Generalized monotone schemes, discrete paths of extrema, and discrete entropy conditions
aa r X i v : . [ m a t h . NA ] N ov GENERALIZED MONOTONE SCHEMES, DISCRETE PATHSOF EXTREMA, AND DISCRETE ENTROPY CONDITIONS
Philippe G. LeFloch and Jian-Guo Liu ABSTRACT. Solutions to conservation laws satisfy the monotonicity property:the number of local extrema is a non-increasing function of time, and local maxi-mum/minimum values decrease/increase monotonically in time. This paper inves-tigates this property from a numerical standpoint. We introduce a class of fullydiscrete in space and time, high order accurate, difference schemes, called gener-alized monotone schemes . Convergence toward the entropy solution is proven viaa new technique of proof, assuming that the initial data has a finite number ofextremum values only, and the flux-function is strictly convex. We define discretepaths of extrema by tracking local extremum values in the approximate solution. Inthe course of the analysis we establish the pointwise convergence of the trace of thesolution along a path of extremum. As a corollary, we obtain a proof of convergencefor a MUSCL-type scheme being second order accurate away from sonic points andextrema.
Published as : Math. of Comput. 68 (1998), 1025–1055. Ecole Polytechnique, Palaiseau, France. CURRENT ADDRESS : Laboratoire Jacques-LouisLions, Centre National de la Recherche Scientifique, Universit´e de Paris 6, 4, Place Jussieu, 75252Paris, France. E-mail address: [email protected]. was supported in parts by the Centre National de la Recherche Scientifique,and by the National Science Foundation under grants DMS-88-06731, DMS 94-01003and DMS 95-02766, and a Faculty Early Career Development award (CAREER). Temple University, Philadelphia, USA. CURRENT ADDRESS : Department of Mathematics,University of Maryland, College Park, MD 20742-4015, USA.J.G.L. was partially supported by DOE grant DE-FG02 88ER-25053.
Mathematics Subject Classification : 35L65, 65M12.
Key words : conservation law, entropy solution, extremum path, monotone scheme, highorder accuracy, MUSCL scheme. Typeset by
AMS -TEX P.G. LEFLOCH AND J.G. LIU
1. Introduction
This paper deals with entropy solutions to the Cauchy problem for a one-dimensional scalar conservation law: ∂ t u + ∂ x f ( u ) = 0 , u ( x, t ) ∈ RI , t > , x ∈ RI , (1.1) u ( x,
0) = u ( x ) , x ∈ RI , (1.2)where the flux f : RI → RI is a given function of class C and the initial data u belongs to the space BV ( RI ) of all integrable functions of bounded total variation.For the main result of this paper, we assume that f is a strictly convex function (1.3)and u has a locally finite number of extrema. (1.4)Solutions to conservation laws are generally discontinuous and an entropy criterionis necessary to single out a unique solution. We refer the reader to Lax [20, 21] forbackground on nonlinear hyperbolic equations and the entropy criterion.As is well-known [19, 37], problem (1.1)-(1.2) admits a unique entropy solution u in the space L ∞ (cid:0) RI + , BV ( RI ) (cid:1) and Lip (cid:0) RI + ; L ( RI ) (cid:1) . (This result holds withoutthe restriction (1.3)-(1.4) and for multidimensional equations as well.) To selectthe solution one can use the distributional entropy inequality ∂ t U ( u ) + ∂ x F ( u ) ≤ , (1.5)where the (Lipschitz continuous) function ( U, F ) : RI → RI is a convex entropy-entropy flux pair, i.e. U is a convex function and F ′ = U ′ f ′ . One can also use theLax shock admissibility inequality u ( x − , t ) ≥ u ( x + , t ) (1.6)for all x and t . Since f is convex and u has bounded variation, a single entropy U in (1.5) is sufficient to ensure uniqueness, and (1.6) is equivalent to (1.5). Atthe discrete level, however, conditions (1.5) and (1.6) leads two drastically differentnotions of consistency with the entropy criterion for a difference scheme. In thepresent paper we will be using both conditions. Indeed in some regions of the ( x, t )plane it is easier to use a discrete version of (1.5), while in other regions (1.6) ismore adapted.We are interested in conservative discretizations of problem (1.1)-(1.2) in thesense of Lax and Wendroff [22]. The monotone schemes and, more generally, theE-schemes are large classes of schemes (including the Godunov and Lax-Friedrichsschemes) for which a convergence analysis is available. The main point is thatmonotone schemes satisfy a discrete version of the entropy inequality (1.5) (see(2.9) below). However they turn out to be first order accurate only, and so have ENERALIZED MONOTONE SCHEMES 3 the disadvantage to introduce a large amount of numerical viscosity that spreadsthe discontinuities over a large number of computational cells.The proof of convergence of the monotone schemes and E-schemes is based onHelly’s and Lax-Wendroff’s theorems. See [3, 6, 8, 11, 16, 26, 31, 35, 23] and thereferences therein. The result holds even for multidimensional equations and/orwhen irregular (non-Cartesian) meshes are used.To get high-order accurate approximations, it is natural to proceed from ana-lytical properties satisfied by the entropy solutions to (1.1), formulate them at thediscrete level, and so design large classes of high-order difference schemes. Onecentral contribution in this direction is due to Harten [13, 14], who introduced theconcept of TVD schemes, for “Total Variation Diminishing”. Harten shows thatconservative, consistent, TVD schemes necessarily converge to a weak solution to(1.1). Moreover such schemes possess sharp numerical shock profiles with no spu-rious oscillations. However, Harten’s notion of TVD scheme is weaker than thenotion of monotone scheme, and a TVD need not converge to the entropy solution.The aim of this paper is precisely to single out a subclass of TVD schemes, refiningHarten’s notion, that are both high-order order accurate and entropy satisfying,cf. Definition 2.2 below.A very large literature is available on the actual design of second-order shock-capturing