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On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws

Author(s): Amiram Harten, Peter D. Lax, Bram Van LeerSource: SIAM Review, Vol. 25, No. 1 (Jan., 1983), pp. 35-61Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2030019.

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SIAM REVIEW ? 1983 Society for Industrial and Applied MathematicsVol. 25, No. 1, January 1983 0036-1445/83/2501-0002 $01.25/0

ON UPSTREAM DIFFERENCING AND GODUNOV-TYPE SCHEMES FORHYPERBOLIC CONSERVATION LAWS*AMIRAM HARTENt, PETER D. LAXt, AND BRAM VAN LEER?

Abstract. This paperreviews some of the recent developmentsin upstream difference schemes throughaunified representation, n order to enable comparisonbetween the various schemes. Special attention is given tothe Godunov-typeschemes that result fromusing an approximatesolutionof the Riemann problem.Forschemesbased on flux splitting, the approximate Riemann solution can be interpreted as a solution of the collisionlessBoltzmann equation.Introduction. Upstream-differencingschemes attempt to discretize hyperbolic par-tial differential equations by using differences biased in the direction determined by thesign of the characteristic speed.In recent years upstream differencing has become very popular, and a multitude ofnew techniques of implementing directionally biased differencing have been suggested.This popularityis primarily due to the robustnessof upstream-differencingschemes, theavailability of an underlying physical model, and the possibility of achieving highresolution of stationary discontinuities. Most of these schemes are an extension of theCourant-Isaacson-Rees scheme [1] to nonlinear conservation laws, and therefore aunifieddescription may be given.The present paper concentrates on reviewing basic concepts and deriving designprinciples; numerical experimentation is not presented. The paper is built up as follows.Section 1 reviews some properties of the equations essential to their proper numericalapproximation.In ?2 we discuss a straightforwardextensionof linearupstreamdifferenc-

ing to nonlinear systems. Section 3 introducesthe physical picture dueto Godunov, usefulto interpret certain schemes from ?2 and to construct new schemes. Finally, in ?4 wecomment on flux splitting, another form of upstream differencing, and relate it to a classof schemes motivated by the Boltzmann equation.1. Weak solutions and their numerical approximation. In this paper we considernumerical solutions of the initial-value problem for hyperbolic systems of conservationlaws

(1. la) Ut+f(u)x = 0, u(x, 0) = u0(x), -00 < x < oc.Here u(x, t) is a column vector of m unknowns and f(u), the flux, is a vector-valuedfunction of m components.We can write (1.1 a) in matrix form(I.lIb) ut + Aux = 0, A(u) =f".(1.1 ) is called hyperbolic if all eigenvalues of the Jacobian matrix A are real. We assumethat the eigenvaluesa1 u), * * * , am(u)are distinct and arrangedin an increasingorder.

*Received by the editors March 29, 1982, and in revisedform May 5, 1982. This research was sponsoredby the National Aeronautics and Space Administration under contract NAS1-15810 while the first and thirdauthors were in residence at ICASE, NASA Langley Research Center, Hampton, Virginia.tTel Aviv University, Ramat Aviv, Israel. The workof this author was supportedin part by the NationalAeronautics and Space Administration under contract NCA2-OR525-001 at NASA Ames Research Center,

Moffett Field, California, and by the U.S. Department of Energy under contract DE-AC02-76ER03077.tCourant Institute of Mathematical Sciences, New York University, New York, New York 10012. Thework of this author was supported in part by the U.S. Department of Energy under contract DE-AC02-76ER03077.?Institute for Computer Applications in Science and Engineering, Leiden State University, Leiden, theNetherlands.35


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36 AMIRAM HARTEN, PETER D. LAX AND BRAM VAN LEERTo allow for discontinuous solutions we admit weak solutions that satisfy ( 1.1 in thesense of distributiontheory, i.e.,

(1.2a) [w,u+ wxf(u)] dxdt+ w(x,O)uo(x)dx = 0,for all C- test functions w(x, t) that vanish forI + t large.Condition (1.2a) is equivalent to requiringthat for all rectangles (a, b) x (tl, 2) therelation obtained by integrating (1.1 a) over the rectangle shouldhold:(1.2b) f u(x, t2) dx f bu(x, tl) dx +t2f(u(b, t)) dtf- t2f(u(a t)) dt =0.

It

Clearly, a piecewise-smooth weak solution of (1.1) satisfies (1.1) pointwise in eachsmooth region; across each curve of discontinuitythe Rankine-Hugoniot relation(1.3) f(UR) -f(UL) = S(UR - UL)holds, where S is the speed of propagation of the discontinuity, and UL and UR are thestates on the left and the right, respectively.Since weak solutions of (1.1) are not uniquely determined by their initial data, weselect physically relevant solutions, defined as those solutions that are limits as ? -- 0 ofsolutions u(e) of the viscous equations(1.4) Ut+ f(U)X = F-Uxx > O.In this paper we consider systems of conservation laws (1.1) that possess an entropyfunction U(u), defined as follows:(i) Uis a convex function of u, i.e., Uuu> 0.(ii) U satisfies(1.5a) Uufu= FUwhere F is some other function called entropyflux; it follows from (1.5a) that everysmooth solution of (1.1 also satisfies(1.5b) U(u)t + F(u)X= 0.Limit solutionsof (1.4) satisfy, in the weak sense, the following inequality:(1.6a) U(u)t + F(u)X - 0;i.e., for all nonnegativesmooth test functionsw(x, t) of compact support(1.6b) -f (wtU + wxF) dxdt - w(x, O)U(uo(x)) dx _ 0.

Condition (1.6b) is equivalent to requiringthat for all rectangles (a, b) x (tl, t2) theinequalityobtainedby integrating (1.6a) overthe rectangle should hold:f U(U(x, t2)) dx - f U(u(x, t1 ) dx(1.6c) a a

+ ft2 F(u(b, t)) dt - ft2 F(u(a, t)) dt _ 0.tI tI

If u is piecewise smooth with discontinuities, then (1.5b) holds pointwise in thesmooth regions,while acrossa discontinuity( 1.6d) F(uR) - F(uL) - S [U(UR) - U(UL)] - 0.


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UPSTREAM DIFFERENCING AND GODUNOV SCHEMES 37Relations (1.6) are called entropyconditions (see [ 12]).In the following we shall describe numerical approximations to weak solutions of(1.1 ) that are obtained by 3-point explicit schemes in conservationform:(1.7a) v1 =; - xf;?1/2 ? XfYl/2 X = T/Awhere( 1.7b) fn+1/ f (Vjn,Vjn+)Here vj'= v(jA, nr), andf(u, v) is a numericalflux. We requirethe numerical flux to beconsistent with the physical flux in the following sense:(1.7c) f(u, u) =f(u).

We say that the difference scheme (1.7) is consistent with the entropy condition(1 .6a) if an inequalityof the followingkind is satisfied:(1.8a) j - XF;j+/12+ XFj>11/2,where the following abbreviationsare used:(1.8b) =Uj+ U(vj+'), Uj = U(Vn),(1.8c) Fj+1/2 = F(Vj, Vj+1);here F(u, v) is a numerical entropyflux, consistent with the entropyflux:(1.8d) F(u, u) = F(u).

