM , between the supersonic state U0 and the subsonic state UM , in the domainbounded by the wall, η = 0, by a sonic parabola PuM , and by the transonic shock SM(free boundary). The natural boundary conditions associated with this problem aregiven by

• no flow through the wall: v = 0,• continuous flow across the sonic line: u = uM , v = vM ,• far-field conditions at ξ = −∞ which guarantee v = vM along PuM , and

• two Rankine–Hugoniot conditions for the position of the transonic shock SM .

The existence of a solution of this problem is still open.

4.2. The RR-wave pattern. The RR-wave pattern denotes a solution whichconsists of a shock S1 meeting with S2 (or a perturbation of S2, S2) at a point on thewall. In this model we distinguish between uniform RR-wave pattern and nonuniformRR-wave pattern or perturbed RR-wave pattern. Uniform RR-wave pattern correspondsto the case in which both the shock S1 and the shock S2 are straight, separating regionsof uniform flow. This wave configuration occurs for the parameter values a and b forwhich the horizontal velocities s1 and s2 are equal. Proposition 3.1 describes thecurve in the a, b-parameter space for which this occurs. We denoted this curve by J .Figure 4.5 shows our numerical simulation of uniform RR-wave pattern for a = 1.833and b = 1.5, at time t = 1, obtained after 1188 time steps. The shocks move with thecorrect speed s = s1 = s2 = 3.859, and at t = 1 they meet at the point x = 3.859 onthe wall.

Perturbed RR-wave pattern consists of a uniform shock S1 and a curved shock S2

(a perturbation of S2). As described in [5], when the uniform shock S1 hits the wall,the shock interaction with the wall can be resolved by solving a quasi-one–dimensionalRiemann problem with Riemann data given by

• the center ΞA = (a2 + 1/2, 0),• a unique space-like line L (vertical line through ΞA), and• the data along L: U1 = (1,−a) along the upper half of L (where η > 0), andU∗1 = (1, a) on the lower half (where η < 0).

Even though this is a hyperbolic problem if a ≥ 1/√

2, it has a solution if and onlyif a ≥ √

2. A simple calculation shows that the following proposition holds (see, forexample, [5]).

Proposition 4.2. The quasi-one–dimensional Riemann problem with the datatriple (U∗1 , U1,ΞA) has a solution if and only if a ≥ √

2. If a ≥ √2, there are two

solutions:

UR = (1 + a2 − a√a2 − 2, 0),

UF = (1 + a2 + a√a2 − 2, 0).

(4.1)

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-30 -20 -10 0 10 20 300

1

2

3

4

5

6

7

8

9

10

S1S2

ξ

η

Fig. 4.5. Numerical simulation of uniform RR-wave pattern with ∆x = 0.068,∆y = 0.083 after1188 time steps; contour plot of v for a = 1.833, b = 1.5.

-15 -10 -5 0 5 100

2

4

6

8

10(a)

ξ

η

P2

P1PR UR

-15 -10 -5 0 5 100

2

4

6

8

10(b)

ξ

η

PR

UR

Fig. 4.6. Geometry of shocks and sonic parabolas in perturbed RR-wave pattern. (a) correspondsto a = 2 and b = 1, and (b) corresponds to a = 1.7 and b = 2.5.

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-15 -10 -5 0 5 10 150

1

2

3

4

5

6

x

y

(b) Perturbed RR-wave pattern; contour plot of v (concave down)

+ 0.00

+ 0.00

+ -1.7

nonuniform subsonic flow -->

-0.3

-0.3 -0.5

-0.5

-15 -10 -5 0 5 10 150

1

2

3

4

5

6

x

y

(a) Perturbed RR-wave pattern; contour plot of v (concave up)

+ 0.00

+ 0.00

+ -2

nonuniform subsonic flow --> 0.2

0.4

Fig. 4.7. Numerical simulation of perturbed RR-wave pattern obtained with ∆x = 0.068, ∆y =0.083 after 2178 time steps; contour plots of v at time t = 1. (a) corresponds to a = 2 and b = 1,and (b) corresponds to a = 1.7 and b = 2.5.

