POSITIVITY PROPERTY OF SECOND-ORDER FLUX-SPLITTING SCHEMES FOR THE COMPRESSIBLE EULER EQUATIONS. Cheng Wang. Jian-Guo Liu

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1 DISCRETE AND CONTINUOUS Website: DYNAMICAL SYSTEMS SERIES B Volume 3 Number May003 pp. 0 8 POSITIVITY PROPERTY OF SECOND-ORDER FLUX-SPLITTING SCHEMES FOR THE COMPRESSIBLE EULER EQUATIONS Cheng Wang Department of Mathematics Indiana University Bloomington IN Jian-Guo Liu Department of Mathematics University of Maryland College Park MD (Communicated by Shouhong Wang Abstract. A class of upwind flux splitting methods in the Euler equations of compressible flow is considered in this paper. Using the property that Euler flux F (U is a homogeneous function of degree one in U wereformulate the splitting fluxes with F + = A + U F = A U and the corresponding matrices are either symmetric or symmetrizable and keep only non-negative and non-positive eigenvalues. That leads to the conclusion that the first order schemes are positive in the sense of Lax-Liu [8] which implies that it is L - stable in some suitable sense. Moreover the second order scheme is a stable perturbation of the first order scheme so that the positivity of the second order schemes is also established under a CFL-like condition. In addition these splitting methods preserve the positivity of density and energy.. Introduction. The general form of a system of hyperbolic conservation laws can be written as t U + x F (U+ y G(U+ z H(U =0 U R n (. such as the governing equations for compressible flow MHD etc. For example the -D Euler equations for gas dynamics is given by t U + x F (U =0 U = ρ m E F = m ρv + p v(e + p (. in which ρ v m = ρv and E stand for density velocity momentum and total energy respectively with the state equation for the pressure p =(γ (E ρv (see [3]. The solutions to these equations as well as the physical phenomenon are very complicated. One of the distinguished futures is the appearance of discontinuous solutions such as shock waves. This imposes a great difficulty in the design of 99 Mathematics Subect Classification. 65M 35L65. Key words and phrases. Conservation laws flux splitting limiter function positivity. 0

2 0 C. WANG AND J. LIU numerical schemes for these systems and also for the mathematical analysis. A scalar equation u t + f(u x =0 (.3 is usually used as a simple model to give the guideline in the corresponding algorithm design and analysis. The concept of Total-variation-diminishing (TVD has played a crucial role in the development of modern shock capturing schemes although the concept of TVD is only valid in the analysis of the scalar equation. Nevertheless direct extension to general systems has been highly successful in the computation of many complicated physical systems. See the relevant references such as the introduction of Monotonic Upstream-centered Scheme for Conservation laws (MUSCL by Van Leer in [ ] the corresponding analysis in [6 5] the discussion of the essentially nonoscillatory (ENO scheme in [8 0 3] etc. A concept of positivity property which was recently proposed by X.-D. Liu and P. Lax in [8] is a natural extension of TV-stable property. In the scalar case any consistent scheme in which the numerical solution u n+ at the time step t n+ can be written as convex (positive combination of u n was proven to be TVD. In the case of a hyperbolic system the extension of a positive coefficient (combination to a positive symmetric matrix gives a positive scheme. Let s have a review of the TVD scheme and its extension to a general system by positivity scheme in the context of flux splitting. The idea can be easily explained by the upwind scheme u n+ u n t + a +n u n un applied for the scalar linear advect equation + a n u n + un =0 (. u t + au x =0 (.5 with the decomposition a = a + + a and a + 0 a 0. The upwind scheme (. can be reformulated so that u n+ is a convex combination of the profile u n namely u n+ = a +n t ( un + a +n t + t a n u n a n t un +. (.6 As a result the stability property of the scheme (. follows directly from a CFLlike condition ( a +n a n t. (.7 For scalar nonlinear conservation laws the flux can be decomposed as u t + f(u x =0 (.8 f = f + + f in which f + 0 f 0. (.9 The upwind scheme can be applied to (.8 by naturally using flux splitting u n+ u n t + f + (u n f + (u n + f (u n + f (u n =0. (.0

3 POSITIVITY PROPERTY OF FLUX-SPLITTING SCHEMES 03 It is obvious that the scheme (.0 can be recast in the form of (.6 with a +n = f + (u n f + (u n u n = f + (ξ un n 0 a n = f (u n + f (u n u n + = f (η un n 0 (. in which ξ n lies between u and u η n between u and u + respectively. Moreover we have a +n t + t a n 0 (. under the following assumption t ( max f + + max f. (.3 Therefore the combination of (. (. and (.6 shows that the first-order flux splitting method (.0 for scalar conservation law is TVD by using the argument in [6] provided that the CFL-like condition (.3 is satisfied. In a second-order approximation the positive and negative fluxes can be approximated by piecewise linear function f +n (x =f +n + s +n (x x f n (x =f n + s n (x x (. for x / <x<x +/ where the notation f +n used. The slopes are determined by a limiter function s ±n = f ±n f ±n = f + (u n φ 0( f ±n + f ±n f ±n f ±n f n = f (u n is. (.5 In this paper we only consider minmod limiter ( φ 0 (θ =max 0min(θ (.6 or V an Leer limiter (see [] φ 0 (θ = θ + θ +θ. (.7 Both limiters are symmetric i.e. aφ 0 ( b a =bφ0 ( a b so that we can write φ 0 (a b =aφ 0 ( b a =bφ0 ( a b (.8 without ambiguity. Consequently the second-order flux splitting scheme is given by u n+ u n + f +n (x +/ f +n (x / t + f (.9 n (x +/ + f +n (x / + =0. Using the notations in (.-(.8 we can rewrite the above scheme as u n+ u n t + ψ +n f +n f +n + ψ n f n + f n =0 (.0a

4 0 C. WANG AND J. LIU with the two coefficients ψ +n ψ n =+ φ0( f +n + f +n f +n f +n φ0( f +n f +n f +n f +n + f n + φ0( f n f n =+ φ0( f n f n + f n f n + f n. (.0b As in (. by applying the intermediate value theorem for f + f the secondorder upwind scheme for the nonlinear problem can still be recast in the form of (.6 namely a +n = ψ +n f + (ξ n a n = ψ n f (η n (. in which ξ n lies between u and u andη n between u and u + respectively. Both the minmod and V an Leer limiter functions satisfy 0 φ0 (θ θ so that the two coefficients ψ +n positivity of a +n CFL-like condition and a n and 0 φ 0 (θ (. and ψ n are between 0 and which results in the. Furthermore (. is satisfied under the following t ( max f + + max f (.3 due to the second property of φ 0 in (.. Therefore the stability and TVD property of the scheme (.0 follows directly from Harten s argument [6]. As a result the numerical scheme converges to a weak solution. It can be observed that the CFL-like assumption (.3 is more strict than the first-order version (.3 due to the usage of a limiter function. In other words the second-order scheme (.0 can be viewed as a stable perturbation of the first-order one (.0 due to the fact that both ψ +n and ψ n are positive coefficients bounded by. This simple stability argument no longer holds for a general system of conservation laws U t + F (U x =0 (. since no TV-stable property can be automatically applied to the analysis of a nonlinear system. Nevertheless a direct extension of the above scheme enoys a great success. The flux vector splitting (FVS method reconstructs the flux F as F (U =F + (U+F (U (.5 where the Jacobians of F + F have only non-negative and non-positive eigenvalues respectively. To achieve high order accuracy we apply the second-order flux vector splitting scheme to each component = U n t +n (F +/ F +n U n+ / t (F n +/ F n / (.6a