The following is an easy (but useful) extensionof an easy (but useful) theoremof Laxand Wendroff [12]:THEOREM.1. Suppose the differencescheme (1.7) is consistent with the conserva-tion law (I.1a), and with the entropy condition (1.6a). Let vjn e a solution of (1.7), withinitial values vj? 0(jA). Extend the lattice function vjno continuous values of x, t bysetting, as usual(1.9) v(x, t) = vjn, j = [x/A], n = [t/r].Suppose thatfor some sequenceAk - 0, r/A = X,the limit

lim v(x, t) = u(x, t)Ak-0exists in the sense of bounded,L" convergence.Then the limit u satisfies the weak form(1.2) of the conservation law, and the weakform (1.6b) of the entropycondition.The proof consists, just as in [12], of multiplying (1.7a) by a test function, summingby partsovern andj, writingthe sum as an integral, and passingto the limit Ak -? 0.Theorem 1.1 remains true, and its proof the same, when the fluxes f and F areallowed to be functionsof 21arguments:(1.1 0) fi+,1/2=f(uJ 1+1,Uj-1+2, . , uj+0)and similarly for Fj+?1/2-

Assume that uo(x) is equal to some reference state u* forIx I arge:(I.lla) uo(x)=u* forlxl>M.Then(1.1llb) v7n=* forAIvI> M+-nA\.


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38 AMIRAM HARTEN, PETER D. LAX AND BRAM VAN LEERThe entropyUmay be altered by adding to it an arbitrary nhomogeneouslinear function;this follows from definition (1.5a). Adding such a linear function to U will not alter itsconvexity, but achieves the following:(1.12) U(u*) =O, UU(u*)=O.Since Uis convex, it follows from (1.12) that U(u) > 0 for u + u*; in fact if Uis strictlyconvex,(1.13) U(u) _ clu - u*12.Now sum (1 .8a) with respect toj over all integersj; we obtain(1.14a) _i iIn other words:total entropyis a decreasingfunction of time. In particular(I. 14b) EU ?i iThis is an a priori inequality for solutions of the difference scheme (1.7), analogous to theenergy inequality for linear symmetric hyperbolic differential and difference equations.Since by (1.13) U is positive for u / u*, this is an a priori estimate for solutions of thedifference scheme (1.7), and indicates that the scheme is stable. However (1.14b) is notstrong enough to provethe pointwise boundedness of solutions of (1.7), nor the existenceof convergentsubsequences.

Theorem 1.1 holds in any numberof space variables. Furthermoremultidimensionalschemes that are composites of one-dimensional fractional steps satisfy the multidimen-sional analogue of the entropy condition (1.8a) if each individual one-dimensional stepsatisfies an entropy inequality of the form (1.8a); see Crandall-Majda [2].A word of caution: when dealing with equations of mathematical physics, inparticular the equations of compressible flow, we must make sure that the differencescheme we are using keeps the variables within their physical range, i.e., that density andpressureare always positive quantities.2. Upstream-differencing chemes. We start our review with the descriptionof thefirst-order-accurateCourant-Isaacson-Rees scheme [1], the simplest upstream differ-

encing scheme for the constant-coefficientscalar equation(2.1) ut + aux = 0, a = constant,fv.a Vj+-vj for a < 0,

(2.2a) Vjn' =vjn- Xa.--VIntroducingthe notation

a =-min (a, 0) - 1/2 a - lal),a+-- max (a, 0) 1/2 (a +lal),

we rewrite (2.2a) as(2.2b) VJ1+=vJn - X[a+(vJ - vJ_1) + a-j(vJ+ - )],which can be rewritten as(2.2c)VJ+ = Aa+VJ_+ X(1 -| laI~ -VJ-a-vn1.


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UPSTREAM DIFFERENCING AND GODUNOV SCHEMES 39Under the Courant-Friedrichs-Lewy condition(2.2d) Xlal I1all coefficientsof v7on the right in (2.2b) are positive. Such a scheme is called monotone,and is stable in the maximumnorm;that is, for a monotonescheme(2.2e) maxIvjn+1 maxvjnEquation (2.2b) can be rewrittenas

Vjn1 x x(2.2f) v=j - - a(vj+I - v,_l) + - IaI (v?+,- 2v,n+ v, ,).2 2This shows that solutionsof (2.2) can be thought of as approximatingsolutionsof(2.3) wt + awx=/2AxJa I(1 -Xlal)wxxto second-orderaccuracy. We observe that the viscosity term in (2.3) vanishes for a = 0;this fact later will allow perfectly resolved stationary shocks but may also result inadmitting entropyviolating discontinuities.We describe now the extension of (2.2) to systems of equations with constantcoefficients:(2.4) ut + Aux = 0, A = constant.

Because of the hyperbolicity assumption, the system (2.4) can be diagonalized by asimilarity transformation(2.5a) w= T-lu, T-'AT=A, Ai=aibij,(2.5b) wt+ Awx= 0.

The components of w are called characteristic variables and (2.5b) is a system ofdecoupledcharacteristicequations.We extend the Courant-Isaacson-Rees scheme to systems by applying the scalarscheme (2.2) to each of the decoupledscalar characteristic equations. In matrix form thiscan be writtenasWn= jnx x Wn(2.6a) w,2 A(w+l - w,,) + -2AI I - 2wj + w, ),where the diagonal matrix IAI s defined by IAi = IaiI i. In the original variables, thescheme (2.6a) takes the formVj+Vnx x(2.6b) v j+ = _2-A(vj+I - v1 I) +2 AI(v,+I - 2v, + v, I),

whereIA = TIA T 1. Clearly, the stability conditionfor (2.6a) and (2.6b) isAt(2.6c) - maxIakI 1,AX k a

In general we definethe matrix x(A) by(2.7) x(A) = TX(A)T-, (X(A))ij = X(ai)bijWe remarkthat underour assumptionof a full set of eigenvectors we can compute x (A)by X(A) =P(A), where P(x) is the Lagrangian interpolation polynomial such that


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40 AMIRAM HARTEN, PETER D. LAX AND BRAM VAN LEERIn the following we shall describe various techniques to extend the upstream-differencingscheme (2.6b) to nonlinearsystems of conservation aws (1. 1).The linear system (2.4) can be regarded as a system of conservation laws (1.1a),where the flux dependslinearly on u:

(2.8a) f(u) = Au.The upstream-differencingscheme (2.6b) is in conservation form (1.7), with numericalfluxgiven by(2.8b) f(u, v) = A+u + A-Vwhere A+ and A- are the positive and negative parts of A, defined by the functionalcalculus (2.7) as(2.8c) A+=X+(A), A- =X-(A)where we set(2.8d) x+(a) = a+, x-(a) = a-.Note that since X+a) + x- (a) = a, and X+a) - X- (a) = I |, we can write(2.9) A+=1/2(A?+AI), A-=1/2(A-IAI).

DEFINITION.A difference scheme in conservation form (1.7) is said to be anupstreamscheme if:(i) For u and v nearbystates, (2.8b) is a linear approximationto the numerical fluxf(u, v).(ii) When all signal speeds associated with the numericalflux (u, v) are > 0,

f(u, v) =f(u).When all signal speeds are < 0,

f(u, v) = f(v).The relevant signal speeds generally differ from the characteristic speeds of the states uand v.

We restate (i) in analytic terms: Suppose u and v are near some reference state u*;then we requirethatf(u, v) =f(u*) + A+(u*)(u - u*) + A-(u*)(v - u*)(2.l1Oa) ?(uuII-*)+ 0(lu - u*l + Iv - u*l).