The state UF is never hyperbolic, while UR is hyperbolic if a ≥ a∗ ≡ (1 +√

5/2)1/2.

The region RR in the bifurcation diagram of Figure 3.1, which lies on the rightof a =

√2 and above the curve M , contains RR-wave pattern solutions. Some con-

figurations of shocks and sonic parabolas for (a, b) belonging to subregions of RR aregiven in Figure 4.6. Figure 4.6a corresponds to (a, b) = (2, 1). This point lies in theconvex region of the bifurcation diagram bounded by the curve J . Figure 4.6b showsthe geometry of shocks and sonic parabolas corresponding to (a, b) = (1.7, 2.5). Thispoint lies above the curve J , inside the region RR. (See the bifurcation diagram in

Figure 3.1.) Notice that shock S2 is transonic along some segment S2. A proof of the

existence of a global solution of the free-boundary problem for the position of S2 isopen.

Numerical simulations of perturbed RR-wave patterns are shown in Figures 4.7aand 4.7b. They were obtained with ∆x = 0.068 and ∆y = 0.083 after 2178 timesteps. Figures 4.7a and 4.7b show contour plots of v corresponding to the parametersof Figures 4.6a and 4.6b. The contours behind the shock S2 in both cases shownonuniform subsonic flow. The numbers in the figures correspond to the values of vtaken at the points denoted by +. We clearly see that the transonic shock S2 is concaveup for a = 2, b = 1 (inside J) and concave down for a = 1.7, b = 2.5 (outside J).

5. The VN-wave pattern. As conjectured in [5], the VN-wave pattern occursfor the parameter values for which

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-3 -2 -1 0 1 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

ξ

η

P1

P2

a =0.5

b =1

U2

U1

U0

S2

S1

Ξ0

Ξ1

Fig. 5.1. Geometry of shocks and parabolas in VN-wave pattern.

• the shock S1 interacts with the subsonic flow before it hits the wall; i.e.,Ξ1 ≡ S1 ∩ P1 is admissible and η > 0. This occurs for a < 1/

√2; and

• ΞI ≡ S1 ∩ S2 is not admissible; i.e., a > I(b).The corresponding geometry of shocks and sonic parabolas is given in Figure 5.1. Tocomplete the global structure of the solution in the subsonic region, a free-boundaryproblem for the quasi-linear, degenerate elliptic equation (1.1) needs to be solved. The

free boundary consists of two parts. One is a continuation of the shock S2, S2, whereit becomes transonic, i.e., from the point Ξ0 ≡ S2∩P2, and the other is a continuationof S1, S1, from the point Ξ1 where S1 becomes transonic, forming a shock stem; seeFigure 5.1. If we introduce a new coordinate in place of ξ so that the sonic parabolasbecome straight lines (ρ = ξ + η2/4), then the shock conditions for S1 and S2 can bewritten as

S1 ...

ρ′s =√ρs − 1

2 u,

ρ′s =η

2− v

u,

S2 ...

ρ′s =√ρs − 1

2 (u+ 1),

ρ′s =η

2− v + a

u− 1.

(5.1)

Here, ρs = ρs(η) denotes the position of a transonic shock, and u and v are traces ofthe functions u and v at the subsonic side of the shock ρs. In [4] it was proved that ifthe shock stem is an increasing and convex function, i.e., if ξ′s(η) ≥ 0 and ξ′′s (η) ≥ 0,

then u is decreasing along S1. Leading order asymptotic analysis, presented in [3],

implies that, if u is decreasing along S1, it has a square-root–type singularity at thedegenerate boundary where S1 intersects P1. Figure 5.2, which shows typical contourplots of u and v in the VN-wave pattern solution, indicates that the assumptions on

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-8 -6 -4 -2 0 2 4 6 80

1

2

3

4

5

6(a) vN-wave pattern; contour plot of v

S S

+-0.0063

+0.000+-0.057+-0.13+-0.16

+-0.2 +0.000

+-0.0063

2 1

ξ

η

-8 -6 -4 -2 0 2 4 6 80

1

2

3

4

5

6(b) vN-wave pattern; contour plot of u

P S S

+1.18

+1.2

+1.2

+1.00

+1.06

+1.18

+1.18

+0.0001 2 1

ξ

η

Ξ1

Fig. 5.2. Numerical simulation of VN-wave pattern for a = 0.2, b = 1 with ∆x = 0.052 and∆y = 0.083, after 1520 time steps; contour plots of v and u at time t = 1.

the geometry of S1 are well founded. Moreover, the results in section 5.1 will showthat our numerical simulations confirm the square-root behavior of u.