schemes. One approach to upgrading a first order scheme was proposedby van Leer [23, 24]: the MUSCL scheme extends the Godunov scheme by replacingthe piecewise constant approximation in the latter with a piecewise affine approxi-mation. The heart of the matter is to avoid the formation of spurious oscillationsnear discontinuities. This is achieved by van Leer via the so-called min-mod limitor.Other classes of high order schemes have been built from the maximum principleand a monotony condition: see the classes of ENO and UNO schemes proposedby Harten, Engquist, Osher, and Chakravarthy [15]. In this paper we concentrateattention on the MUSCL scheme, but our main convergence theorem, Theorem 2.3below, should also be applicable to other schemes.After Osher’s pioneering result [32] and the systematic study of high-order schemesby Osher and Tadmor [33] (which however required a technical condition on theslopes of the affine reconstructions), Lions and Sougadinis [27, 28] and, indepen-dently, Yang [38] established the convergence of the MUSCL scheme. Both proofsapply to an arbitrary BV initial data (so (1.4) is not assumed); the flux-function isassumed to be convex in [38] and strictly convex in [28]. In both papers significantlynew techniques of proof are introduced by the authors. In [38], Yang develops amethod for tracking local extremum values. In [28], Lions and Sougadinis elegantlyre-formulate the MUSCL scheme at the level of the Hamilton-Jacobi equation asso-ciated with (1.1) and rely on Crandall-Lions’ theory of viscosity solutions for suchequations. A large class of difference approximations is treated in [28]. Techniquesin both papers are restricted in an essential way to semi-discrete schemes, in whichthe time variable is kept continuous. In [25] the present authors announced a proofof the convergence of a class of fully discrete schemes that include the MUSCLscheme. Independently, Yang [39] also extended his approach to a large class offully discrete methods.
P.G. LEFLOCH AND J.G. LIU
Several other works deal with the convergence of the van Leer’s scheme or vari-ants of it. A discrete version of inequality (1.6) was established by Brenier andOsher [2] and Goodman and LeVeque [12], the latter dealing with both first andsecond order methods. Nessyahu, Tadmor, and Tamir [30] establish both the con-vergence and error estimates for a variety of Godunov-type schemes. Various Ap-proaches to upgrading Lax-Friedrichs scheme are actively developed by Tadmor andco-authors; see, for instance, Nessyahu and Tadmor [29]. An extensive discussion ofthe discretization of the entropy inequality (1.5) is found in Osher and Tadmor [33]and the many references cited theirein. See also Bouchut, Bourdarias and Perthame[1], and Coquel and LeFloch [4].The objective of the present paper is to provide a framework to prove the conver-gence of high order accurate and fully discrete difference schemes. We will stronglyrely on a property shared by all entropy solutions to (1.1): the monotonicity prop-erty . Given an arbitrary entropy solution u , the number of extrema in u ( t ) is anon-increasing function of t , and local maximum/minimum values decrease/increasemonotonically in time. (See the Appendix for rigorous statements.) This propertywas studied first by Harten [13] from the numerical standpoint, in order to arriveat his notion of TVD scheme. It was also essential in [28, 38] and in Tadmor [36].Observe that monotone or TVD schemes do not necessarily satisfy the mono-tonicity property. For instance the Lax-Friedrichs scheme may increase the num-ber of extremum values! This motivates the introduction of a subclass of TVDschemes, guaranteeing this property together with the high order of accuracy. Thescheme then closely mimics an important property of the solutions to the con-tinuous equation. Further requirement is necessary to ensure that the scheme isentropy-satisfying.Based on the monotonicity property, we thus introduce the notion of generalizedmonotone scheme (Definition 2.2) characterized by three conditions: maximumprinciple, entropy consistency, and local behavior at extrema. The (first order)Godunov method and the (second order) MUSCL method are prototype examples.Our first requirement is a (strong) version of the local maximum principle , whichwill allow us to show a discrete analogue of the monotonicity property. In particularour condition prevents the formation of spurious numerical oscillations and thescheme is TVD in the sense of Harten.The second requirement is motivated by the following observation made by Os-her for semi-discrete schemes [31]: in the non-decreasing parts of the approximatesolutions, it is possible to simultaneously achieve second order accuracy and the ex-istence of one cell entropy inequality –i.e. a discrete analogue of (1.5). Our secondcondition therefore requires a cell entropy inequality in the non-decreasing regions.Finally, in order to prevent cusp-like behavior near extrema, which may leadto entropy violating discontinuities, the scheme should be well-behaved near localextrema. To this end, we introduce a condition referred to as the quadratic decayproperty at local extrema . It reflects the nonlinear behavior of the numerical flux-function near extrema, and assumption (1.3) again is essential. For simplicity, itis also assumed that the scheme reduces to a three points, first order scheme atextrema. ENERALIZED MONOTONE SCHEMES 5