A natural choice for u* is (u + v)/2; setting this into (2.1Oa),makinguse of (2.9) to setA+ -A- =IAI

and noting thatf (u + v) f(u) + f(v) + o(lu - vI)

we get(2.lOb) f(U,v)- f(u) +f(v) (u )|(v-u)?o(Iu-vI).


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UPSTREAM DIFFERENCING AND GODUNOV SCHEMES 41We can write any numerical flux in the form

(2.1la) f(u, v)= f(u) +f(v) 1 d(u v);2 2for the sake of consistency (1 .7c) we need(2.1 lb) d(u, u) = 0.The upstream condition (2.1 Ob)can be expressedthen as(2.1lc) d(u,v) = A (u ) (v-u) + o(Iu-vI).

Formula (2.6a) shows that linear upstream difference schemes contain a large doseof artificial viscosity, except for those components where ak is small, in particular whereak = 0. The same appears to be true for all upstream difference schemes for nonlinearconservation laws: when all characteristic speeds are not zero, each component is treatedlike a scheme with a hefty amount of artificialviscosity, smearing discontinuities. There ishowever quite a distinction among the schemes when one of the characteristic speeds iszero; this showsup in the way each scheme resolves a stationary shock, centered transonicrarefaction wave, and stationary contact discontinuity. We turn now to examining thesematters.The most critical difference in performance occurs in resolving a stationary shock(see (1.6d)):

u, x0,The lack of numerical dissipation allows the design of schemes that perfectly resolvestationary shocks, i.e., (2.1 2a) is a steady solution of the numerical scheme. The conditionfor that is(2.12b) d(u, v) = 0 iff(u) =f(v), F(v) < F(u).On the other hand

u, x F(u)lV, x>0,is not an admissible discontinuity and should not be a steady solution of the finitedifferencescheme, i.e., we requirethat(2.12d) d(u, v) # 0 iff(u) =f(v), F(v) > F(u).

We remark that the danger that a given upstream scheme selects a nonphysicalsolutionwill occuronly for stationaryor near-stationary discontinuities;otherwise there isenough numerical viscosity in (2.3) to enforce the selection of a physically relevantsolution. Hence there are two options in designing an upstream-differencingscheme forsolving problemswith discontinuoussolution:(1) to switch direction of differencing in a way that will effectively introducenonlineardissipationat the expense of slightly spreadingthe shock;(2) to satisfy (2.12) and thus get perfect resolution of a stationary shock, but to adda mechanism for checking the admissibility of the discontinuity.


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42 AMIRAM HARTEN, PETER D. LAX AND BRAM VAN LEERWe turn now to describingvarious forms of d(u, v) in (2.11 a). The most straightfor-wardway to generate such functions is by

(2.13a) d(u, v) = A I u,v)(v -U),whereIA I u, v) is a matrix function of u and v that has nonnegativeeigenvalues,and suchthat(2.13b) IAI u, u) =IA(u) I;IA(u) I s definedby (2.7).The simplest formsof (2.13) are(2.14a) IAI(u,v)= A( u )|or(2.14b) JAI(u,v)= 1/2 [IA(u)I+ JA(v)I].The latter was used by van Leer in [13], and introduces some nonlinear numericaldissipation that somewhat smears stationary shocks but on the other hand excludesnonphysicaldiscontinuities.Another formof (2.13a) with (2.14a) has been suggested by Huang [10]:(2.15) d(u, v) = sgn (A (u + v)) [f(v) -f(u)];here sgn (x) is the sign of x, and sgn (A) is definedby (2.7).Yet anothertype of scheme has been designed by Roe [ 19]. His scheme is of the form(2.13a), where the matrix function A(u, v) is requiredto have these properties:(i)(2.16a) f(v) -f(u) = A(u,v)(v - u).

(ii) A(u, v) has real eigenvalues and a complete set of eigenvectors.(iii)(2.16b) A(u, u) = A(u).For the Euler equations of compressibleflow Roe [ 19] has constructed a linearization ofform (2.16a) having these properties.We show now that such a linearization exists quitegenerally:

THEOREM .1 (Harten-Lax). Suppose (1.l a) has an entropyfunction. Then (1.l a)has a Roe-type linearization.Proof. We shall construct an A satisfying (2.16a) which is of form A = BP, Bsymmetric, P positive definite. Clearly, such an A is similar to the symmetric matrixP'12BP'/2 and so has property(ii). In our constructionwe use the entropy function U(u);since U is convex, the mapping u -- w = Uu s one-to-one;we introducew as new variablein place of u. Let u, and u2be two arbitrary states, w, = w(ul), w2= w(u2),Af =f(ul),f2 ==f(U2). Then

2 -f= df(w2 + (1 - O)wj) dO(2.17) d)


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UPSTREAM DIFFERENCING AND GODUNOV SCHEMES 43We claim thatf, is a symmetric matrix; this implies that B is symmetric. To see aboutf",we use relation (1.5a); differentiating with respect to u we get

Uuufu + Uufuu = FUUThe second term on the left is a linear combination of symmetric matricesf'u; the rightside, FUU lso is symmetric.Thereforeso is the first term

Uuu uIt follows then that also(2.18) uuis symmetric.

Differentiatingw=u=

with respect to u shows that wu= Uuu;hereforeUj) = uw.

Substituting this into (2.18) shows thatfuu UW= fw

is symmetric, as asserted.Next we express-w, =f dw(Ou2 + (1 - O)ul)dO(2.19) 0 O

= wUdO(u2 UO)= P(u2 - Ul).Using wu,= Uuu nd the convexity of U we conclude that P as definedby (2.19) is positivedefinite.Combining (2.17) and (2.19) gives

f2 -f1 = BP(u2 -ul) = A(u2 - ul).Thus the A we have constructed can be factored as BP, as asserted. Condition (2.16b) isclearly satisfied. DNote that B depends symmetrically on w1and w2,and P symmetrically on ul and u2.This shows that A is a symmetric function of ul and u2.A similar result holds for systems of conservation laws in any number of spacevariablesas long as there is an entropy (see Harten [8]).Having constructedA, we can define its absolutevalue by (2.7); then we set(2.20) d(u, v) =I A(u, v) I v - u).

When u and v correspondto a stationary discontinuity (2.12), then it follows fromf(v) - f(u) = 0 and (2.16a) that v - u is a null vector of A(u, v), and consequentlyin thenull space of IA u, v) I; thus d(u, v) = 0, whether or not the entropy conditionF(v) - F(u) < 0 is satisfied. Therefore the corresponding upstream differencing mayadmit nonphysical solutions. In an appendix to [9], Harten and Hyman describe aviscosity-like term, i.e., of the form (3.33a), that can be added to (2.15) and (2.16) to


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44 AMIRAM HARTEN, PETER D. LAX AND BRAM VAN LEERreject inadmissiblediscontinuities without affecting the perfect resolution of the physicalones.