Regarding the structure of S2 we have the following observations. As for theshock stem, we first assume certain geometry of S2 and then use it to conclude thebehavior of the subsonic solution. Namely, we would like to prove that the strengthof S2 decreases toward S1 and becomes zero at the point of interaction with S1. Theassumptions we employ are the following. If the shock S2 moves in the positive time-like direction, then ρ′s ≥ 0. This behavior is also confirmed by numerical simulations;

see Figure 5.2. Numerical simulations also indicate that the shock S2 is concavedown in the ξ, η-physical space. This implies ξ′′s (η) ≤ 0, or equivalently, ρ′′s (η) ≤ 1

2 .Now, using these two assumptions, namely, ρ′s ≥ 0 and ρ′′s (η) ≤ 1

2 , and the Rankine–

Hugoniot conditions (5.1) for S2, we obtain that u is decreasing from Ξ0 toward S1;

i.e., u′(η) ≥ 0. What is left is to show that the strength of S2 is zero at Ξ1 = S1 ∩P1.

This can be verified in several ways. One way is to check that the shock S2 betweenthe state U1 and a subsonic state U is tangent to the parabola P1 at the point Ξ1

where it intersects S1. Namely, the following proposition holds.

Proposition 5.1. A transonic shock, Si, between a uniform state, Ui, and asubsonic state, U , is tangent to the sonic parabola Pi at Ξi if and only if the strength

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-0.200.20.40.60.811.21.41.6

0

1

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

x

y

u

P1Ξ1

U0

U1

Fig. 5.3. Detail of a mesh surface plot of u for a = 0.2, b = 1. This numerical simulation wasobtained with ∆x = 0.052 and ∆y = 0.083 after 1520 times steps at ∆t = 0.00066.

of Si is zero at Ξi.Proof. The Rankine–Hugoniot conditions for the position of a transonic shock Si

are given by

ρ′s =

√ρs − 1

2(u+ ui),

ρ′s =η

2− v − viu− ui

.

(5.2)

From the first condition we see that ρ′s = 0 at a point Ξi = (ρi, ηi) = (ui, ηi) ∈ Pi ifand only if u = ui at Ξi.

Figure 5.2 shows typical contour plots of v and u corresponding to the VN-wavepattern solution, obtained after 1520 time steps with ∆x = 0.052 and ∆y = 0.083.We added the parabola P1 in the contour plot of u to show that the shock S2 is indeedtangent to P1. We took a closer look at the behavior of u and v near Ξ1 by studyingtheir behavior in the boxed region shown in Figure 5.2b. We describe our findings inthe following section.

5.1. Square-root–type singularity at Ξ1. To study local behavior of u andv near P1 at Ξ1 we took the traces of u and v along a straight line normal to theparabola P1 at the point Ξ1, shown in Figure 5.3. We took the data for u and v at

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numerical solutionnonlinear fit

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.41

1.05

1.1

1.15

1.2

1.25

ρ1 − ρ

u

Fig. 5.4. Nonlinear least squares fit of the data for the behavior of u near the sonic point Ξ1.The data for u lies above the normal to P1 at Ξ1, shown in Figure 5.3.