The rest of this section is devoted to comments upon the proof of our main resultthat any generalized monotone scheme converges to the entropy solution of (1.1)-(1.2) provided (1.3)-(1.4) holds. Our approach was driven by Yang’s paper [38] onsemi-discrete schemes. However, our technical arguments differ substantially fromthe ones in [38]. In particular we restrict attention to initial data having a locallyfinite number of extrema, making the tracking of paths of extremum values almosta trivial matter. The main part of our proof is studying the convergence of thetraces of the approximate solution along it. We make use of the quadratic decayproperty above to exclude the formation of “cusp” near extremum values, whichcould lead to entropy-violating shock. This is the main contribution of this presentpaper. We do not believe that the extension of our proof to arbitrary BV data isstraightforward. The result we obtain seems satisfactory however since condition(1.4) covers “generic” initial data.The proof distinguishes between the non-increasing parts and non-decreasingparts of the solution, and is based on several observations as follows. We use thenotation u h for the approximate solutions, v for the limiting solution, and h for themesh size.First of all, the strong maximum principle ensures that the u h ’s are total vari-ation diminishing in time, so of uniformly bounded total variation. The strongmaximum principle enables to easily define a discrete path and each time step thediscrete path move at most one grid points. As a consequence, the discrete pathis Lipschitz continuous. (This is a major difference between the present paper andYang’s paper in which the construction of the path and the limiting paths are muchmore involved.)By Helly’s Theorem, the scheme converges in the strong L topology to a limit-ing function, say v , which according to Lax-Wendroff’s theorem is a weak solutionto (1.1)-(1.2). It remains to prove that v is the entropy solution. We show thatthe strong maximum principle in fact implies a discrete analogue of the monotonic-ity property. Relying on (1.4), we construct a (locally) finite family of Lipschitzcontinuous paths in the plane by tracking the local extrema in u h . The paths areshown to converge in the uniform topology to limiting curves.Next we make the following two observations. On one hand, in a non-increasingregion for u h , the function v is also non-increasing, and so can only admit non-increasing jumps. Thus v satisfies the Lax shock admissibility inequality (1.6) inthe non-increasing regions. On the other hand, a discrete cell entropy inequality,by assumption, holds in the non-decreasing regions of u h . So v satisfies (1.5) in thenon-decreasing regions.It remains to prove that v has only non-increasing jumps along any path ofextrema. This is the most interesting part of the proof. Let ψ h be an approximatepath of extrema, and let ψ be its uniform limit. The path ψ is the boundaryseparating two regions where the analysis in the paragraph above applies. Indeed(1.5) holds in the side where v is non-increasing and (1.6) holds in the side where v is non-decreasing. A specific proof must be provided to determine the behaviorof v along the path . We analyze the entropy production in a small region of theplane limited on one side by the path ψ h . In the course of this proof we derive a P.G. LEFLOCH AND J.G. LIU uniform bound for the time integral of the local oscillation in space along the path ψ h , which is a direct consequence of the quadratic decay property mentioned above.For the sake of simplicity, we use here the assumption that the scheme reduces toa three points, first order scheme at extrema.Note that assumption (1.3) is not used in the construction of the extremumpaths, but is essential in the convergence analysis which strongly relies upon (1.6),only valid for convex fluxes.Our analysis shows that, for a class of difference schemes, certain approximategeneralized characteristics –those issued from an extremum point of the initial data–can be constructed for scalar conservation laws with convex flux. Constructingapproximate generalized characteristics issued from an arbitrary point remains achallenging open problem. Recall that, for the random choice scheme, Glimm-Lax [10] did obtain a general theory of approximate generalized characteristics(applicable to systems, as well).The organization of this paper is as follows. In Section 2, we define the class ofgeneralized monotone schemes and state the main result of convergence, cf. Theo-rem 2.3. Section 3 contains the proof of the main result. In Section 4, we applyTheorem 2.3 to the MUSCL scheme and provide some additional remarks.
2. Generalized Monotone Schemes
This section introduces a class of TVD schemes which are built to closely mimican essential property of the entropy solutions to (1.1), i.e. the monotonicity prop-erty: the number of local extrema is a non-increasing function of time, and localmaximum/minimum values decrease/increase monotonically in time. See Appen-dix for a precise statement. We investigate here this property from a numericalstandpoint. Monotone and TVD schemes actually do not necessarily satisfy thisproperty (see Section 4 for an example), and a more restricted class, the generalizedmonotone schemes , is natural.We consider a (2 k + 1)–point difference scheme in conservation form for theapproximation of (1.1)-(1.2): u n +1 j = u nj − λ (cid:0) g nj +1 / − g nj − / (cid:1) , n ≥ , j ∈ Z , (2.1)where we use the notation g nj +1 / = g (cid:0) u nj − k +1 , . . . , u nj + k (cid:1) and λ = τ /h for the ratioof the time-increment τ by the space-increment h . We set t n = nτ , x j = jh , and x j +1 / = ( j + 1 / h . The value u nj presumably is an approximation of the exactsolution at the point ( x j , t n ). As is usual, the numerical flux g : RI k → RI is assumedto be locally Lipschitz continuous and consistent with f , i.e. g ( v, . . . , v ) = f ( v ) forall v . Note that g may depend on λ . For definiteness, we set u j = 1 h Z x j +1 / x j − / u ( y ) dy. (2.2)This is sufficient for second order accuracy. For higher orders, one should use aRunge-Kutta time-step method (Shu [34]). We also define the piecewise constant ENERALIZED MONOTONE SCHEMES 7 function u h : RI × RI + → RI by u h ( x, t ) = u nj , t n ≤ t < t n +1 , x j − / ≤ x < x j +1 / . (2.3)By construction u h is a right continuous function. For simplicity we assume thefollowing CFL restriction: λ sup v | f ′ ( v ) | ≤ / . (2.4)Several of the properties below would still hold if, in (2.4), one replaces 1 / v and w , we define R ( . ; v, w ) to be the entropy solution to (1.1)-(1.2) with, here, u ( x ) = v if x < , w if x > . As is well-known, R ( · ; v, w ) depends on the self similarity variable x/t only, and isgiven by a closed formula. If v ≤ w , R is a rarefaction wave and, if v > w , a shockwave. More important R ( xt ; v, w ) is a monotone function connecting v to w. (2.5)In Godunov scheme, one solves Riemann problems and, at each time level, oneprojects the solution on the space of piecewise constant functions. If u h ( t n ) isknown, let ˜ u ( x, t ) for t ≥ t n be the entropy solution to (1.1) assuming the Cauchydata ˜ u ( t n +) = u h ( t n +) . Since u h ( t n +) is a piecewise constant function, ˜ u is obtained explicitly by glue-ing together the Riemann solutions R ( . ; u nj , u nj +1 ). In view of (2.4), there is nointeraction between two nearby solutions, at least for t ∈ ( t n , t n +1 ). Set u h ( x, t n +1 +) = 1 h Z x j +1 / x j − / ˜ u ( y, t n +1 ) dy, x j − / ≤ x < x j +1 / . Using the conservative form of (1.1), the scheme can be written in the form (2.1)with k = 1 and g = g G given by g G ( v, w ) = f ( R (0+; v, w )) for all v and w. (2.6)The proofs of (2.7)–(2.10) stated below are classical matter; e.g. [3, 6, 16, 26].On one hand, one can think of the Godunov scheme geometrically as a two-stepmethod: a marching step based on exact (Riemann) solutions and an L projectionstep. In view of (2.5) and the monotonicity property of the L projection, one P.G. LEFLOCH AND J.G. LIU easily have a simple geometrical proof of the properties listed below. On the otherhand, an algebraic approach is based on the explicit formula deduced from (2.6).The Godunov scheme is monotone: the function g G is non-decreasing with re-spect to its first argument, and non-increasing with respect to its second one. Thisproperty implies that the scheme is monotonicity preserving , i.e.,if u nj , u nj +1 , . . . , u nj is a non-increasing (resp. non-decreasing) sequencefor some indices j < j , so is the sequence u n +1 j +1 , u n +1 j +2 , . . . , u n +1 j − . (2.7)The Godunov scheme satisfies the local maximum principle , i.e.,min (cid:0) u nj − , u nj , u nj +1 (cid:1) ≤ u n +1 j ≤ max (cid:0) u nj − , u nj , u nj +1 (cid:1) (2.8)for all n ≥ j ∈ Z . In fact (2.7) and (2.8) are shared by both steps in theGodunov scheme. It will convenient to us to rewrite (2.8) in term of the jumps of u h at the endpoints of a cell:min (cid:0) u nj +1 − u nj , u nj − − u nj , (cid:1) ≤ u n +1 j − u nj ≤ max (cid:0) u nj +1 − u nj , u nj − − u nj , (cid:1) . (2.8’)Any monotone scheme –in particular the Godunov scheme– satisfies a discreteanalogue of (1.5) for every convex entropy pair ( U, F ): U ( u n +1 j ) − U ( u nj ) − λ (cid:0) G nj +1 / − G nj − / (cid:1) ≤ , n ≥ , j ∈ Z . (2.9)In (2.9), G nj +1 / = G ( u nj − k +1 , . . . , u nj + k ), and G is a numerical entropy flux consis-tent with F , that is G ( v, v . . . , v ) = F ( v ) for all v .Finally concerning the local behavior of u h in the neighborhood of local extrema,it is known that, say for local maximum, u n +1 j ≤ u nj . (2.10)A similar property holds for local minima.In fact the classical properties (2.8) and (2.10) can be improved as follows. Proposition 2.1.
Under assumptions (1 . and (2 . , the Godunov scheme satis-fies the following two properties: (1) the strong local maximum principle :
12 min (cid:0) u nj +1 − u nj , u nj − − u nj , (cid:1) ≤ u n +1 j − u nj ≤
12 max (cid:0) u nj +1 − u nj , u nj − − u nj , (cid:1) (2.11)(2) and the quadratic decay property at local extrema , that is e.g. for a maxima:If u nj is a local maximum value , u n +1 j ≤ u nj − α min ± (cid:0) ( u nj − u nj ± ) (cid:1) . (2.12) ENERALIZED MONOTONE SCHEMES 9 with α = λ inf f ′′ / . (cid:3) The proof of (2.11) is straightforward from a geometrical standpoint. If alsofollows from Proposition 4.1 established later in Section 4. Note that the coefficient1 / u n +1 j − u nj ) of the solution in thecell j in term of the jumps at the endpoints: the values u nj evolves “slowly” as t n increases. As we shall see, this property implies that the scheme satisfies a discreteanalogue of the monotonicity property.Estimate (2.12) is stronger than (2.10) and shows that the decrease/increase ofa maximum/minimum is controlled by the quadratic oscillation of u h nearby thisextremum. It is a truly nonlinear property of the Godunov flux. It will be usedbelow to prove that cusp can not form near extremum points. For convenience, theproof of (2.12) is postponed to Section 4, where second-order approximations aretreated as well.In [12], Goodman and LeVeque derive for the Godunov method a discrete versionof the Oleinik entropy inequality. In particular, this shows that the Godunov schemespreads rarefaction waves at the correct rate. Our estimate (2.12) is, at least inspririt, similar to this spreading estimate, and expresses the spreading of extremumvalues.We are now ready to introduce a class of high-order schemes based on the prop-erties derived in Proposition 2.1. Definition 2.2.
The scheme (2 . is said to be a generalized monotone scheme ifany sequence (cid:8) u nj (cid:9) generated by (2 . satisfies the following three conditions: (1) the strong local maximum principle (2 . , (2) the cell entropy inequality (2 . for one strictly convex pair ( U, F ) in anynon-decreasing region, including local extrema, (3) the quadratic decay property at local extrema (2 . for some constant α > .It is also assumed that the numerical flux and the numerical entropy flux are es-sentially two-point functions at local extrema. (cid:3) According Proposition 2.1, the (first order) Godunov scheme belongs to the classdescribed in Definition 2.2. Section 4 will show that there exist high order accurateschemes satisfying the conditions in Definition 2.2. Our main convergence result is:
Theorem 2.3.