Yet anotherway to construct d in (2.11c) is(2.21) d(u, v) = f IA(w)I dw,where the integration in (2.21) is carriedout on a path in state-space connecting u and v.Osher in [16] suggests a path of integration r that is piecewise parallel to the righteigenvectorsRk of A:

m(2.22a) r= U rk,

k=i

dUk = Rk(u ), 0 < I < Ik, k = 1, ,(2.22b) rk: dluk (lk+I?) = Uk (O),

(2.22c) u'(O) = u, uV(1,)= v.Existence of a unique solution to (2.22b)-(2.22c) is guaranteed if Ilu vii s sufficientlysmall [16]. A consequence of this choice of path is that d in (2.21) decouples intocharacteristiccontributions

m(2.23) d(u, v) = I/ lak(U(l)) IRk(uA())dl.Osher shows that limit solutions of (2.11) with (2.23) satisfy the entropy conditionand that a stationary contact discontinuity is perfectly resolved;the representation of astationaryshock requiresat least one intermediate state.3. Godunov-type chemes.A. The generaltheory. Godunov,in his constructionof the "best"monotonescheme[4], has used the exact solutions of local Riemann problems to obtain an upstream-differencingscheme.The solution of the Riemann problem

UL, X 0,dependsonly on the states ULand URand the ratiox/t; it will be denotedby u(x/t; UL, UR).Since signals propagatewith finitevelocity,(3.2a) u(x/t; UL, UR) = UL for x/t - aL,(3.2b) u(x/t; UL, UR) = UR for x/t _ aR;aL and aR are the smallest and largest signal velocity.Godunov derives his scheme by consideringthe numerical approximationv(x, tJ) ofthe discrete time levels tn,n = 0, 1, . . . , to be a piecewise constant functionin x, i.e.,(3.3a) v(x, tn) = v7n for x in Ij = ((j - 1/2), (j + 1/2)A).

To calculate the numerical approximation at the next time level tn+ = tn + i, we firstsolve exactly the initial-value problem


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UPSTREAM DIFFERENCING AND GODUNOV SCHEMES 45for tn _ t _ tn+ -, and denote its solution by un(x, t). Each discontinuity in v(x, tj)constitutes locally a Riemann problem. If we keep XI max /2whereI Iis the largestsignal speed, then because of (3.2) there is no interaction between neighboring Riemannproblems, and un(x, t) can be expressed exactly in terms of the solutions of local Riemannproblems:(3.3c) Un(X,) = a ( ; vJ vj+ for A < x < (j + I)A\, tn- t -C t,+,.Godunov obtains a piecewise-constant approximation v(x, tn+1) by averaging un(x,tn+l),i.e., he sets(3.3d) vUn+l , un(x, tn + r)dx.We can rewrite (3.3d) in terms of the solutions to the local Riemann problem as(3.4a) vu, = t u(x/; vj,_, v) dx + u x/1; v,vj?) dxSince un s an exact solution of the conservationlaws (3.1), we can evaluate the integraldefining vjn+7n (3.3d) by applying (1.2b) over Ij x (tn, tn+,); we get(3.4b) vj = vj - X[ (vJ+1/2)-f(v1-l/2)],where(3.4c) vj+/2 = U(OV;Vj+1)-This showsthat (3.4b) is in conservationform, with(3.4d) f(v, w) = f(u(O; v, w)).

The exact solution un(x,t) of the Riemann problem satisfies the entropy condition(1.6c):

J U(Un(X,n+1)) dx _ zAU(vjn)TF(V+1/2) + TF(i>112).Since Uis a convex function, Jensen's inequalityholds:

u(Z u(x, t) dx) A U(u(x, t)) dx.Combining the last two inequalities, we deduce that Godunov's scheme satisfies theentropy inequality (1.8a).The description (3.3c) makes sense only if the local Riemann problems don'tinteract, i.e., if

Xlamaxl< 1/2.

On the other hand, (3.4b) remains consistentwith (3.3d) as long as the waves issuingfrom(j ? 1/2)A do not reach (j T 1/2)A during the time interval tn_ t _ tn+l.This will be thecase as long asXlamaxI1.

It follows from the Rankine-Hugoniot relation (1.3) that f(u(s; v, w)) - su(s; v, w)


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46 AMIRAM HARTEN, PETER D. LAX AND BRAM VAN LEERis a continuous function; however it is only piecewise differentiable. It follows that theGodunov flux functionf(v, w) definedby (3.4d) is only piecewise differentiable.Godunov'sscheme satisfies criterion (ii) for upstream schemes because of (3.2). To

dx - adxdx da dt adxdx Idt +

U2 U3

UO UL I Um UR0.x

FIG,. .

verify that it also satisfies criterion (i) we shall show that Godunov's scheme, whenappliedto linear equations,reducesto (2.6b).Consider(3.5a) ut Aux = O,A a constant matrix. Here the solution of the Riemann problemis composedof constantstates separated by a fan of m characteristic lines (see Fig. 1).(3.5b) U(X/t;UL,UR) =Uk for ak


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UPSTREAM DIFFERENCING AND GODUNOV SCHEMES 47Let N be an integer such that

aN < O < aN+1-Then by (3.5e)

U(O; UL, UR) = UN = UL + YN(A)(UR - UL)(3.6) = U(I N(A))UL + UN(A)UR.Since by (2.8a), in the linear casef(u) = Au, substituting (3.6) into (3.4d) gives

f(u, v) = A(I -UN(A))u + AMN(A)v.From the functionalcalculus (2.7) and (2.8c) we deduce that

A(I -UN(A)) = A+, AUN(A) = A-,so that

f(u, v) = A+u + A-v,in full agreement with (2.8b). Thus in the linear case Godunov's scheme reduces to theupstreamscheme (2.6b).The solution to the Riemann problem (3.1) has a rather complicated structure:as inthe constant coefficient case (3.5), the solution to (3.1) depends on x/t and consists ofconstant states Uk,k = O, , m; uO= UL, Um = UR, separated by a fan of waves. Unlikein the constant coefficient case, the k-wave separating Ukl and Uk s not necessarily asingle line having a characteristic speed ak. If the kth characteristic field is genuinelynonlinear then the k-wave is either a rarefaction wave (ak(Ukl) < ak(Uk)) or a shockpropagating with speed S, (ak(Uk1l)> S> ak(Uk))- If the kth characteristic field islinearly degenerate then the k-wave is a contact discontinuity propagating with speedak(Ukl) = ak(Uk) (see [12]).It is evident from (3.3d) that, due to averaging, the Godunovscheme does not makeuse of all the information contained in the exact solution of the Riemann problem. Wetherefore considerreplacing the exact solution to the Riemann problemu(x/t; UL, UR) in(3.4a) by an approximationw(x/t; UL, UR); the latter can have a much simpler structureas long as it does not violate the essential properties of conservation and entropyinequality. The following theorem due to Harten and Lax [7, Thm. 2.1] shows that thistype of approximation s consistent:

THEOREM .1 (Harten-Lax). Let w(x/t; UL, UR) be an approximation to the solutionof the Riemann problem that satisfies thefollowing conditions:(i) consistency with the integralform of the conservationlaw in the sense that(3.7a) A/2 w(x/t; UL, UR) dx = - (UL + UR) - rfR + rfL-A/2 2for A/2 > Tmax l ak 1, where

fR =f(UR), fL =f(UL);

(ii) consistency with the integralform of the entropycondition in the sense that(3.7b) J U(w(x/t; UL, UR)) dx --A (UL+ UR) - TFR + TFL-A2 2for A\/2 > T maxltakl, where

FR = F(UR), FL = F(UL).