12 equidistant points along this line, where the first point lies above Ξ1. Since thesepoints do not necessarily correspond to the mesh points, we used two-dimensionallinear interpolation to calculate the corresponding values of u and v. Figure 5.3shows a magnified surface plot of u at t = 1 above the region [−0.2, 1.4]× [0.2, 2] andthe interpolated 12 values of u above the normal to the parabola P1. The rectangleson the surface of u correspond to the mesh blocks. The precise values of u obtainedat the 12 points along the normal to P1 are shown in Figure 5.4. The distance fromthe “triple point” Ξ1 along the normal is ρ1−ρ. To study the local behavior of u andv as functions of ρ1 − ρ we calculated a nonlinear least squares fit with functions ofthe form

z(x;C,Λ) =

m∑j=1

Cjxλj , x ∈ Rn,(5.3)

where x corresponds to ρ1 − ρ and C = (C1, . . . , Cm), Λ = (λ1, . . . , λm). For thehorizontal velocity component u, with m = 2, we obtained C = (1.0166, 0.3217)and Λ = (0, 0.5098), with least square error of 0.0201. The best nonlinear fit is thefunction z = 1.0166 + 0.3217x0.5098. The data and the nonlinear least squares fit areshown in Figure 5.4. We draw two conclusions. First, the behavior of u near the sonicpoint Ξ1 can best be described as the square root of the distance from the degenerateboundary, as predicted in [4]; second, the strength of S2 tends to zero near the pointof interaction with S1 since the value of u near Ξ1 approaches the value of u1 = 1.

Similarly, for the vertical velocity component v, with m = 2, we obtained C =(−0.1901, 0.2066) and Λ = (0, 0.7269), with the least square error of 0.0065. Thisimplies that v appears to be “smoother” near the sonic line than the function u.

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-5 0 5 100

2

4

6

8

10

12

x

y


v0

v1v2

+0.00

+-0.8

+0.00

-0.031

-0.114

-0.197

-0.031

Fig. 6.1. Numerical solution of TMR-wave pattern for a = 0.8, b = 0.15, obtained with ∆x =0.05,∆y = 0.1 after 1000 time steps; contour plot of v at t = 1.

6. The TMR- and KMR-wave patterns. To complete the description of thestructure of solutions for the parameter values in the bifurcation diagram shown inFigure 3.1, there are two regions left; they are denoted by TMR and KMR.

The region TMR is bounded from below by the curve I, on the right by the curvea = 1/

√2, and from above by the curve M . The solution in this region presents a

continuous transition from the solution which occurs for (a, b) ∈ I and other typesof reflection. The solution structure for (a, b) ∈ I was first described in [5]. Forcompleteness, we summarize the description here. The case when (a, b) ∈ I is thelimiting case in which ΞI is still an admissible intersection (see Definition 3.4). Theconfiguration of the corresponding sonic curves and the shock waves S1 and S2 isshown in Figure 3.3a. The shock S2 is sonic at ΞI , and a “degenerate” hyperbolicwave interaction takes place at ΞI . A local solution to this degenerate wave interactionwas obtained in [5]. It consists of a shock S1 between U0 and an intermediate stateUM , followed by a thin rarefaction wave between UM and U2, where the length of thecharacteristics in the rarefaction wave tends to zero as U → U2. It was proven in [5]that this solution is unique.

When (a, b) ∈ TMR, a perturbation of this solution takes place. In this casethe uniform shocks S1 and S2 intersect slightly inside the sonic parabola P2, and S2

becomes transonic from Ξ0 to ΞI . The solution structure in this parameter regime hasbeen a subject of many discussions. At a first glance the numerical pictures suggest atriple shock configuration; see Figure 6.1. Since the triple shock configuration cannotoccur in this model [2], the presence of a logarithmic singularity at the triple point wassuspected. However, the proof presented in [7] shows that such a subsonic solutioncannot be matched with the rest of the flow and therefore has to be discarded. Thework in [5] resolves this problem in the following way. The conjecture is that the

solution u decreases along the shock S2 and becomes sonic or supersonic at the pointof interaction with S1, giving rise to a quasi-one–dimensional Riemann problem at

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WALL

space-like linerarefaction wave

S1

P2

S2

U1

U0

U2

UD

Fig. 6.2. Sketch of the TMR-wave pattern.

P2

UP

SUBSONIC

FLOW

HYPERBOLIC STATE

U2U1

U0UD

SONICSTATE

HYPERBOLICSTATE

HYPERBOLICSTATE

HYPERBOLICSTATE

RAREFACTIONWAVE

mm

(ELLIPTIC REGION)

Fig. 6.3. Magnified region in TMR-wave pattern near the interaction of S1 and S2.