Let (2 . be a generalized monotone scheme. Assume that assump-tions (1 . - (1 . hold together with (2 . . Then the scheme (2 . is L ∞ stable, i.e., inf l ∈ Z u nl ≤ u n +1 j ≤ sup l ∈ Z u nl , n ≥ , j ∈ Z , (2.13)(2) is total variation diminishing, i.e., X j ∈ Z (cid:12)(cid:12) u n +1 j +1 − u n +1 j (cid:12)(cid:12) ≤ X j ∈ Z (cid:12)(cid:12) u nj +1 − u nj (cid:12)(cid:12) , n ≥ , (2.14) (3) and converges in the L p loc strong topology for all p ∈ [1 , ∞ ) to the entropysolution of (1 . - (1 . . (cid:3) The proof of Theorem 2.3 is given in Section 3.Theorem 2.3 is satisfactory for a practical standpoint. Suppose that u is anarbitrary BV function, and we wish to compute an approximation to the solution u of (1.1)-(1.2) of order ε > L norm. Let us determine first an approximationof u , say u ,ε , that has a finite number of local extrema and such that k u ,ε − u k L ( RI ) ≤ ε. Applying a generalized monotone scheme to the initial condition u ,ε yields anapproximate solution u hε that, in view of Theorem 2.3, satisfies k u hε − u ε k L ( RI ) ≤ o ( h ) ≤ ε for h is small enough, where u ε is the entropy solution associated with the initialcondition u ,ε . Since the semi-group of solutions associated with (1.1) satisfies the L contraction property, one has k u ε − u k L ( RI ) ≤ k u ,ε − u k L ( RI ) ≤ ε, and therefore k u hε − u k L ( RI ) ≤ k u hε − u ε k L ( RI ) + k u ε − u k L ( RI ) ≤ ε.
3. Convergence Analysis
The proof of Theorem 2.3 is decomposed into several lemmas, Lemmas 3.1–3.12.For the whole of this section, we assume that the hypotheses made in Theorem 2.3are satisfied.We introduce first some notation and terminology. We call u nj a local maximumor a local minimum if there exist two indices j ∗ and j ∗ with j ∗ ≤ j ≤ j ∗ such that u nj ∗ = u nj ∗ +1 = · · · = u nj ∗ > max( u nj ∗ − , u nj ∗ +1 )or u nj ∗ = u nj ∗ +1 = · · · = u nj ∗ < min( u nj ∗ − , u nj ∗ ) . In such a case, there is no need to distinguish between the extrema u nj ∗ , u nj ∗ +1 , . . . , u nj ∗ . Based on the strong maximum principle (2.11), we show in Lemmas 3.1and 3.2 that the scheme satisfies a discrete form of the monotonicity property. Weconstruct a family of paths in the ( x, t )-plane by tracing in time the points wherethe approximate solution u h ( t ) achieves its local extremum values. One difficultyis proving that the interaction of two (or more) paths does not create new paths,so the total number of paths at any given time remains less or equal to the initialnumber of local extrema in u . In passing we observe that an extremum pointmoves one grid point at each time-step, at most. ENERALIZED MONOTONE SCHEMES 11
Lemma 3.1.
For some j ∗ < j ∗ , suppose that the sequences u nj ∗ − , u nj ∗ − , u nj ∗ − , u nj ∗ and u nj ∗ , u nj ∗ +1 , u nj ∗ +2 , u nj ∗ +3 are two monotone sequences, no specific assump-tion being made on the values u nj , j ∗ ≤ j ≤ j ∗ . Then the number ν ′ of extrema inthe sequence S n +1 := (cid:0) u n +1 j (cid:1) j ∗ − ≤ j ≤ j ∗ +2 is less or equal to the number ν of extrema in S n := (cid:0) u nj (cid:1) j ∗ − ≤ j ≤ j ∗ +2 . When ν ′ ≥ , there exists a one-to-one correspondence between ν ′ local extremaof S n and the ν ′ local extrema of S n +1 with the following property: if a maxi-mum/minimum u nj is associated with a maximum/minimum u n +1 j ′ , then | j ′ − j | ≤ and u n +1 j ′ ≤ u nj , resp. u n +1 j ′ ≥ u nj . (3.1) (cid:3) Proof.