5 POSITIVITY PROPERTY OF FLUX-SPLITTING SCHEMES 05 where the p th component of F ± +/ F ±(p +/ is given by F +n(p +/ = F +n(p + +n(p (F F +n(p F n(p +/ = F n(p + + n(p (F+ F n(p φ 0( F +n(p + F +n(p F +n(p F +n(p φ 0( F n(p + F n(p + F n(p + F n(p (.6b and the limiter function φ 0 is chosen as either (.6 or (.7. The approach of the FVS scheme goes back to Van Leer []. See also [3]. The second order version (.6 is in fact the same as the convex ENO scheme discussed by Liu and Osher in [9]. Meanwhile a framework to provide some theoretical guides for the general systems of conservation laws has been proposed by P. Lax and X.-D. Liu in [8] with the concept of positive scheme which is motivated by the fact that the only functional known to be bounded for solutions of linear hyperbolic equation is energy as proven by Friedrichs in []. A conservative scheme of the form U n+ = U + t (F +/ F / (.7 where F is a consistent numerical flux is called positive if U n+ can be written as = C K U+K n (.8 K so that the coefficient matrices C K which themselves depend on all the U +K that occur in (.7 have the following properties: each C K is symmetric and positive i.e. C K 0; (.9a C K = I (.9b U n+ K C K =0 except for a finite set of K. (.9c Then they argued that for positive schemes the numerical solution U n is L stable in some suitable sense. Note that for the case of a scalar equation (.8 the positivity (.9 is reduced to the convex combination form (.6 while the combination coefficients c c c 0 t correspond to a +n a n a +n + a n respectively. In the second-order flux vector splitting scheme the general form can be written as (3.5 in Section 3 yet the form of the combination matrices C K depends on the concrete splitting. It should be noted that the symmetry of the coefficient matrices C K plays an important role in the L -stability of the positivity scheme [8]. However a symmetric Jacobian matrix can hardly be found for a system of hyperbolic conservation laws in terms of physical variables. Instead a symmetrizable system is taken into consideration in [8] B 0 (U t U + B(U x U =0 (.30 in which B is symmetric B 0 is positive and its symmetric square root is denoted as S i.e. B 0 = S. Multiplying (.30 by B0 gives t t U + A(U x U =0 with A = B0 B. (.3

6 06 C. WANG AND J. LIU Moreover it turns out that SAS is symmetric. For such a symmetrizable system the numerical scheme is defined as positive if it can be recast in the form of (.8 and the matrices D K (J =S(JC K (JS (J (.3 satisfy the following conditions D K differs by O( t from a symmetric matrix (.33a the symmetric part of D K has non-negative eigenvalues D K = I K (.33b (.33c D K =0 except for a finite set of K. (.33d The L -stability of the positive scheme satisfying (.33 was established in [8]. Due to the fact that the compressible Euler equations are not symmetric but symmetrizable they constructed a second-order positive scheme in the article by using Roe matrix decomposition which was proven to be very efficient. In this paper we show the positivity and weak stability of the second-order limiter schemes with some well-known FVS in the computation of the compressible Euler equation including Steger-Warming and Van Leer splittings proposed in [3] and []. We recall that the gas equation is symmetrizable only in terms of the velocity pressure and entropy variables. Its conservative form which is comprised of the dynamic equations for the density momentum and the total energy is not. Yet we have to keep the shock-capturing scheme in the conservative form (.7 consistent with the conservation laws to get correct shock speed in view of Lax- Wendroff theorem. The key point in this article is the reformulation of the positive and negative fluxes in the Steger-Warming and Van Leer splitting so that the corresponding matrices are either symmetric or symmetrizable and keep non-negative and non-positive eigenvalues respectively. This fact is reported in Proposition. in Section below. The basic idea is to use the property of Euler flux that it is a homogeneous function of degree one in the variable. A direct consequence of it is the positivity of the first order scheme which comes from the rewritten forms of the splitting fluxes F + F. It is shown in Section 3 that the second order limiter scheme is a stable perturbation of the first order one. The corresponding second order scheme using Steger-Warming or Van Leer splitting is proven to be positive under some CFL-like conditions in Section and 5 due to the representation formula that the numerical U n+ is a positive combination of U n. Furthermore both splitting schemes preserve the positivity of density and energy variables. To achieve this the fluxes F + and F need to be formulated in another way so that the corresponding matrices A + A are diagonal with respect to the first and third components. The diagonal elements are also bounded by fluid velocity and sound speeds. Then the density and energy at time step t n+ is shown to be a positive combination of their values at time step t n which indicates the positivity preserving property of the scheme under a CFL-like condition. A variety of formulations of F + and F are possible because of the nonlinearity of Euler flux. Moreover we note that condition (.33b is in fact a consistency requirement. Since the system is nonlinear we can write (.7 in several different forms. To simplify our presentation by observing that all the flux splitting schemes we consider here are in a consistent conservative form we modify the condition (.33c

7 POSITIVITY PROPERTY OF FLUX-SPLITTING SCHEMES 07 by C K = I + O( t if U n is Lipschitz continuous at time step n. (.3 K Note that in the scalar case condition (.9a plus C K =+O( t (.35 K can ensure the TV-stable property of the numerical scheme. In this article the requirement of positivity is set to be each C K is either symmetric or symmetrizable and C K 0 (.36a up to O( t difference ; C K = I + O( t if U n is Lipschitz continuous at time step n ; (.36b K C K =0 except for a finite set of K. (.36c. The compressible Euler equations and symmetrizable flux splitting. The -D system of the Euler equations is given by t U + x F (U =0 U = ρ m m F = ρv + p (. E v(e + p where ρ v m = ρv and E are density velocity momentum and total energy respectively and the state equation for the pressure is given by p =(γ (E ρv. The eigenvalues of the Jacobian matrix of F (U are λ = v c λ = v λ 3 = v + c (. with the sound speed c = γp ρ. The constant γ depends on the gas and usually ranges from to 3. For example such constant can be taken as γ =. for the air. Our analysis below is valid for any <γ<3... Steger-Warming flux splitting. Through the similarity transformation on the flux vector using the property F (U =A(UU A(U =F (U due to the fact that the flux vector is a homogeneous function of degree one the Steger-Warming splitting of F in [3] is given by F (U =F + (U+F (U =Q Γ + QU + Q Γ QU (.3 where A = Q ΓQ and Γ ± = diag(λ ± λ± λ± 3 λ± i =(λ i ± λ i /i= 3. The corresponding F + and F for the subsonic case 0 v c read F + SWS = ρ γ F SWS = ρ γ γv + c v (γ v +(v + c (γ v 3 + (v + c3 + 3 γ (γ (v + cc v c (v c (v c3 + 3 γ (γ (v cc. (.a In the supersonic case v>c F + = F F =0. (.b