ΞI = S1 ∩ S2. This is similar to what happens for (a, b) ∈ I. The corresponding

data triple is given by (U0, UD, ΞI), where U0 is the state on the right of ΞI andUD the state on the left. The existence and uniqueness of a local solution under theassumption that the state on the left of S2, UD, is sonic at ΞI were proved in [5].Namely, the following proposition holds.

Proposition 6.1 (see [5]). Let a ≥ √2 and suppose that ΞI is any point on

S1. Then there exists a unique value UD on S+(U1, ΞI) which is sonic at ΞI . (Thefirst condition means that UD belongs to the +-shock branch of the downstream locusof U1 at ΞI .) Furthermore, there is a unique solution of the quasi-one–dimensional

Riemann problem at ΞI with data U0 and UD, consisting of a shock between U0 andan intermediate state Um and a rarefaction wave between Um and UD.

The local solution is similar to the one occurring for (a, b) ∈ I. Figures 6.2 and 6.3,show sketches of the solution configuration. The supersonic state Um becomes sonicat the parabola Pm and subsonic on the left of it; see Figure 6.3. Similarly, each stateUα in the thin rarefaction wave in which Uα changes from Um to UD becomes sonic at

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Contour plot of u and the sonic boundary

subsonic region

boundary of the subsonic region ---->

region where the flow is

sonic or supersonic S2

ξ

η

Ξi

shock stem

Fig. 6.4. TMR-wave pattern; one contour of u and the sonic boundary.

the point where the characteristic along which Uα is constant becomes tangent to thecorresponding parabola Pα and subsonic on the left of it. Therefore, there is a smallhyperbolic region near by the point where S1 and S2 interact, which is “embedded”in the infinite elliptic region.

In this section we show that the state behind S2 at the point of interaction withS1 is indeed sonic and that there is a region behind S2 which contains points thatare sonic or supersonic. In Figure 6.1 we show a contour plot of v corresponding to(a, b) = (0.8, 0.15), which belongs to the region TMR. The values of the contours of vare taken at the positions denoted by +. The contour plot of v in Figure 6.1 showsa structure that indicates the presence of a rarefaction wave in a small hyperbolicregion behind S2, although the structure is too small to be well resolved numerically,especially in the first velocity component u for which the jumps in the states across theshock are of greater magnitude than in the vertical velocity component v. However,to see that there is a small sonic/supersonic region behind S2 and the shock stem, in

Figure 6.4 we draw the left-most contour tracing shock S2 and the left-most contourtracing the shock stem. We also show the boundary of the subsonic region which isnumerically calculated based on the values of u, ξ, and η. Figure 6.4 indicates thepresence of a small sonic/supersonic region behind S2 and the shock stem, near Ξ1.

To show that a degenerate quasi-one–dimensional Riemann problem takes placeat ΞI , we compared the computed value of u at the “triple point,” ΞI , and the corre-sponding sonic value. We have u(ΞI) = 5.288 and usonic(ΞI) = 5.2153. Consideringthat the point ΞI is in the region of high gradients, where the computational error isat its worst, we conclude that the error of 7× 10−2 is within the error bounds of thenumerical method.

The structure of the KMR-wave pattern is similar to the solution occurring forthe parameter values in region TMR. As conjectured in [5], the solution consists ofa constant state on the left of S2 until S2 reaches the sonic parabola P2. Then S2

becomes nonuniform, reaching S1 at a point above the wall since, by Proposition 4.2,

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-15 -10 -5 0 5 10 150

1

2

3

4

5

6

7

8

9

10

ξ

η Ξ0

PM

S+

1.4

+1.16

+1.00

+0.386

+0.00

+1.54

2 S1

ΞM

<- - - ΞI

Fig. 6.5. KMR-wave pattern for a = 0.8, b = 2 obtained with ∆x = 0.05,∆y = 0.1 after 1000time steps; contour plot of u with minimal number of contours.