We distinguish between various cases depending on the number of localextrema in the sequence S n and construct the one-to-one correspondence.If S n has no local extremum, for instance is non-decreasing, then S n +1 is alsonon-decreasing. This indeed follows from inequalities (2.11) which reduce in thiscase to · · · ≤ (cid:0) u nj − + u nj (cid:1) ≤ u n +1 j ≤ (cid:0) u nj +1 + u nj (cid:1) ≤ u n +1 j +1 ≤ · · · Consider next the case that S n has exactly one local extremum, say a localmaximum at some u nl . The same argument as above shows that the sequences (cid:8) u n +1 j (cid:9) j ∗ ≤ j ≤ l − and (cid:8) u n +1 j (cid:9) l +1 ≤ j ≤ j ∗ are non-decreasing and non-increasing re-spectively. Therefore we only need to exclude the case that both u n +1 l − > u n +1 l and u n +1 l +1 > u n +1 l , which would violate the monotonicity property since S n +1 inthis case would have two local maximum and one local minimum, so two new ex-trema. Indeed assume that the latter would hold, then using (2.11) at the points l − l , and l + 1 gives us u n +1 l − ≤ (cid:0) u nl − + u nl (cid:1) ,u nl + 12 min (cid:0) u nl ± − u nl (cid:1) ≤ u n +1 l , and u n +1 l +1 ≤ (cid:0) u nl +1 + u nl (cid:1) , which are incompatible with the inequalities u n +1 l − > u n +1 l and u n +1 l +1 > u n +1 l .Consider now the case that S n has two local extrema, say one local maximumat l and one local minimum at m with l < m . We distinguish between three cases: If l < m −
2, then the two extrema can not “interact” and the arguments beforeshow that the solution at time t n +1 has the same properties.If l = m −
2, the two extrema can interact. Using (2.11) at each point j = l − , · · · , l + 3, one gets u n +1 l − ≤ (cid:0) u nl − + u nl (cid:1) ,u nl + 12 min (cid:0) u nl ± − u nl (cid:1) ≤ u n +1 l , and u n +1 l +1 ≤ (cid:0) u nl +1 + u nl (cid:1) , and u n +1 l +1 ≤ (cid:0) u nl +1 + u nl +2 (cid:1) ,u nl +2 + 12 min (cid:0) u nl +2 ± − u nl +2 (cid:1) ≤ u n +1 l +2 , and u n +1 l +3 ≤ (cid:0) u nl +3 + u nl +2 (cid:1) . It is not hard to see that these inequalities imply that S n +1 (1) either has one maximum at either j = l − , l, l + 1 and a minimum at j = l + 1 , l + 2 , l + 3,(2) or is non-decreasing.In the first case, we achieve the property we wanted. In the second case, there isno extremum at the time t n +1 .This analysis can be extended to the case that several extrema can “interact”; weomit the details. Property (3.1) is a consequence of condition (2.4): an extremumpoint can only move up to one grid point at each time step. (cid:3) Consider the initial condition u and its approximation u h (0) defined by L projection, cf. (2.2). Locate the minimum and maximum values in the initial data u . For h much smaller than the minimal distance between two extrema, u h (0) hasthe same number of extrema as u and the same increasing/decreasing behavior as u . Indeed, there exist indices J hq (0) for q in a set of consecutive integers E ( u )depending on u but not on h , such that u j is non-decreasing for J h p (0) ≤ j ≤ J h p +1 (0) u j is non-increasing for J h p − (0) ≤ j ≤ J h p (0) . (3.2)Those indices are not uniquely determined in the case that u is constant on aninterval associated with a local extremum. Since u has a locally finite numberof local extrema, there exists a partition of Z into intervals ( j ∗ , j ∗ ) in which thehypothesis of Lemma 3.1 holds. It is an easy matter to use the one-to-one corre-spondence in Lemma 3.1 and trace forward in time up to time t = τ the locations ENERALIZED MONOTONE SCHEMES 13 of the extrema in ( j ∗ , j ∗ ) . At each time level a (possibly new) partition of Z isconsidered and Lemma 3.1 is used again. Indeed the values J hq ( n + 1) in Lemma3.2 below are defined from the J hq ( n )’s according to the one-to-one correspondenceestablished in Lemma 3.1. Finally piecewise affine and continuous paths are ob-tained by connecting together the points of local extrema. It may happen thatthe number of extrema decreases from time t n to t n +1 . In such a case, one path,at least, can no longer be further extended in time and so, for that purpose, weintroduce a “stopping time”, denoted by T hq = t n .The following lemma is established. Lemma 3.2.
There exist continuous and piecewise affine curves ψ hq : [0 , T hq ] → RI for q ∈ E ( u ) , passing through the mesh points (cid:0) x J hq ( n ) , t n (cid:1) and having the followingproperties: ψ hq ( t ) = x J hq ( n ) + t − t n τ (cid:0) x J hq ( n +1) − x J hq ( n ) (cid:1) , t ∈ [ t n , t n +1 ] , (3.3) for each n = 0 , , , . . . , N hq with T hq = N hq τ ≤ ∞ , ψ hq ≤ ψ hq +1 , | x J hq ( n ) − x J hq ( n +1) | ≤ h ; (3.4) there is only a finite number (uniformly bounded w.r.t. h) of curves ψ hq on each compact set (3.5) and x ∈ ( ψ h p ( t n ) , ψ h p +1 ( t n )) u h ( x, t n ) is non-decreasing, x ∈ ( ψ h p − ( t n ) , ψ h p ( t n )) u h ( x, t n ) is non-increasing. (3.6) Furthermore, the functions w hq : [0 , T hq ] → RI defined by w hq ( t ) = u nJ hq ( n ) for t n ≤ t < t n +1 (3.7) are non-decreasing if q is even, and non-increasing if q is odd. (cid:3) Remark 3.3.
1) The definition (3.3) is not essential. All the results below stillhold if ψ hq is replaced by any (uniformly) Lipschitz continuous curve passing throughthe mesh points (cid:0) x J hq ( n ) , t n (cid:1) . As a matter of fact, it is an open problem to showthe strong convergence of the derivatives of approximate paths. By comparison, forthe approximate solutions built by the random choice scheme, Glimm and Lax [10]prove the a.e. convergence of the first order derivatives of the paths.2) Introducing the stopping times T hq is necessary. At those times, certain pathscross each other and their extension in time is not well-defined. For instance apath of maximum and a path of minimum can cross and “cancel out”. The caseof exact solutions (Cf. the appendix) is simpler in this respect: the paths can bedefined to be characteristic curves for all times, even when they are no longer pathsof extrema.
3) It is not interesting to trace the minimal (or maximal) paths of extrema in theapproximate solution. Such paths would not converge to the paths obtained in thecontinuous case. (cid:3)
By construction, cf. (3.4), a path “jumps” up to one grid point at each time-step.So the slope of a path remains uniformly bounded by 1 /λ and the curves ψ hq arebounded in the W , ∞ l oc norm, uniformly with respect to h and q . On the other hand,Lemma 3.3 implies that the scheme is TVD so T V ( u h ( t n )) is uniformly bounded.We thus conclude that the approximate solutions and the paths of extrema arestrongly convergent, as stated in the following Lemmas 3.4 and 3.5. For simplicity,we keep the same notation for a sequence and a subsequence. Lemma 3.4.