8 08 C. WANG AND J. LIU The subvectors F + F for other cases including the subsonic case c v 0and supersonic case v< c can be obtained in a similar way... Van Leer Splitting (VLS. In [] Van Leer developed the splitting which is differentiable even at sonic points in terms of the local Mach number M M = u/c. For M the eigenvalues of Jacobian of F are all positive and thus F + VLS = F F VLS = 0. Similarly F + VLS =0andF VLS = F for M. The Van Leer splitting is given as follows for <M< f ± F ± VLS = f ± ((γ v ± c/γ (.5 f ± ((γ v ± c /(γ where f ± = ±ρc(m ± /. The Jacobian of the positive flux (F + VLS has two positive and one zero eigenvalues while (F VLS has two negative and one zero eigenvalues..3. Symmetrizable reformulation of the splitting fluxes F + and F. Note that the positive and negative matrices Q Γ + Q Q Γ Q of the Steger-Warming splitting are not symmetrizable in its original form. The same is for the Van Leer splitting. The key observation in this section is that the representation form of such matrices is not unique due to the nonlinearity of the fluxes. The matrices can be reformulated to be symmetrizable and keep positive and negative eigenvalues respectively. The following proposition is crucial to the positivity argument of both flux splittings presented in later sections. Proposition.. The fluxes F + F in either Steger-Warming splitting (. or Van Leer splitting (.5 can be represented as F + = A + U F = A U (.6a in which A + A are either symmetric or symmetrizable and A + 0 A 0. (.6b Moreover A + A is a diagonal matrix with non-negative elements. A direct consequence of Proposition. is the positivity of the first order method using either flux splitting. Corollary.. The first order flux splitting method = U n t +n (F F +n U n+ t (F n + F n (.7 with the positive and negative fluxes F + F given by either Steger Warming (. or Van Leer (.5 is positive under some CFL-like conditions namely (.-(. or (.9 below. Proof. The insertion of (.6a into the first order scheme (.7 shows that U n+ = U n λ(a +n U n A +n U n λ(a n + U + n A n U n = λa +n U n +( λa n + U + n +(I λa +n + λa n U n with the constant λ = t. By taking the notation C = λa +n (.8 C = λa n + C 0 = I λ(a +n A n (.9a

9 POSITIVITY PROPERTY OF FLUX-SPLITTING SCHEMES 09 we arrive at U n+ = C U n + C 0 U n + C U+ n. (.9b It is indicated by Proposition. that both C and C are either symmetric or symmetrizable and keep non-negative eigenvalues. In addition since A + A is a diagonal matrix with non-negative elements C 0 is also positive symmetric if λ is bounded by the inverse of the eigenvalues which is shown to be a CFL-like condition as can be seen in Remarks below. The property that C +C 0 +C = I + O( t can be verified by the smoothness of the positive and negative matrices A + A with respect to the physical variables under the assumption that the solution is Lipschitz continuous. The proof of Proposition. is a constructive one and is based on the following lemma. Its proof is straightforward and we omit it. ( a a Lemma.. A matrix A = is symmetrizable and keeps non-negative a a eigenvalues if a > 0 a > 0 a a > 0 a a a a ; (.0a it is symmetrizable and keeps non-positive eigenvalues if a < 0 a < 0 a a > 0 a a a a. (.0b Proof of Proposition. for the case of Steger-Warming splitting: (a In the subsonic case 0 v<c using the state equation E = ρv + and p = γ ρc we can rewrite F + F as F + = = F = = γ γ ρv + γ ρc γ γ ρv + γ ρvc + γ ρc γ γ ρv3 + 3 γ ρv c + (γ ρvc + ( γ γ v + γ cρ γ v + γ γ cρv + 8 (v c ρv + a + 33 E ( 5γ γ 3 γ E γ ρv γ ρc γ ρv γ ρvc + γ ρc γ(γ ρc3 γ ρv3 3 γ ρv c + (γ ρvc γ γ γ(γ ρc3 (c vρ [ c ( γ + γv ] ρv ( γ E 3 γ 8 (v c ρv + a 33 E p γ (. (. so that F + F can be represented as F + = A + U F = A U with a + A a = 0 a + γ A 0 0 = 0 a γ 3 γ 0 8 (v c a + 3 γ (v c a 33 (.3

10 0 C. WANG AND J. LIU in which the coefficients have the following estimates a + = γ v + γ γ c a+ = 5γ γ v + γ γ c γ a + 33 = γ ρv3 + 3 γ ρv c + (γ ρvc + γ(γ ρc3 3 γ 8 E 3 γ γ c a+ 33 γ +γ v + γ +3 γ γ c a = γ (c v a = γ ρv(v c [ c ( γ + γv ] (c v γ a 33 = γ ρv3 3 γ ρv c + (γ ρvc γ(γ ρc3 3 γ 8 ρv(v c E γ (v c a 33 3 γ (v c < 0. γ + (. The verification of the above estimates is straightforward by algebraic calculation of the physical variables. The detail is omitted here. From (. we have (γ (3 γ 36 3 γ γ(γ + (v c γ a + a γ γ c a a 33 c γ 3 γ 8 (v c 3 γ 8 (v c (.5 for <γ<3. The application of Lemma. shows that A + A are symmetrizable and keep non-negative and non-positive eigenvalues since the criteria (.0a (.0b can be verified by the usage of (.5. It can be seen that the matrix A + A has the form A + A = a + a a + a a + 33 a 33 (.6 which is diagonal and keeps only non-negative eigenvalues. Moreover the estimate (. leads to a + a = γ γ v + γ c a+ a = γ v + γ γ c a + 33 a 33 γ +γ v + γ +γ 3 (.7 γ γ(γ c so that the eigenvalues are bounded by the linear combination of the velocity and sound speed. This was used in the proof for the positivity of the first order method. The other subsonic case c <v 0 can be dealt with in the same manner. That finishes the proof of Proposition. in the subsonic case of the Steger-Warming splitting. (b In the supersonic region v c the positive and negative fluxes are given by F + = F = ρv ρv +p ρv m v(e+p E E F =0=A U (.8