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48 AMIRAM HARTEN, PETER D. LAX AND BRAM VAN LEERUsing the approximation w to the Riemann problem, we can define a Godunov-typescheme asfollows:

(3.8) vj =if;/w(x/t; v,, v, )dxj A w(x/t; vj,v,+) dx.Assertion. If conditions (3.7a) and (3.7b) are satisfied, the scheme (3.8) is inconservation orm consistent with (3.1), and satisfies the entropyinequality (1 .8a).Forproof, see [7, after Thm. 2.1]. It is shown there that the Godunov-typeaveragingcan be replaced by Glimm-type sampling. [7] contains a good account of this techniqueand an analysis of its accuracy.Theorem 3.1 shows that Godunov-typeschemes that satisfy conditions (i) and (ii)above satisfy the hypotheses of Theorem 1.1; this shows that if such a scheme converges,

the limit satisfies the conservation law and the entropy condition in the weak sense.We note that Godunov's scheme is of Godunov type.We have shown at the beginning of this section that Godunov'sscheme (3.3) can alsobe written as a scheme (3.4) in conservation form. The appropriate numerical flux wasobtained from the integral conservation laws (1.2b). We show now that all schemes ofGodunov type can be expressed in conservation form; we can obtain the appropriatenumerical flux by applying the integral conservation law (1.2b) to the approximatesolution of the Riemann problem over the rectangle (-1A/2, 0) x (0, T):(3.9a) Jo W(XT; UL, UR) dx - AUL+ r[fLR -M = ?-A/2 2where

fLR =f(UL, UR)-This gives(3.9b) fLR =fL - 0 W(X/T; UL, UR) dx + -UL--A/2 2rIf we apply the integral conservationlaw (1.2b) over the rectangle (0, A/2) x (0, T), weobtain(3.9c) f LR =R + A/2 W(X/T; UL, UR) dx - AURJo 27-URThe equality of (3.9b) and (3.9c) is just the content of the consistency relation (3.7a).Using formula (3.9b) forfJ+1/2n (1.7b) and (3.9c) forfj1/2 in (1.7b) and setting theresulting expressions into (1.7a) gives (3.8), i.e., it puts the Godunov-type scheme inconservation form (1.7a):(3.10) = - [f(vi, v+1) -f(v>_1, vj)].

If all signal speeds are positive, then w(s; UL, UR) = uLfor s < 0; according to (3.9b)inthiscasefLR =fL. Similarly,f all signalspeedsarenegative, hen w(s; UL, UR) = UR fors > 0; it follows from (3.9c) that in this casefLR ==fR. This is property (ii) of upstreamschemes, which is thus satisfied by all Godunov-typeschemes.In a scheme of Godunovtype we can incorporateinto the numerical flux all physicalinsights that we can put into the approximatesolution of the Riemann problem. Also, asHarten and Lax pointedout in [7], a Godunov-typescheme (3 8) can be used just as easily


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UPSTREAM DIFFERENCING AND GODUNOV SCHEMES 49on a grid that varies in time, by adjusting the intervals of integrationon the right in (3.8).This makes these schemes the natural choice for adaptive grids; further developmentofsuch algorithmsand numericalexperimentsare describedin Harten and Hyman [9].B. Construction of particular Riemann solvers. We turn now to describing twodifferent approximateRiemann solvers, and the Godunov-type schemes correspondingtothem. The first, due to Roe, is based on a linearization notion of type (2.16). Roeapproximates solutions of the Riemann problem for (3.1) by exact solutions of theRiemann problemfor the following linear hyperbolicequationwith constant coefficients:

UL, X < 0,(3.1 Ia) wt+ ALRWX =, w(x, ) = IUR, X>0O.Here ALR is a matrix that satisfies (2.16a)-(2.16b) and has properties (i)-(iii) listedthere. Combining (2.16a) with (3.5c) yields(3.1 Ib) fR -fL = ALR(UR - UL) = E aiJiRi,where ai are the eigenvalues of ALR, Ri the correspondingright eigenvectors, and Ji thecoefficientsin the resolution(3.5c):(3.11c) UR - UL= E JiRi.The approximateRiemann solver is given by (3.5b), with Ukdefinedby (3.5d).The numerical flux associated with an approximate Riemann solver is given by(3.9b); substituting (3.5d) into (3.9b) we get(3.12a) fLR =RfL + E a,JiRwhere

a- =min (a,0) = 112(a- jai).Substituting this into (3.12) and using (3.1 lb) gives

fLR = /2(fL + R) - 1/2 ZaIJR(3.12b)= 1/2(fL +fR) - 1/2ALRI (UR - UL);

in the last step we have used the definition of IA las given by (2.7). Indeed, (3.12b) is Roe'sscheme definedin (2.1 la), (2.20).As already pointed out in ?2, Roe's scheme admits nonphysical, i.e. entropy-violating, stationarydiscontinuities. In an appendixto [9], Harten and Hyman show howto modify Roe's scheme to eliminate such entropy-violatingdiscontinuitieswhile retainingthose that satisfy the entropylaw.We note that the numerical flux (3.1 2b) of Roe's scheme resembles Osher's scheme(2.23). There thejumps Jk in the characteristic state variables are representedby the pathlength lk* Osher's scheme, however, is not of Godunov type in the sense of (3.9) since theintegration path r in state-space does not correspond to a univalued approximateRiemann solutionw(x/t, UL, UR) as in (3.8).Roe's Riemann solver contains a great amount of detail: m -1 intermediate states.


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50 AMIRAM HARTEN, PETER D. LAX AND BRAM VAN LEERWe describe next a hierarchy of Riemann solvers where much of this detail is lumpedtogether. The simplest of these schemes contains only one intermediate state.

i) Denote by aL and aR lower and upper bounds, respectively, for the smallest andlargest signal velocity, calculated according to some algorithm. Define the approximateRiemann solver byUL forx/t < aL,

(3.13) u(x/t; UL, UR) = ULR foraL < x/t< aR,LUR foraR < x/t,

where the state ULRs determinedfrom the conservationlaw (3.7a):(iTaL+ A/2)uL + T(aR - aL)uLR + (A/2 - TaR)uR =

A(UL + UR) -T fR fL]A2

This gives(3.14) ULR= aR UR- aLUL fRf-fLaR-aL aR-aLWe turn now to the determination of the associated numerical flux. We substitute (3.14)into (3.13), and then into (3.9b):

fL when 0< aL,aL-aL aR aLaR(3.15a)fLR fR? fL + (UR - UL) when aL < ? < UR,aR-aL aR-aL aR-aL

fR when aR


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UPSTREAM DIFFERENCING AND GODUNOV SCHEMES 51c) The entropylaw is satisfied.Such an approximateRiemann solver was constructed in [7]; the one presentedherediffers from it in some important details. We are grateful to Paul Woodward for asuggestion which has been incorporated n the scheme.Let the velocities aL and aRbe defined as in approximationi) described above. Wedefine a velocity V as follows: Let U be an entropy function defined by (1.5); denote itsgradientby w = U, and introduce the abbreviation

(3.17a) W(UR) - W(UL) = ILRWe set(3.17b) V = lLR * (fR -fL)/lLR (UR - UL)where the dot denotes the Euclidean scalar product.

Next we show that Vis well defined,and derive its salient properties:LEMMA 3.2: i) The denominatorin (3.17b) is positive for UL : URii) Vis uniformly bounded.iii) If UL and URsatisfy the Rankine-Hugoniot condition (1.3),

(3.18) fR -fL = S(UR - UL),then V = S.