-15 -10 -5 0 5 10 150

1

2

3

4

5

6

7

8

9

10

x

y

+

0.00

+-0.8

+

0.00


-0.2

-0.3

-0.5

-0.7

-0.2

-0.2

Fig. 6.6. KMR-wave pattern for a = 0.8, b = 2, obtained with ∆x = 0.05,∆y = 0.1 after 1000time steps; contour plot of v at t = 1.

it is impossible to have an RR-type solution (with S1 straight all the way to thewall) in the region KMR. Moreover, because a >

√2, the strength of the shock at

the point ΞI , where S2 meets S1 cannot be zero. The conjecture for the solution inthis case is that the shock S2 becomes sonic at the point where it meets S1 and aquasi-one–dimensional Riemann problem at ΞI has a solution consisting of a shockand a rarefaction wave. The difference between this solution and the TMR-wave

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pattern is that the shock S2 becomes tangent to a certain sonic curve on its way fromΞ0 = S2 ∩ P2 to ΞI ; this is shown in Figure 6.5. The point of tangency is denotedby ΞM . The derivative of the position of the shock S2, given by the equation ρs(η),

changes sign at ΞM . The shock S2 moves “backward in time” on the right of ΞM .Expressed in ρ, η-coordinates, the continuation of S2, ρs = ρ(η) is concave up in the

neighborhood of ΞM and concave down at ΞI . Figure 6.5 shows a contour plot of u,the point Ξ0 where S2 intersects P2, the parabola PM , and the point of tangency, ΞM ,between S2 and PM . Figure 6.6 shows the corresponding contour plot of v. Noticehow this solution interpolates between the solution occurring in the RR region (where

the continuation of S2 can be thought of as moving “backward” in time), the solutionoccurring for the parameter values in the TMR region (where the structure of the localsolution at ΞI is identical), and the solution corresponding to the VN region (wherethe continuation of S2 is tangent to the parabola PM = P1 and ΞM = Ξ1).

7. Conclusions. In this work we provided a complete description of possiblewave interactions arising in a two-parameter family of oblique shock initial valueproblems (1.1) and (1.2) for the UTSD equation. Coupled with several theoreticalresults, our numerical study reveals global structure of solutions arising in this model.

Several questions remain open. They relate to the proofs of global existenceof solutions numerically studied in this work, and to the relevance of the solutionstructures studied here to the actual problem of shock reflection.

Appendix. To see how well our numerical method behaves when severe singu-larities are present in the initial data, we tested the method on the problem of (real)regular reflection. In contrast with the initial data consisting of two shocks S1 andS2, we computed our solution with a single initial shock (incident shock) at an anglewhich should give rise to a regular reflection wave pattern. More precisely, our initialdata consist of an incident shock x = ay with a = 1.833 between the states U0 = (0, 0)and U1 = (1,−a) in the upper half-plane and “no flow through the wall” boundarycondition v = 0; see Figure A.1. Shock polar analysis implies that there are twosolutions, UR and UL, given by

UR = (1 + a2 − a√a2 − 2, 0),

UF = (1 + a2 + a√a2 − 2, 0),

(A.1)

which give rise to a regular reflection pattern solution (cf. Proposition 4.2). For the fullset of Euler equations these are known as the “weak shock solution” and the “strongshock solution.” For a = 1.833 we obtain UR = (2.222, 0.0) and UF = (6.497, 0.0).None of these solutions can be discarded on theoretical grounds. However, it is anexperimental fact that the weak shock solution is the one which usually occurs (see[1]). Indeed, our numerical simulation produces the solution with the state UR =(2.222, 0.0) immediately behind the reflected wave near the point where the two shocksmeet at the wall. Namely, we ran our code on the shock reflection initial data withspatial discretization ∆x = 0.035,∆y = 0.058. We show the computed solution attime t = 1 obtained after 2115 time steps in Figure A.1. Figure A.1 shows a contourplot of v with the values along the contours indicated at the points denoted by +.The point where the two shocks meet at the wall corresponds to the exact value ofx = 3.859. To see what is the value of u behind the reflected shock close to the wall(y = 0), we plotted a cross-section of u at the distance ∆y = 0.058 above the wallin Figure A.2. Figure A.2 shows a region with some oscillation around the constant

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-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 300

2

4

6

8

x

y


+ -1.83

+ 0.00

-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 300

2

4

6

8

x

y


+ 0.00

+ -1.83

+ 0.00 -0.26

-0.39

-0.52

-0.65

-0.78

Fig. A.1. Numerical solution for regular reflection initial data with ∆x = 0.035,∆y = 0.058,after 2115 time steps. The figures show contour plots of v at time t = 0 and at t = 1.