11 POSITIVITY PROPERTY OF FLUX-SPLITTING SCHEMES where A = 0. Then we can rewrite F + F in the same form as F + = A + U F = A U with A + = a + = v a a a + 33 a+ = ρv + p ρv A =0 a + 33 = v(e + p E. (.9 The following estimate for the components in F + is also straightforward by the algebraic calculation 0 a + = v 0 a+ = ρv + p ρv v + γ c 0 v(e + p a+ 33 v + E γ c. (.0 In this case A + A is exactly A + presented in (.9 which is diagonal and its eigenvalue estimate is given in (.0. For another supersonic case v c F + =0andF can be similarly represented and an analogous estimate to (.0 can be obtained. Then we finish the proof of Proposition. for the Steger-Warming splitting. Remark. As can be seen in both the subsonic and supersonic cases the matrix C 0 = I λ(a + A is diagonal and its diagonal elements are non-negative under the constraint t ( a + ii a ii i = 3. (. With the usage of the estimates (.7 (.0 the constraint (. is satisfied provided that t ( γ max +γ γ γ γ +γ 3 ( v + c. (. γ(γ As a result the first order Steger-Warming splitting method is positive under the CFL-like condition (.. Proof of Proposition. for the case of Van Leer splitting:

12 C. WANG AND J. LIU In the subsonic case 0 M< F + and F can be represented as F + = = = ρc(m +M +/ ρc M +M+ (γ v+c γ ρc M +M+ [(γ v+c] (γ ( v + M + cρ γ γ (M +ρv + γ ρv + γ+3 γ ρvc + γ ρc [ γ+3 = A + U ρc M +M+ [(γ v+c] (γ ( v + M + ( cρ γ c + γ γ (M ++ γ γ (3 γ 7 (v c + a + 33 E ] v ρv + γ E (.3 F = = = = A U. ρc M M+ ρc M M+ (γ v c γ [(γ v c] (γ ρc M M+ [ c + ( Mv] ρ γ γ ( Mρv + γ ρv γ+3 γ ρvc + γ ρc ρc M M+ [(γ v c] (γ [ ( γ+3 γ c [ c + ( Mv] ρ ] γ γ ( M v (3 γ 7 (v c ρv + a 33 E γ + γ ρv ( γ E (. in which the matrices A + A have the form a A + = 0 a + γ (3 γ 0 7 (v c a + 33 a 0 0 A = 0 a γ (3 γ 0 7 (v c a 33 (.5a

13 POSITIVITY PROPERTY OF FLUX-SPLITTING SCHEMES 3 a + = v + M + a + = γ +3 ( γ γ c + γ a = c + ( Mv a [ γ +3 ( a = γ c c a + 33 = F +(3 (3 γ 7 (v c ρv E (M ++ γ γ γ + γ γ v 33 = F (3 (3 γ 7 (v c ρv E ] ( M γ v. (.5b The following estimates can be verified by a careful calculation. omitted. The detail is 0 a + ( v + c γ +3 γ c 3 γ v a + γ γ v + γ +3 γ γ c c a+ 33 γ(γ +γ 3 v + γ (γ + γ + c (c +3 v a 0 ( γ γ v + γ +3 ( γ +3 γ γ c a γ c 3 γ v [ γ(γ +γ 3 ( γ (3 ] γ 33 γ v + + c a (3 (γ + γ + 6 (γ + c( M. (.6 Similarly the estimate (.6 results in a + a+ 33 a a 33 ( γ+3 γ 3 γ (3 ( γ γ +3 (γ + γ c (3 γ (γ c γ (3 γ (γ (v c 7 3 γ c ( M γ (3 γ (v c 7 (.7 for <γ<3. Moreover by applying Lemma. and using the estimate (.7 we conclude that A + A are symmetrizable and keep non-negative and non-positive eigenvalues respectively. It can be seen that the matrix A + A has the form A + A = a + a a + a a + 33 a 33 (.8 which is diagonal and keeps only non-negative eigenvalues. The eigenvalues are bounded by the linear combination of the velocity and sound speed a + a v + 3 c a+ a γ γ v + γ c a + 33 a 33 γ(γ +γ 3 ( γ (3 (.9 γ v + + c. γ + γ + 3

14 C. WANG AND J. LIU For another subsonic case <M 0 F + and F can be similarly represented and an analogous estimate to (.5 can be obtained. In the supersonic case either M orm the representation and the estimates (.8 (.9 are also valid. Then we finish the analysis of the Van Leer splitting thus complete the proof of Proposition.. Remark. In both the subsonic and supersonic cases the matrix C 0 = I λ(a + A is diagonal and its diagonal elements are non-negative under the constraint (. which is assured to be satisfied if t ( max γ(γ +γ 3 γ + γ γ γ + (3 γ + ( v + c (.30 3 with the usage of the estimates (.9 (.0. As a result the first order Van Leer splitting method is positive under the CFL-like condition ( Analysis of second-order flux splitting. In this section we give a general analysis for the second-order FVS scheme (.6 applied to a system of conservation laws (.. We need to rewrite the scheme for the convenience of the positivity proof in the following sections. Substituting (.8 into (.6b gives φ 0 (F +(p + F +(p φ 0 (F (p + F (p F +(p F (p F +(p F (p +(p = (F + F +(p =(F +(p (p = (F+ F (p =(F (p φ 0( F +(p F +(p F +(p + F +(p F +(p φ0( F +(p + F +(p F +(p F +(p φ 0( F (p F (p F (p + F (p F (p φ0( F (p + F (p F (p F (p (3.a (3.b in which F ±n(p denotes the p-th component of the positive (negative flux F ±n. Consequently the positive flux terms F +n +/ F +n / can be rewritten as F +n +/ = F +n + φ0( F +n + F +n F +n F +n = F +n + φ0 (F +n + F +n F +n F +n = F +n + Φ+ +n +/ (F F +n (3. where Φ + +/ = diag(φ+( +/ φ+( +/...φ+(m +/ and φ +(p +/ = φ0( F +n(p + F +n(p for p m. (3.3 F +n(p F +n(p

15 POSITIVITY PROPERTY OF FLUX-SPLITTING SCHEMES 5 Similarly we can express F +n / as F +n / = F +n + φ0 (F +n F +n F+n F +n = F +n + (3. Φ+ +n / (F F +n with the diagonal matrix Φ + / = diag(φ+( / φ+( /...φ+(m / and φ +(p / = φ0( F +n F +n F +n F +n for p m. (3.5 It is obvious from the TVD property (. that 0 φ +(p +/ φ+(p / for p m. (3.6 Therefore a combination of (3. and (3. gives F +n +/ F +n +n / =(F F +n + +n Φ(F F +n (3.7 where Φ =diag( φ +( φ +( φ +(3 and φ +(p = φ +(p +/ φ+(p / p =... m. (3.8 Clearly we have φ +(p p =... m. Now (3.7 becomes F +n +/ F +n / =(I + Φ + (F +n F +n =Ψ+ (F +n F +n (3.9 where Ψ + =(I + Φ + =diag(ψ +( ψ +( ψ +(3 and ψ +(p =+ φ +(p p =... m. (3.0 Combining the results of (3.6 (3.8 (3.0 shows that 0 Ψ + I. A similar procedure can be applied to rewrite F which gives F n +/ = F n + φ0 (F n + F n + F n + F n = F n + Φ n +/ (F+ F n (3.a F n / = F n = F n φ0 (F n + F n F n Φ n / (F+ F n F n (3.b with Φ +/ = diag(φ ( +/ φ ( +/...φ (m +/ Φ / = diag(φ ( / φ ( /...φ (m / and for p =... m +/ = φ0( F n(p + F n(p + φ (p F n(p + F n(p φ (p / = φ0( F n(p F n(p F n(p + F n(p (3.c. (3.