Proof. i) Combining(3.17a) and (2.19),(3.19) lLR * (UR - UL) = P(UR - UL) * (UR - UL);since P is positivedefinite, the abovequantity is positive for UR : UL.ii) Use (2.17), (3.17a) and (2.19) to expressthe numeratorof (3.17b), and (3.19) forthe denominator;we get

V P(UR - UL) * BP(UR - UL)(UR - UL) P(UR -UL)

This is a ratio of two quadratic forms and therefore lies between the smallest and largesteigenvalue of P112BP'12.These are equal to the eigenvalues aj of A = BP constructedforTheorem2.1. In fact, Vcan be representedas a weighted averageof the eigenvaluesof A.iii) Setting (3.18) into (3.17b) givesV= S.This completes the proofof Lemma 3.2.We now outline two methods for constructing approximate Riemann solversw(x/t; UL, UR) with two intermediatestates u *and uR separatedby the line dx/dt = V; .e., wis of the form

UL, x/t < aL,UL* aL < x/t < V,(3.20) w(x/t; UL, UR) = R x/t


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52 AMIRAM HARTEN, PETER D. LAX AND BRAM VAN LEER

dxdt |\ ? dxdx dtdt aLI

\ \ I u

UL UR

xFIG.2.

We introducea numerical fluxacross the line x = Vt,denoted asfv(UL, UR).

We require consistency with the exact flux:(3.22) fM(u, u) =fv(u) =f(u) - Vu.

Having introduced flux (3.21) across lines, we can write approximate conservationlaws for the triangularregions boundedby t =Ti, x = Vt, and x= aLt or x = aRtrespectively:(3.23a) (V- aL)UL + fv(UL, UR) -faL(UL) = 0and(3.23b) (aR - V)UR + faR(UR) -fv(UL, UR) = 0;UL* nd uR can be determined from (3.23). Clearly, since (3.23a,b) are conservation laws,the resulting scheme (3.20) satisfies the consistency relation (3.7a). Thus requirement a)on conservation is fulfilled.We turn now to requirement b), the exact resolution of single shocks and contactdiscontinuities. A shock or contact discontinuity is characterized by the Rankine-Hugoniot condition (3.18) and the entropy condition (1.6d). Using the notation (3.21),these can be written as follows:(3.24) fs(u) =fs(UR), FS(UL) - FS(UR)where(3.25) Fs(u) = F(u) - SV.We denote byfv a vector function that has the following property:

Iffv(uL) =fv(UR), then(3.26) fIV(uL, UR) = fV(UL) = fV(UR)-


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UPSTREAM DIFFERENCING AND GODUNOV SCHEMES 53Below we shall specify two distinct ways of constructingsuch anfv. We now set(3.27) fv(UL, UR) =JV(UL, UR) - f(UL, UR)(UR - UL),where 3has the followingproperty:(3.28) 1(UL, UR) =0 when (3.24) holds.We take (3.27) as our numerical flux. It follows from (3.26) that this flux satisfies theconsistency condition (3.22), and from (3.26), (3.28) and part iii) of Lemma 3.2 thatsingle shocksand contact discontinuitiesare resolved exactly.Here are our choices forfv and d: we defineAR and bL by(3.29a) 6 aR- V L- V- aLaR-aL aR-aLNote that(3.29b) O -AR, -AL, AR + AL =l IThen we define(3.30a) f'V(UL, UR) = aRfV(UL) + bLfV(UR)and(3.30b) fV(UL, UR) =fv(ULR) -f(LUL + 6RUR) + aLfL + aRfR,where ULRis defined by (3.14).It can be verifiedimmediately thatf asatisfies (3.26). To verify it forf bwe note thatiffv(UL) =fv(uR), then ULR = 6R UR + 6L U ; setting this into the right side of (3.30b) we seethat it equals the right side of (3.30a).We define( as follows:(3.31a) d= CIf3 + C232where(3.31b)

31(UL, UR) = [FV(uR) - FV(UL) -/2 (UL + UR) (v(UR) -fv(UL))M IUR - ULIwherep+ denotesmax(0,p), and(3.31c) f2(UL, UR) = (aR - aL) 11v(UR) -fv(UL) 11211 R - UL -The analysis in [7, ?4] shows Alis a boundedfunction.By construction,Al= 0 when the shock condition (3.24) is satisfied; 2 = 0 when theRankine-Hugoniot condition (3.18) alone is satisfied. Thus our choice of a satisfies(3.28), and so requirementb) is fulfilled.An analysis similar to that carriedout in [7, ?4] shows that the positiveconstants Cland C2 n (3.31a) can be so chosen that the entropy condition (3.7b) is satisfied. Thus isrequirement c) fulfilled.To derive the numerical flux associated with the above Godunov-type scheme wesubstitute (3.20) into (3.9b), using (3.23) to express uL*andu*:

fLR = /2 {f L + fR + YL [ fV(UL) -fv(UL, UR)]+ YR[ V(UL, UR) -fV(UR)] -I VI (UR -UL)1


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54 AMIRAM HARTEN, PETER D. LAX AND BRAM VAN LEERwhere

(3.32b) VI aLl Y aR I VIV- aL aR -VOne can easily verify that when aL ' 0,fLR =fR, and that when V= , fLR = fv(UL, UR).If on the right side of (3.32a) we substitute (3.27) forMf(uL, UR), the following termcontainingf appears:(3.33a) -I/2 (YR - 'YL) f(UR - UL).From (3.32b)

0 if aL> Oor aR


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UPSTREAM DIFFERENCING AND GODUNOV SCHEMES 55It is easy to see that (4.1)-(4.2) reduces to (2.6b) in the constant-coefficientcase if andonly iff a(w) becomesIA Iw.Steger and Warming [22] introducedthe notion of flux splitting for the equationsofgas dynamics. They took advantage of the fact that in gas dynamicsf is a homogeneousfunction of w of degree one. Then Euler's identity holds:(4.3a) f(w) = A(w)w, A =f,.Steger and Warmingdefine(4.3b) f +(w) = A+(w)w, f -(w) = A-(w)w,where A' andA- are defined by (2.7), (2.9). Clearly (4.1a) is satisfied;(4.2a) becomes(4.3c) f a(w) =IA (w)Iw,whereIA w) I s defined by (2.7).Substituting (4.3c) into (4.2c) gives, after rearrangement,(4.3c') d(u,v)= /2h(IA(u)I IA(v)I)(v-u) +/2(IA(v)I-IA(u)I)(u +v).

The right-handside is of the upstreamform (2.1 1c), except for those nearbyvalues ofu and v for which sgn ak(u) # sgn ak(v) for some k. The consequence of this nonsmooth-ness is a kink in computed solutions near such transitions,e.g. near sonic points.This canbe rectified to some extent if one replaces IAIby x(A), where x(s) is a smoothapproximationto Is1; ee Steger [23] or Harten [5]. Van Leer [15] has derived a smoothflux splitting for the Euler equationswithout referenceto (4.3a).We describe now a way of splitting whenf(u) is not a homogeneous function of u, byintroducing a reference state uoand applying the mean-value linearization (2.16a) to uand uo:(4.4a) f(u) -f(uo) = A(uO,u)(u - UO)We consider u - uoto be a new state variable, andf(u) - f(uo) to be a new flux function;denote them again by u andf(u), respectively. Thus any solution of the conservationlaw(1.1) satisfies(4.4b) ut + [A(uo,U)u]= .The new flux(4.5a) f(u) = A(uo,u)ucan be split as(4.5b) f(u) = A+(uO,u)u + A-(uOu)u =f +(u) +f -(u),exactly as in the homogeneouscase (4.3a).We turn now to a class of upstream schemes that are a natural generalization ofSteger and Warming'sflux splitting. These schemes are obtained by approximatingeachconservationlaw in (1.1 a) by a collisionless Boltzmannequation.Let 4(x, t, q) be a vector function whose ith component denotes the density of aparticle of ith kind at positionx and time t, travellingwith velocity q. We assume that theparticles stream freely, i.e. that Asatisfies the collisionless Boltzmannequation:(4.6a) 4f,+ qfx= 0.