-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30

0

0.5

1

1.5

2

2.5

3

3.5u(x,dy) at t = 1

x

u(x,d

y)

← supersonic flow →← subsonic flow →

uR = 2.22

Fig. A.2. Cross-section of u at y = dy = 0.058140. The horizontal line uR = 2.22 showsthe exact value of u behind the reflected wave near the wall. The vertical line denotes the sonicboundary. It divides the plot in the region where the flow is subsonic, and therefore not necessarilyuniform, and the region of supersonic flow which contains constant state uR and a shock betweenuR and u0 = 0.

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state uR = 2.22, a shock from uR to u0 = 0, and a region of (nonuniform) subsonicflow behind (on the left of) uR. We conclude that our numerical results are in goodagreement with the exact solution.

Acknowledgments. We are grateful to Barbara L. Keyfitz for reading themanuscript and for making many helpful suggestions. We also thank the refereesfor their helpful comments.

The authors appreciate the hospitality of James Glimm and the Department ofApplied Mathematics and Statistics at SUNY Stony Brook, which both authors visitedwhile writing the final version of the manuscript.

REFERENCES

[1] G. Ben-Dor, Shock Wave Reflection Phenomena, Springer-Verlag, New York, 1992.[2] M. Brio and J. K. Hunter, Mach reflection for the two dimensional Burgers equation,

Phys. D, 60 (1992), pp. 194–207.[3] S. Canic and B. L. Keyfitz, An elliptic problem arising from the unsteady transonic small

disturbance equation, J. Differential Equations, 125 (1996), pp. 548–574.[4] S. Canic and B. L. Keyfitz, Oblique shock interactions and the von Neumann paradox, in

Shock Waves, B. Sturtevant, J. E. Schepherd, and H. G. Hornung, eds., World Scientific,Singapore, 1996, pp. 435–440.

[5] S. Canic and B. L. Keyfitz, Riemann problems for the two-dimensional unsteady transonicsmall disturbance equation, SIAM J. Appl. Math., 58 (1998), pp. 636–665.

[6] S. Canic, B. L. Keyfitz, and D. H. Wagner, A bifurcation diagram for oblique shock inter-actions in the unsteady transonic small disturbance equation, in Proc. Fifth InternationalConference on Hyperbolic Problems: Theory, Numerics, and Applications, J. Glimm et al.,eds., World Scientific, Singapore, 1996, pp. 178–187.

[7] I. Gamba, R. R. Rosales, and E. Tabak, Constraints on Formation of Singularities for theTransonic Small Disturbance Equations, preprint, University of Texas, Austin, TX.

[8] E. Harabetian, Diffraction of a weak shock by a wedge, Comm. Pure Appl. Math., XL (1987),pp. 849–863.

[9] A. Harten, S. Osher, B. Engquist, and S. R. Chakravathy, Some results on uniformlyhigh order accurate essentially nonoscillatory schemes, Appl. Numer. Math., 2 (1986),pp. 347–377.

[10] R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhauser-Verlag, Basel, 1992.[11] C. S. Morawetz, Potential theory for regular and Mach reflection of a shock at a wedge,

Comm. Pure Appl. Math., XLVII (1994), pp. 593–624.[12] S. Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM J.

Numer. Anal., 21 (1984), pp. 217–235.[13] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms

based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), pp. 12–49.[14] S. Osher and C.-W. Shu, High-order essentially nonoscillatory schemes for Hamilton–Jacobi

equations, SIAM J. Numer. Anal., 28 (1991), pp. 907–922.[15] C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-

capturing schemes, J. Comput. Phys., 77 (1988), pp. 439–471.[16] E. G. Tabak and R. R. Rosales, Focusing of weak shock waves and the von Neumann paradox

of oblique shock reflection, Phys. Fluids A, 6 (1994), pp. 1874–1892.

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a numerical study of riemann problems for the two-dimensional unsteady transonic small disturbance...

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