16 6 C. WANG AND J. LIU Furthermore a similar argument shows that 0 Φ ±/ I. As a result we arrive at F n +/ F n / = = Ψ (F n + F n with the diagonal matrix F n + F n + (Φ / Φ n +/ (F+ F n (3.3 0 Ψ = I + (Φ / Φ +/ I. (3. Finally the combination of (.6 (3.9 (3. leads to U n+ = U n t +n (F +/ F +n / t n (F +/ F n / = U n t Ψ+ (F +n F +n t Ψ (F n + F n (3.5 where 0 Ψ ± I. The above derivation shows that the second-order FVS method (3.5 is also a stable perturbation (with the addition of Ψ ± between 0 and I of the first-order scheme in the case of a nonlinear system. Its positivity comes from the splitting of the flux and the CFL-like conditions stated below.. Positivity property of Steger-Warming splitting (SWS. Applying the flux splitting F + F into (.6 we get the second-order Steger-Warming splitting scheme: U n+ = U n t +n (F +/ F +n / t n (F +/ F n / (.a F +n +/ = F +n + φ0( F +n + F +n F +n F +n F n +/ = F n + + φ0( F n + F n + F n + F n (.b where F + and F are given by (.. Here φ 0 is the minmod limiter or V an Leer limiter and φ 0 (a b for each component. Theorem.. The second-order Steger-Warming splitting scheme (.a (.b is positive under a CFL-like condition where a = γ +γ γ t ( max a v n + b c n + γ γ b = γ + γ +γ 3 γ(γ (.a. (.b Proof of Theorem. First applying the argument in Section 3 gives U n+ = U n t +n (F +/ F +n / t n (F +/ F n / = U n t Ψ+ (F +n F +n t Ψ (F n + F n (.3

17 POSITIVITY PROPERTY OF FLUX-SPLITTING SCHEMES 7 where 0 Ψ ± I as indicated in (3.5. The rewritten forms of F + and F as given in Proposition. can be used to substitute into (.3. For conciseness and without loss of generality we only consider the three cases: ( subsonic region with the assumption 0 vi n <c n i i= + ; ( supersonic region with the assumption vi n c n i i= +; (3 sonic transient region with the assumption v n cn and 0 vn i <c n i for i = +. The other subsonic supersonic or transient regions can be dealt with in the same manner due to the detailed reformulation of F + F given in Section... Subsonic region. Substitution of (.-(. into (.3 gives U n+ = U n λψ + (A +n U n A +n U n λψ (A n + U + n A n U n = λψ + A +n U n +( λψ A n + U + n +(I λψ + A +n = C U n + C 0 U n + C U n + where in the last step we denoted C = λψ + A +n + λψ A n U n (.a C = λψ A n + C 0 = I λψ + A +n + λψ A n. (.b Then the remaining work is to confirm that the matrices C C 0 C satisfy the positivity property given by (.36a (.36b (.36c. It can be seen that C = c c c 3 0 c 3 c 33 c = λψ+( ( γ v n + γ γ cn c = λψ+( ( 5γ γ v n + γ γ cn c 3 = γ λψ+( c 3 = 3 γ λψ+(3 8 (vn c n c 33 = λψ+(3 (a + 33 n (.5a (.5b which is symmetrizable and has only non-negative eigenvalues by the estimates (. (.5 and Lemma.. The matrix C has a similar form to that of C C = c = λψ ( γ (cn + v+ n c = λψ ( γ c c c 3 c 3 c 33 [ c n + ( γ + γv n +] c 3 = λψ ( γ (.6a (.6b c 3 = λψ (3 3 γ 8 (vn + c n + c 33 = λψ (3 (a 33 n + Again the estimates (. (.5 and Lemma. indicate that C is symmetrizable and has only non-negative eigenvalues.

18 8 C. WANG AND J. LIU The matrix C 0 can be represented as C 0 = c c 0 c 0 3 c 0 3 c 0 33 (.7a where c 0 = λψ +(( γ v n + γ γ cn λψ ( γ (cn v n c 0 = λψ +(( 5γ γ v n + γ γ cn c 0 3 = λ γ ( ψ +( + ψ ( c 0 33 = λψ +(3 (a + 33 n λψ (3 (a 33 n. λψ ( γ ( c n ( γ + γv n c 0 3 = λ 3 γ 8 (vn c n ( ψ +(3 + ψ (3 (.7b The three diagonal elements of C 0 : c 0 c 0 c 0 33 can be controlled by the following argument with the usage of the preliminary estimate (.7 ( (γ c 0 0 if max λ v n + n γ γ cn ( (γ c 0 0 if max λ v n + n γ γ cn ( γ c if max λ +γ v n + γ +γ 3 c n. n γ γ(γ (.8 In addition we note that both ψ (3 ψ +(3 and ψ ( ψ +( are O( so that c 0 3 c 0 3 are O( t if some suitable continuity assumption for the numerical solution is satisfied. Then we conclude that C 0 differs O( t from a diagonal positive matrix under such condition. In the general case we can still get the positivity and symmetrizable property of C 0 by adusting the coefficients in c 0 3. The technical detail is omitted. Next we verify (.36b. The direct calculation shows that C + C 0 + C = s s 0 0 s 3 s 33 (.9a