There exist times T q ∈ [0 , + ∞ ] and Lipschitz continuous curves ψ q :(0 , T q ) → RI such that T hq → T q as h → , (3.8) and ψ hq → ψ q uniformly on each compact subset of C ([0 , T q )) . (3.9) (cid:3) Lemma 3.5.
The sequence u h satisfies estimates (2 . - (2 . , and so is uniformlystable in the L ∞ ([0 , ∞ ) , BV ( RI )) and Lip ([0 , ∞ ) , L ( RI )) norms. There exists afunction v in the same spaces such that u h ( x, t ) → v ( x, t ) for all times t ≥ and almost every x ∈ RI , (3.10) and there exist functions w q in BV ((0 , T q ) , RI ) such that w hq → w q almost everywhere on (0 , T q ) (3.11) for all q ∈ E ( u ) . (cid:3) The convergence results (3.10) and (3.11) hold in particular at each point ofcontinuity of v ( t ) and w q , respectively. Introduce now the following three sets,which provide us with a partition of the ( x, t )-plane into increasing/decreasingregions for v :Ω ( v ) = (cid:8) ( x, t ) / ψ p ( t ) < x < ψ p +1 ( t ) , p ≤ p , t < T p , t < T p +1 , and t > T q , for all q = 2 p + 1 , ..., p (cid:9) , Ω ( v ) = (cid:8) ( x, t ) / ψ p − ( t ) < x < ψ p ( t ) , p ≤ p , t < T p − , t < T p , and t > T q , for all q = 2 p , ..., p − (cid:9) , Ω ( v ) = Closure (cid:8) ( ψ q ( t ) , t ) , for all relevant values of t and q (cid:9) . The set Ω ( v ), by construction, contains all of the curves ψ q including their endpoints. The sets Ω ( v ) and Ω ( v ) are open and contain regions limited by curvesin Ω ( v ). These definitions take into account the fact that the path need not bedefined for all times. Observe also that an arbitrary point in Ω ( v ) need not be apoint of extremum value for v . The decomposition under consideration is not quitethe obvious partition of the ( x, t )-plane into regions of monotonicity for v . Strictlyspeaking, the sets Ω j ( v ) may not be determined from the function v alone.Using Lemmas 3.2, 3.4, and 3.5, we immediately check that: ENERALIZED MONOTONE SCHEMES 15
Lemma 3.6.
The limiting functions satisfy the properties: v / Ω ( v ) ( t ) is non-decreasing in each subcomponent of Ω ( v ) ,v / Ω ( v ) ( t ) is non-increasing in each subcomponent of Ω ( v ) , (3.12) and w q is non-decreasing if q is even and non-increasing if q is odd. (3.13) (cid:3) Since the scheme is consistent, conservative, and converges in the L strongnorm, we can pass to the limit in (2.1). It follows that v is a weak solution to (1.1).We note that, in the set Ω ( v ), the functions u h and, so v , are non-increasing.The Lax shock inequality holds for both u h and v . On the other hand, the cellentropy inequality (2.9) holds for u h in the non-decreasing regions, i.e., in Ω ( v ).The passage to the limit in (2.9) is a classical matter. Lemma 3.7.
The function v is a weak solution to equation (1 . and satisfies v ( x − , t ) ≥ v ( x + , t ) in the set Ω ( v ) (3.14) and ∂ t U ( v ) + ∂ x F ( v ) ≤ in the set Ω ( v ) . (3.15) (cid:3) The rest of this section is devoted to proving that the Lax shock inequality holdsalong the paths ψ q which we will attain in Lemma 3.10. In a first stage, we prove: Lemma 3.8.
Along each path of extremum ψ q and for almost every t ∈ (0 , T q ) ,one of the followings hold: w q ( t ) = v ( ψ q ( t ) − , t ) or w q ( t ) = v ( ψ q ( t )+ , t ) . (3.16) (cid:3) Roughly speaking (3.16) means that that no cusp-like layer can form in thescheme nearby local extrema. The idea of the proof of Lemma 3.8 is as follows:we are going to integrate the discrete form of the conservation law (2.1) on a(small) domain limited on one side by an approximate path of extremum, then weshall integrate by parts and pass to the limit as h →
0. Finally, we shall let thedomain shrink and reduce to the path itself. To determine the limits of the relevantboundary terms as h →
0, we have to justify the passage to the limit in particularin the numerical fluxes evaluated along the approximate path. Lemma 3.9 belowprovides us with an a priori estimate for the oscillation of u h along the path, whichfollows from the quadratic decay property (2.12). Lemma 3.9.
Along a path of extremum values ψ hq , we have β n = n + X n = n − min (cid:0) | u nJ hq ( n ) ± − u nJ hq ( n ) | , | u nJ hq ( n ) ± − u nJ hq ( n ) | (cid:1) ≤ u n − J hq ( n − ) − u n + J hq ( n + )+1 , (3.17) for all ≤ n − ≤ n + ≤ N hq , where β = min( α, / . (3.18) (cid:3) Proof of Lemma 3.8.