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56 AMIRAM HARTEN, PETER D. LAX AND BRAM VAN LEERNote that the densities of the different kindsof particles are completely decoupled.Using equation (4.6a) we can determinethe value of 0 for t > 0 in terms of X0(x,q) =4(x, 0, q):(4.6b) (x, t, q) = (x-qt, q).

We denote by z and g the total density and fluxassociated with the density ?:(4.7a) z(x, t) = f4(x, t, q) dq,(4.7b) g(x, t) = f q(x, t, q) dq.These satisfy the conservationlaw obtained by integrating (4.6a) with respectto q:(4.8) Z,+ gx = 0.The initial values of z andg can be obtained by setting t = 0 in (4.7):(4.9) zo(x) = 0fo(x, q) dq, go(x) = fq 00(x, q) dq.If the initial values of z and g are equal to those of u andf(u):(4.10) zo(x) = uo(x), go(x) =f(uo(x));then for t small enough the solution z of (4.8) would be a reasonableapproximationto thesolution u of (1.1 a). We show now how to choose 00 so that (4.10) holds:we introduce avector distribution u(q, u), dependingon a vector parameter u, satisfying(4.1 la) f(q, u) dq = u,(4.1 Ib) fq,A(q, u) dq = f(u).Then we simply set(4.12a) 00(x, q) = Au(q, o(x)).Substituting (4.12a) into (4.9) and using (4.1 1 shows that (4.10) is satisfied.Equation (4.8) can be solved explicitly; as in the Godunov-typeschemes, averages ofthese explicit solutionswill be used to approximatesolutionsof (1.1a).

Let v(x) be an approximationat t, to the solution u of (1.1 a). We define the initialvalue 00by (4.12a):(4.12b) 00(x, q) = Au(q,(x)).Using this in formula (4.6b) for the exact solutionof equation (4.6a) gives(4.12c) ?(x, t, q) = ,u(q, v(x - qt)).Substituting this into (4.7a) results in(4.13) z(x, t) = fAt(q, v(x - qt)) dq.We assume that v = v"is piecewise constant. We define an approximation vn+l to u attn+ = tn + r that is piecewise constant on each interval Ij = (xj11/2, xj+ /2) by definingthevalue vjn+Iof Vn+ Ion Ij to be the averageof z(x, 7) overIj. Using (4.13) we get

(4.14) V2+ l= Ij l-I ffy(, n(X-\Jdqdx


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UPSTREAM DIFFERENCING AND GODUNOV SCHEMES 57We show next how to express (4.14) as a scheme in conservation form. We integrate(4.6a) over the rectangle Ij x (0, T); we get

T T ~~Xi+/2=0t Xdx |+ JTq(A dt | =0.0 0 Xi- /2

We integrate this with respect to q, obtaining the conservationform(4.15a) IjI vjI - vj7) + rf(f'+l/2 fn /2) = 0,where vjn+7, jnare the values of Vn+ vnon Ij, andf n+2 is defined by

nf+1/2 = ff q4(xj+ 1/2 t, q) dt dq,withfnI2 definedsimilarly. Using (4.12c) to express0 on the right-hand side we get(4.15b) n = f T q (q, vn(Xj+ -qt)) dt dq.This is the numerical fluxassociated with the scheme (4.14).We show now that this numerical flux is consistent with the fluxf in (1.1 a). Let'stake the case, sure to be satisfied in any scheme of practical significance, that Pt hasbounded q-support. Then it follows from (4.15b), and the fact that v' is piecewiseconstant, that there is an integer N, whosevalue dependson T, such that

fn =f (Vjn N+ . .* jn)X+1/2fvN+l, 'Vj+N\).Suppose Vj+k= v, k = -N + 1, , N; then vn(Xj+112 - qt) on the right-hand sideof (4.1 5b) equals v on the q-supportof ,u;using (4.1 lb) we see that the right-handside of(4.1 5b) equals Tf(v). This proves consistency.We turn nowto the task of determiningthe distributionA.Clearly, since only the firsttwo moments of ,uare specified by conditions (4.1 1), there is a great deal of leeway. Forguidance we turn to the linear case,

f(u) = Au, A constant.The solution of the linear equation(4.16a) ut + Au, = 0,with initial values u(x, 0) = u0(x), has the form(4.16b) u(x, t) = ZPiu0(x - ait).Here aj are the eigenvalues of A, and Pi is projectiononto the line spanned by the righteigenvector Ri. On the other hand, substituting (4.6b) into (4.7a) gives the followingexpressionfor z:(4.17) z(x, t) = f00(x - qt, q) dq.Clearly, comparing (4.16b) and (4.17) we see that(4.18a) z (x, t) u (X, t)if(4.18b) ~00(x, q) = ,, Pi uo(x) 6(q - at).


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58 AMIRAM HARTEN, PETER D. LAX AND BRAM VAN LEERIt is not hardto showthat (4.18a) holds for no otherchoice of 00whose support s boundedin x and q. Setting (4.18b) for k0 nto (4.12a) we conclude that ,umust be of the form(4.19) ,u(q, u) = E 6(q - ai)Piu.

We turn nowto the nonlinear case. As we have shown earlier in this section, the fluxcan be put in form (4.5a):f(u) = A(u)u

where the matrix A(u) has real eigenvalues ai(u) and a complete set of eigenvectors.According to the spectral theoryof matrices,

(4.20) EPi = I, aiPi==A.It follows from this that the distribution, definedby (4.19) satisfies relations (4.11) evenwhen a, and Pi are functions of u.Since relations (4.11) are linear, the most general ,uthat satisfies (4.11) is of theform(4.21) ,u(q, u) = ,(q, u) + q q, u),where, is any distributionwhose first two q-momentsare zero for all values of u.We call schemes of form (4.14) Boltzmann-type;we list some of their properties.(a) Suppose the support of the distribution ,uis contained in IqI_ Q. Suppose forsimplicity that the mesh over which we discretize is uniform, i.e., that each intervalI, hasthe same length A. Then it follows easily that (4.14) is a three-pointscheme if r is chosenso that(4.22) TQ _ A.

For Boltzmann-type schemes, the flux (4.15b) splits naturally into two parts. Wedefinej,u(q,u) forq _ 0,

+(qu)=0 for q < 0,(4.23) 0 forq> 0,4=(q,u) for q


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UPSTREAM DIFFERENCING AND GODUNOV SCHEMES 59where A' and A- are the positive and negative partsof the matrix A definedby (2.7) and(2.9). This is the Steger-Warming scheme (4.3).

(c) Considerthe equationsof compressibleflowin Euler coordinates. In this case thethree components of 4 describe the transport of mass, momentum and energy. Sincemomentum is mass x velocity, and q is velocity, it is reasonable to stipulate that thesecond componentof / be q times its firstcomponent.In view of (4.12a), this wouldbe thecase if and only if the same relationholds for the componentsof ,:(2) (1A(2 = qii(l).