19 POSITIVITY PROPERTY OF FLUX-SPLITTING SCHEMES 9 in which the diagonal elements have the following form s =+λψ +(( F +n( ρ n F +n( ρ n s =+λψ +(( (a + n (a + n + λψ (( F n( ρ n F n( + ρ n + + λψ (( (a n (a n + s 3 = λ 3 γ 8 ψ+(3( (v n c n (v n c n +λ 3 γ 8 ψ (3( (v n c n (v n + c n + s 33 =+λψ +(3( F +n(3 E n F +n(3 E n + λψ (3( F n(3 E n F n(3 + E+ n. (.9b Since F + F are smooth functions of the fluid variables ρ m E it can be concluded that C + C 0 + C = I + O( t if the numerical solution is Lipschitz continuous at time t n thus (.36b is satisfied. In addition C k = 0 except for K = 0 which gives that the scheme with the flux splitting (3. is positive in the subsonic region... Supersonic case. Substitution of (.8 (.9 into (.3 gives the formula (. with the same notation of C C 0 C except that the form of F ± A ± has been changed. Since F n =0A n = 0 we actually have where U n+ = C U n + C 0 U n (.0a C = λψ + A +n C 0 = I λψ A +n. (.0b In more detail λψ +( v n 0 0 +n( C = +( F 0 λψ 0 ρ n vn (. 0 0 λψ +(3 (a + 33 n λψ +( v n 0 0 +n( C 0 = +( F 0 λψ 0 ρ n vn. (. 0 0 λψ +(3 (a 33 n The estimate (.0 and the fact that 0 Ψ ± I ensure the positivity of C and C 0 if ( λ max v n + n γ cn (. which implies that the CFL-like assumption (. is a sufficient condition for (.36a. For the verification of (.36b a careful computation shows that C + C 0 = s s 0 (.5a 0 0 s 33

20 0 C. WANG AND J. LIU with the diagonal elements s =+λψ +(( F +n( ρ n s =+λψ +(( F +n( ρ n vn s 33 =+λψ +(3( F +n(3 E n F +n( ρ n F +n( ρ n vn F +n(3 E n. (.5b Therefore C + C 0 = I + O( t ifu n is Lipschitz continuous with respect to x. Thus the condition (.36b is guaranteed. The condition (.36c is obvious since C K = 0 except for K = 0. Then we finish the proof of the positivity property in the supersonic region. Remark 3. We observe that both C and C 0 are diagonal hence symmetric positive in the supersonic region..3. Sonic transient region. We consider the case such that v n cn and 0 v n i <c n i for i = +. The previous argument shows that F +n( ρ n = v n + F +n( + ρ n = γ + γ F +n( = F +n( + = F +n(3 F +n( ρ n = γ v n + γ γ cn 0 F +n( m n v n + γ cn v+ n + γ cn + ( 5γ γ v n + γ γ cn m n + γ E n ( 5γ γ v n + γ γ cn m n + + γ E+ n = 3 γ 8 (vn c n +(a + 33 n E n F +n(3 + = 3 γ 8 (vn + c n + +(a + 33 n +E+ n 0 F +n(3 E n v n + 3 γ γ cn γ cn (a + 33 n γ +γ γ 3 γ γ cn + (a + 33 n + γ +γ v+ n + γ +3 γ γ cn + v n + γ +3 γ cn (.6a

21 POSITIVITY PROPERTY OF FLUX-SPLITTING SCHEMES F n( = γ (cn v n ρ n F n( = [ c n γ ( γ + γv n F n( + = γ F n(3 F n( + = γ (cn + v n +ρ n + ] m n + γ E n [ c n + ( γ + γv+] n m n + + γ E+ n = 3 γ 8 (vn c n +(a 33 n E n F n(3 + = 3 γ 8 (vn + c n + +(a 33 n +E+ n γ (vn c n (a 33 n 3 γ γ + (vn c n < 0 γ (vn + c n + (a 33 n + 3 γ γ + (vn + c n + < 0 F n =0. (.6b The formula for U n+ in (.a involves F +n F +n F n and F n +. At the supersonic point and the subsonic point +F + and F can be expressed as F +n = A+n U n F n + = A n + U n + (.7 in which A +n A+n + have the same form as in (.9 (.3 respectively so that the estimates in (.6 can be applied. At the transient point we assume that ɛ vn c n if the solution is Lipschitz continuous. To facilitate the positivity argument below we express F n form as that of F +n i.e. in a similar form as that of F n + F +n in a similar F +n = A +n U n F n = A n U n (.8 in which A n + An take the form of (a + A n n 0 + = 0 (a + n (a + 33 n (a A n n 0 0 = 0 (a n γ 3 γ 0 8 (vn cn (a 33 n (.9a (a + n = γ v n + γ γ cn (a + n = F +n( m n (a + 33 n = F +n(3 E n (a n = γ (cn v n (a n = [ c n γ ( γ + γv n ] γ (vn c n (a 33 n 3 γ γ + (vn c n < 0. (.9b

22 C. WANG AND J. LIU The estimates in (.6 are valid to control the above terms except for (a + n.the assumption of Lipschitz continuity of the solution shows that F +n( m n = F +n( m n + O(h which leads to 0 F +n( m n v n + γ cn + O(h. (.0 Similar to the subsonic and supersonic cases we have U n+ = C U n + C 0 U n + C U+ n (. if we denote C = λψ + A +n C = λψ A n + C 0 = I λψ + A +n + λψ A n. (. In more detail the matrices C C 0 C take the following forms λψ +( v n 0 0 +n( C = +( F 0 λψ 0 (.3 ρ n vn 0 0 λψ +(3 (a + 33 n c 0 0 C = 0 c c 3 0 c 3 c 33 c = λψ ( γ (cn + v+ n c = λψ ( [ c n γ + ( γ + γv+] n c 3 = λψ ( γ c 3 = λψ (3 3 γ 8 (vn + c n + c 33 = λψ (3 (a 33 n + with C 0 = c c 0 c 0 3 c 0 3 c 0 33 c 0 = λψ +(( γ v n + γ γ cn λψ ( γ (cn v n +n( c 0 F +( = λψ m n λψ ( [ c n γ ( γ + γv n ] c 0 3 = λψ ( γ c 0 3 = λψ (3 3 γ 8 (vn c n +n(3 c 0 F +(3 33 = λψ E n λψ (3 (a 33 n. (.a (.b (.5a (.5b Using a similar argument as in the subsonic and supersonic regions we conclude that C C 0 C are symmetric or symmetrizable and keep non-negative eigenvalues and C + C 0 + C = I + O( t ifu n is Lipschitz continuous. Thus the scheme (. with Steger-Warming flux splitting is also positive in the transient region provided the CFL-like condition (. is satisfied. Theorem. is proven.

23 POSITIVITY PROPERTY OF FLUX-SPLITTING SCHEMES 3.. Positivity of density and total energy. In this subsection we prove the second order Steger-Warming splitting method preserves positivity of hydrodynamic variables including density and total energy. To achieve this we need to formulate F + F so that the corresponding matrices A + A are diagonal with respect to the first and third components. At a supersonic point the form of A + A is still the same as (.9 and the estimate (.0 is valid. At a subsonic point 0 v<c the corresponding matrices can take another form a + A a = 0 a + γ A 0 0 = 0 a γ 0 0 a + ( a 33 where a ± a± are the same as in (.. The third diagonal element a± 33 reads a + 33 = F +(3 E = a 33 = F (3 E = γ ρv3 3 γ γ ρv3 + 3 γ ρv c + (γ ρvc + γ(γ ρc3 E γ ρv c + (γ ρvc γ(γ ρc3 E (.7 and the following estimate can be derived 0 a + 33 = F +(3 E γ +γ v + γ +3 γ γ c c a 33 = F (3 E 0. (.8 Then the numerical scheme can be written in the form of either (. (.0 or (.-(. at subsonic supersonic or transient region respectively. An important fact we should mention is that in all three cases subsonic supersonic or transient the three matrices C C 0 and C are diagonal with respect to the first and third components. As a result ρ n+ and E n+ can be written as positive combinations of ρ n ρn ρn + anden En En + respectively i.e. ρ n+ = λ ρ n + λ 0 ρ n + λ ρ n + (.9 E n+ = λ 3 E n + λ 03 E n + λ 3 E+ n (.30 where λ = λψ +( (a + n λ = λψ ( (a n + λ 0 = λψ +( (a + n + λψ ( (a n λ 3 = λψ +(3 (a + 33 n (.3 λ 3 = λψ (3 (a 33 n + λ 03 = λψ +(3 (a + 33 n + λψ (3 (a 33 n. It can be observed that λ i i = 0 andλ i3 i = 0 are non-negative provided the CFL-like condition t ( max a v n + b c n (.3a ( γ a = max γ +γ ( b = max γ γ γ + 3 (.3b γ with the usage of the estimates (. (.0 and (.8. Then the positivity of ρ n and En implies the positivity of ρn+ and E n+. In other words the numerical scheme preserves the positivity of density and total energy. However preservation of positivity for the pressure is not so obvious. It is an inferior result to [8] [5] in which the positivity of both the density and the internal energy was established.

24 C. WANG AND J. LIU Remark. We note that a variety of formulations of F + and F are possible because of the nonlinearity of Euler flux. The matrices A + and A as shown in (.6 in the subsonic case are not symmetrizable by Lemma. so that they cannot be used in the proof of Proposition.. However such a formulation is very useful in the proof of the positivity preserving property for the density and energy variables since the matrices are diagonal with respect to the first and third components and the diagonal elements are also bounded by fluid velocity and sound speeds as shown in ( Positivity of Van Leer Splitting (VLS. The main theorem in this section is stated below. Theorem 5.. The second-order scheme (.a(.b using Van Leer splitting (.5 is positive under a CFL-like condition t ( max a v n + b c n (5.a where ( a = max γ + γ(γ +γ 6 b = γ γ + γ + 8γ (3 γ +. γ + 3 (5.b Proof of Theorem 5. A similar analysis as in Section can be carried out using the same notation except for the difference of F + F. The solution U n+ is expressed as U n+ = U n t Ψ+ (F +n F +n t Ψ (F n + F n (5. with 0 Ψ ± I. Denote C = λψ + A +n then we have C = λψ A n + C 0 = I λψ + A +n λψ A n (5.3 U n+ = C U n + C 0 U n + C U+ n. (5. The positivity of the second order scheme is based on the rewritten forms of F + F as established in Proposition.. Similar to the presentation in Section we consider the following three cases: ( subsonic region with 0 vi n <c n i i= + ; ( supersonic region with vi n c n i i = + ; (3 sonic transient region with v n cn and 0 vn i <c n i for i = +. The other subsonic supersonic or transient regions can be treated in the same way. 5.. Subsonic region. In this case the matrices C C 0 C read λψ +( (a + n 0 0 C = 0 λψ +( (a + n +( γ λψ 0 λψ +(3 (3 γ 7 (v n cn λψ +(3 (a + 33 n (5.5 C = λψ ( (a n λψ ( (a n + ( γ λψ 0 λψ (3 (3 γ 7 (v+ n cn + λψ (3 (a 33 n + (5.6

25 with POSITIVITY PROPERTY OF FLUX-SPLITTING SCHEMES 5 C 0 = c c 0 c 0 3 c 0 3 c 0 33 c 0 = λψ +( (a + n λψ ( (a n (5.7a c 0 = λψ +( (a + n λψ ( (a n c 0 3 = λψ +( γ + λψ ( γ c 0 33 = λψ +(3 (a + 33 n λψ (3 (a 33 n c 0 3 = λ (3 γ (v n c n ( ψ +(3 + ψ (3 7 (5.7b in which a ± ii i = 3 are given by (.5b. Using the same argument as in Section and applying the estimate (.6 we conclude that C C 0 C are diagonal or symmetrizable and C K 0 (5.8 up to O( t difference C + C 0 + C = I + O( t if U n is Lipschitz continuous (5.9 under the following conditions ( max n vn + 3 ( γ cn max v n + n γ γ cn [ γ(γ max +γ 6 ( 8γ v n (3 ] (5.0 γ + + c n n γ + γ + 3 which can be assured by the CFL-like assumption (5.b. Thus the positivity is proven. 5.. Supersonic region. In this case the splitting flux F + F have exactly the same form as in the Steger-Warming splitting scheme. Therefore the argument in Section can be applied here. We omit the detail Sonic transient region. Without loss of generality we assume v n cn and 0 vi n <c n i for i = +. Using a similar argument as in the analysis of this case in Section we can represent C C 0 and C as the following λψ +( v n 0 0 C = 0 λψ +( (a + n 0 ( λψ +(3 (a + 33 n λψ ( (a n C = 0 λψ ( (a n ( γ + λψ 0 λψ (3 (3 γ 7 (v+ n cn + λψ (3 (a 33 n + (5. c C 0 = 0 c 0 c 0 3 (5.3a 0 c 0 3 c 0 33

26 6 C. WANG AND J. LIU with +n( c 0 F +( = λψ ρ n λψ ( (a n +n( c 0 F +( = λψ m n λψ ( (a n c 0 3 = λψ ( γ +n(3 c 0 F +(3 33 = λψ E n λψ (3 (a 33 n c 0 (3 (3 γ 3 = λψ (v n c n. 7 (5.3b It is straightforward to verify that (5.8 and (5.9 are valid under the CFL-like assumption (5.b. Therefore Theorem 5. is proven. 5.. Positivity of density and total energy. Once again we have to rewrite the fluxes F + F to make the corresponding matrices A + A are diagonal with respect to the first and third components in the proof of the property that the second order Van Leer splitting method preserves positivity of density and total energy. At a supersonic point the form of A + A is still the same as (.9 and the estimate (.0 is valid. At a subsonic point 0 v<c the corresponding matrices can take the same form as in (.6 and we have the following estimates 0 a + ( v + c (c +3 v a 0 0 a + 33 = F +(3 E γ(γ +γ 3 v + (γ + ( γ(γ +γ 3 v + (γ + γ γ + c γ γ + c a 33 = F (3 E 0. (5. As a result the numerical scheme can be recast in the form of either (. (.0 or (.-(.. We note that the three matrices C C 0 and C are diagonal with respect to the first and third components in all three cases: subsonic supersonic or transient. Consequently ρ n+ and E n+ can be written as positive combinations of ρ n and E n respectively i.e. ρ n+ = λ ρ n + λ 0 ρ n + λ ρ n + (5.5 E n+ = λ 3 E n + λ 03 E n + λ 3 E+ n (5.6 with λ = λψ +( (a + n λ = λψ ( (a n + λ 0 = λψ +( (a + n + λψ ( (a n λ 3 = λψ +(3 (a + 33 n (5.7 λ 3 = λψ (3 (a 33 n + λ 03 = λψ +(3 (a + 33 n + λψ (3 (a 33 n. The combination coefficients λ i i = 0 λ i3 i = 0 are non-negative under the following CFL-like condition t ( max a v n + b c n (5.8a