We claim that this is true for , as given by (4.19). ForPiu = wiRi

and so, by (4.19),(4.24a) = (q - ai)wiRi(4.24b) q4 = E 6(q - ai)aiwiRi.

For the equations of compressible flow, the first componentf(l) of flux and thesecond conserved quantity u(2) are equal. This implies that the first row of A =f, is(0, 1, 0). It follows then from the eigenvalue equationARi= aiRithat the secondcomponent of Ri is ai times its firstcomponent.This shows that the secondcomponentof (4.24a) equals the firstcomponent of (4.24b), as asserted.The use of , as given in (4.19) in a Boltzmann-likedifference scheme for the Eulerequations goes back to Sanders and Prendergast [21]. Other choices of ,uare found in[11], [17], and [18].(d) We analyze now the stability of Boltzmann-type schemes for linear equationswith constant coefficients. In this case we take ,u o dependlinearly on u,

(4.25) Au(q, ) = M(q) u,M a matrix-valued function. The consistencyconditions (4.1 1) become(4.26) fM(q) dq = I, qM(q) dq = A.The scheme (4.13) is

z(x, t) = M(q) v(x - qt) dq.Taking the Fouriertransformwe get

( t) = M(t)v ().A condition for stability is that M(t) should be powerbounded,uniformlyfor all t. Notethat there is no stability restriction of the time step t; the reason for this is thatBoltzmann-typeschemes automatically adjusttheir domainof dependence.Note that we have analyzed here the stability of scheme (4.13). The full scheme(4.14) is a combination of (4.13) and projection onto the space of piecewise-constant


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60 AMIRAM HARTEN, PETER D. LAX AND BRAM VAN LEERfunctions. The latter decreases every weighted L2 norm. Therefore if (4.13) decreasessome weighted L2norm,the combinedscheme (4.14) is L2-stable.

For the case (4.19), we haveM(q) = E 3(q - ai)Pi and M(t) eia

If A is symmetric, the Pj are orthogonal projections,and IIMQ)I-1; so in this case thescheme is stable. For A nonsymmetric, stability can be proved by replacingthe Euclideannorm by some matrix-weightednorm.(e) We have not carried out any stability analysis in the nonlinearcase, nor studiedthe interesting questionof how to assurethe entropycondition.(f) We conclude by observing that flux-splitting schemes cannot exactly resolvestationary discontinuities. For suppose that the stationary Rankine-Hugoniot conditionf(u) =f(v) is satisfied. It does not follow from this that also(4.27) f a(U) =f a(V).But for a split-flux scheme, (4.2b) shows that (4.27) is necessary for the exact resolutionof stationary discontinuities.The split-flux scheme of van Leer [15] for the Euler equations can represent astationary shock with, at best, one intermediate state, just as the Osher [16] scheme. Acontact discontinuity, however, will keep on spreading with the use of any split fluxscheme. This follows immediately from the diffusive nature of the underlying Boltzmannmodel.

REFERENCES[1] R. COURANT,E. ISAACSONAND M. REES, On the solution of nonlinearhyperbolicdifferential equations,Comm. PureAppl. Math., 5 (1952), pp. 243-255.[2] M. G. CRANDALLAND A. MAJDA, Monotone difference approximations for scalar conservation laws,Math. Comp., 34 (1980), pp. 1-21.[3] J. GLIMM, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl.Math., 18 (1965), pp.697-715.[4] S. K. GODUNOV,A differenceschemefor numerical computation of discontinuous solutions of equationsoffluid dynamics, Math. Sbornik,47 (1959), pp. 271-306. (In Russian.)[5] A. HARTEN, High resolution schemes for hyperbolic conservation laws, Report DOE/ER/03077-.67,New York Univ., March 1982.[6] A. HARTEN, J. M. HYMAN AND P. D. LAX, On inite-difference approximations and entropy conditionsfor shocks, Comm. PureAppl. Math., 29 (1976), pp. 297-322.[7] A. HARTENAND P. D. LAX, A random choicefinite-difference scheme for hyperbolic conservation laws,SIAM J. Numer. Anal., 18 (1981), pp. 289-315.[8] A. HARTEN, On the symmetric form of systems of conservation laws with entropy, ICASE Report 81-34,ICASE, Hampton, VA, 1981.[9] A. HARTENANDJ. M. HYMAN,A self-adjustinggridfor the computation of weak solutions of hyperbolicconservationlaws, Report LA9105, Center for Nonlinear Studies, Theoretical Division, Los AlamosNational Lab., Los Alamos, NM, 1981.[10] L. C. HUANG, Pseudo-unsteady difference schemes for discontinuous solutions of steady-state one-

dimensionalfluid dynamics problems, J. Comp. Phys., 42 (1981), pp. 195-211.[11] S. KANIEL AND J. FALCOVITZ,Transportapproach for compressible flow, Proc. 4th InternationalIRIASymposium on Computing Methods in Science and Engineering, Versailles, Dec. 1979, R. Glowinskiand P. L. Lions, eds., North Holland, Amsterdam, 1981.[12] P. D. LAX, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves,CBMS Regional Conference Series in Applied Mathematics 11, Society for Industrial and AppliedMathematics, Philadelphia, 1972.


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UPSTREAM DIFFERENCING AND GODUNOV SCHEMES 61[13] B. VAN LEER, Towards the ultimate conservativedifference scheme. III, J. Comp. Phys., 23 (1977) pp.263-275.[14] , Upwind differencing for hyperbolic systems of conservation laws, in Numerical Methods forEngineering, Vol. 1, Dunod,Paris, 1980, pp. 137-149.[15] , Flux-vector splitting for the Euler equations, presented at the 8th Internat. Conference onNumerical Methods for Engineering, Aachen, June, 1982.[16] S. OSHER, Numerical solution of singular perturbationproblems and hyperbolic systems of conservationlaws, in Mathematics Studies, 47, North-Holland, Amsterdam, 1981, pp. 179-205.[17] D. J. PULLIN, Direct simulation methodsfor compressible inviscid ideal gas flow, J. Comp. Phys., 34(1980), pp. 231-244.[18] R. D. RIETZ, One-dimensionalcompressiblegas dynamics calculations using the Boltzmann equation, J.Comp. Phys., 42 (1981), pp. 108-123.[19] P. L. ROE, The use of the Riemann problem in finite-difference schemes, in Proc. 7th InternationalConference on Numerical Methods in Fluid Dynamics, Stanford/NASA Ames, June 1980, LectureNotes in Physics, 141, Springer-Verlag,New York, 1981, pp. 354-359.[20] , Approximate Riemann solvers, parameter vectors, and differenceschemes, J. Comp. Phys., 43(1981), pp.357-372.[21] R. H. SANDERS AND K. H. PRENDERGAST, On the origin of the 3-kiloparsec arm, Ap. J., 188 (1974), pp.489-500.[22] J. L. STEGER AND R. F. WARMING, Flux-vector splitting of the inviscid gas dynamic equations withapplications tofinite-differencemethods, J. Comp. Phys., 40 (1981), pp. 263-293.[23] J. L. STEGER, A preliminary study of relaxation methods for the inviscid conservative gas dynamicsequations usingflux-vector splitting, Report 80-4, Flow Simulations, Inc., Sunnyvale, CA, August1980.

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