27 ( γ γ a = max γ POSITIVITY PROPERTY OF FLUX-SPLITTING SCHEMES 7 γ(γ +γ 6 5 ( b = max γ + γ 8γ γ + γ +3 γ (5.8b with the usage of the estimate (5.. Then the scheme preserves the positivity of density and energy. Once again the positivity of internal energy (henceforth the pressure field cannot be obtained from the above derivation thus the result reported in this section is inferior to that in [8] [5]. Acknowledgments. The research of J.-G. Liu was supported by the National Science Foundation under Grant DMS REFERENCES [] W. Anderson J. Thomas and B. Van Leer Comparison of finite volume flux vector splitting for the Euler equations AIAA Journal ( [] B. Engquist and S. Osher One-sided difference approximations for nonlinear conservation laws Math. Comp. 36 ( [3] H.ChoiandJ.-G.LiuThe reconstruction of upwind fluxes for conservation laws its behavior in dynamic and steady state calculations J. Comput. Phys. ( [] K. Friedrichs Symmetric hyperbolic linear differential equations Comm. Pure Appl. Math. 7 ( [5] J. Gressier P. Villedieu and J. M. Moschetta Positivity of flux vector splitting schemes J. Comput. Phys. 55 ( [6] A. Harten High resolution schemes for hyperbolic conservation laws J. Comput. Phys. 9 ( [7] A. Harten On a class of high resolution total-varational-stable finite-difference schemes SIAM J. Numer. Anal. : (98 3. [8] A. Harten B. Engquist S. Osher and S. Chakravarthy Some results on uniformly nonoscillatory schemes Appl. Numer. Math. ( [9] A. Harten P. Lax and B. Van Leer On upstream differencing and Godunov-type schemes for hyperbolic conservation laws SIAM Review 5 ( [0] A. Jameson and P. Lax Conditions for the construction of multi-point total variation diminishing difference schemes Princeton University report MAE [] S. Jin and J.-G. Liu The effects of numerical viscosities. I. Slowly moving shocks J. Comput. Phys. 6 ( [] S. Jin and Z. Xin The relaxation schemes for systems of conservation laws in arbitrary space dimensions Comm. Pure Appl. Math. 55 ( [3] P. Lax Hyperbolic systems of conservation laws and the mathematical theory of shock waves SIAM Philadelphia 973. [] P. Lefloch and J.-G. Liu Generalized monotone schemes discrete paths of extrema and discrete entropy conditions Math. Comp. 68 ( [5] R. LeVeque Numerical methods for conservation laws Birkhauser Verlag second edition 99. [6] P.-L. Lions and P. Souganidis Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton-Jacobi equations Numer. Math. 69 ( [7] M.-S. Liou and C. J. Steffen A new flux splitting scheme J. Comput. Phys. 07 ( [8] X.-D. Liu and P. Lax Positive schemes for solving multi-dimensional hyperbolic systems of conservation laws CFD J. 5: ( [9] X.-D. Liu and S. Osher Weighted essentially non-oscillatory schemes J. Comput. Phys. ( [0] X.-D. Liu S. Osher and T. Chan Weighted essentially non-oscillatory schemes J. Comput. Phys. 5 ( [] J.-G. Liu and Z. Xin Nonlinear stability of discrete shocks for systems of conservation laws Arch. Rational Mech. Anal. 5 (

28 8 C. WANG AND J. LIU [] P. Lyra K. Morgan and J. Periro TVD algorithms for the solution of the compressible Euler equations on unstructured meshes Int. J. Numer. Met. in Fluids 9 ( [3] J. Mandel and S. Deshpande Kinematic flux vector splitting for Euler equations Fluids 3 ( [] H. Nessyahu and E. Tadmor Non-oscillatory central differencing for hyperbolic conservation laws J. Comput. Phys 87 ( [5] S. Osher Convergence of generalized MUSCL schemes SIAM J. Numer. Anal. ( [6] S. Osher and F. Solomon Upwind difference schemes for hyperbolic systems of conservation laws Math. Comp. 38 ( [7] B. Perthame Second order Boltzmann schemes for compressible Euler equations in one and two space dimensions SIAM J. Numer. Anal. 9 (99 9. [8] B. Perthame and C.-W. Shu On positive preserving finite volume schemes for compressible Euler equations Numer. Math. 73 ( [9] R. Sanders and K. Prendergast The possible relation of the three-kiloparsec arm to explosions in the galactic nucleus Astrophysical Journal 88 (97. [30] C.-W. Shu Total-Variation-Diminishing time discretizations SIAM J. Stat. Comput. 9:6 ( [3] C.-W. Shu and S. Osher Efficient implementation of essentially non-ocsillatory shock capturing schemes II J. Comput. Phys. 83 ( [3] J. L. Steger and R. F. Warming Flux vector splitting of the inviscid gasdynamic equations with application to finite difference methods J. Comput. Phys. 0 ( [33] E. Tadmor Numerical viscosity and the entropy condition for conservative difference schemes Math. Comp. 3 ( [3] B. Van Leer Towards the ultimate conservative difference scheme I. The quest of monotonicity Springer lecture notes in physics 8 ( [35] B. Van Leer Towards the ultimate conservative difference scheme II. Monotonicity and conservation combined in a second order scheme J. Comput. Phys. ( [36] B. Van Leer Towards the ultimate conservative difference scheme III. Upstream centered finite-difference schemes for ideal incompressible flow J. Comput. Phys. 3 ( [37] B. Van Leer Towards the ultimate conservative difference scheme IV. A new approach to numerical convection J. Comput. Phys. 3 ( [38] B. Van Leer Towards the ultimate conservative difference scheme V. A second sequel to Godunov s method J. Comput. Phys. 3 ( [39] P. Woodward and P. Colella The numerical simulation of two dimensional fluid flow with strong shocks J. Comput. Phys. 5 ( Received April 00; revised August 00; final version January address: cwang@indiana.edu address: liu@math.umd.edu