We will prove that, for almost every t in (0 , T q ), thefollowing three Rankine-Hugoniot like relations hold: − dψ q dt ( t ) (cid:0) v ( ψ q ( t )+ , t ) − v ( ψ q ( t ) − , t ) (cid:1) + f ( v ( ψ q ( t )+) , t ) − f ( v ( ψ q ( t ) − , t )) = 0 , (3.19) − dψ q dt ( t ) (cid:0) v ( ψ q ( t ) ± , t ) − w q ( t ) (cid:1) + f ( v ( ψ q ( t ) ± , t )) − f ( w q ( t )) = 0 . (3.20)Since there is only one non-trivial pair of values that achieves a Rankine-Hugoniotrelation for a scalar conservation law with a strictly convex flux and a given shockspeed dψ q ( t ) /dt , the desired conclusion (3.16) follows immediately from (3.19)-(3.20).Observe that (3.19) is nothing but the standard Rankine-Hugoniot relation since v is a weak solution to (1.1) and ψ q is Lipschitz continuous. For definiteness weprove (3.20) in the case of the “+” sign. The proof of (3.20) with “ − ” sign isentirely similar. (Actually only one of the two relations in (3.20) suffice for thepresent proof.)Let θ ( x, t ) be a test-function having its support included in a neighborhood ofthe curve ψ q and included in the strip RI × (0 , T q ). So for h small enough the supportof θ is included into RI × (0 , T hq ) and all the quantities to be considered below makesense. Let us set θ nj = θ ( x j , t n ). To make use of estimate (3.17), it is necessaryto define a “shifted” path ˜ ψ hq , to be used instead of ψ hq . So we consider the set ofindices P hq = (cid:8) ( j, n ) / j ≥ J hq ( n ) + ǫ hq ( n ) (cid:9) , where ǫ hq ( n ) = 0 (respectively ǫ hq ( n ) = 1) if J hq ( n ) − J hq ( n ) + 1) achievesthe minimum in the left hand side of (3.17). A shifted path is defined by˜ ψ hq ( t ) = x J hq ( n )+ ǫ hq ( n ) + t − t n τ (cid:0) x J hq ( n +1)+ ǫ hq ( n +1) − x J hq ( n )+ ǫ hq ( n ) (cid:1) , t ∈ [ t n , t n +1 ] . Introducing the shifts ǫ hq ( n ) does not modify the convergence properties of the path.It is not hard to see, using solely the fact that the path is uniformly bounded inLipschitz norm, that as h → ψ hq → ψ q W , ∞ weak- ⋆ . (3.21) ENERALIZED MONOTONE SCHEMES 17
We also set ˜ w hq ( t ) = u nJ hq ( n )+ ǫ hq ( n ) for t n ≤ t < t n +1 . Using (3.17) and (3.7) of w hq , it is checked that˜ w hq → w q L strongly. (3.22)Consider I h ( θ ) ≡ − X ( n,j ) ∈P hq (cid:0) u n +1 j − u nj + λ ( g nj +1 / − g nj − / ) (cid:1) θ nj h = 0 , (3.23)which vanishes identically in view of (2.1). Using summation by parts gives I h ( θ ) = − X ( n,j ) ∈P hq (cid:0) u n +1 j θ n +1 j − u nj θ nj (cid:1) h + X n g nJ hq ( n )+ ǫ hq ( n ) − / θ nJ hq ( n ) τ + X ( n,j ) ∈P hq u n +1 j ( θ n +1 j − θ nj ) h + g nj +1 / ( θ nj +1 − θ nj ) τ = I h ( θ ) + I h ( θ ) + I h ( θ ) . (3.24)The passage to the limit in I h ( θ ) is an easy matter, since it has the classical formmet, for instance, in the Lax-Wendroff theorem. We find I h ( θ ) → I ( θ ) = Z Z(cid:8) x ≥ ψ q ( t ) (cid:9) (cid:0) v∂ t θ + f ( v ) ∂ x θ (cid:1) dxdt. (3.25)To deal with I h ( θ ), we recall that the flux g nJ hq ( n )+ ǫ hq ( n ) − / depend on two argumentsso satisfies g nJ hq ( n )+ ǫ hq ( n ) − / = f ( u nJ hq ( n )+ ǫ hq ( n ) ) + O (cid:0) | u nJ hq ( n ) − ǫ hq ( n ) − u nJ hq ( n )+ ǫ hq ( n ) | (cid:1) = f ( ˜ w hq ( t n )) + O (cid:0) | u nJ hq ( n ) − ǫ hq ( n ) − u nJ hq ( n )+ ǫ hq ( n ) | (cid:1) . (3.26)Indeed, by construction, the point J hq ( n ) + ǫ hq ( n ) − / ǫ hq ( n ) ∈ { , } , so J hq ( n ) + ǫ hq ( n ) − / ∈{ J hq ( n ) − / , J hq ( n ) + 1 / } . Using (3.26), estimate (3.17), the Cauchy-Schwartzinequality, and finally Lebesgue convergence theorem, it is not hard to prove that I h ( θ ) → I ( θ ) = Z RI + f ( w q ( t )) θ ( ψ q ( t ) , t ) dt. (3.27)It remains to prove that I h ( θ ) → I ( θ ) = − Z RI + dψ q dt ( t ) w q ( t ) θ ( ψ q ( t ) , t ) dt. (3.28) We return to the definition of the modified path and define e hq ( n + 1) by J hq ( n + 1) + ǫ hq ( n + 1) = J hq ( n ) + ǫ hq ( n ) + e hq ( n + 1) . Using only (3.21), one can prove X n e hq ( n ) u nJ hq ( n )+ ǫ hq ( n ) θ nJ hq ( n )+ ǫ hq ( n ) h → − Z RI + dψ q dt ( t ) θ ( ψ q ( t ) , t ) dt. (3.29)We claim that e hq ( n + 1) ∈ (cid:8) − , , (cid:9) . (3.30)Namely, using (2.11) and the definition of ǫ hq ( n + 1), we have either ǫ hq ( n ) = 1 J hq ( n + 1) = J hq ( n ) or J hq ( n + 1) + 1 , and e hq ( n + 1) = 1 or 1or ǫ hq ( n ) = 1 J hq ( n + 1) = J hq ( n ) or J hq ( n + 1) + 1 , and e hq ( n + 1) = − . The term I h ( θ ) then can be rewritten in the form I h ( θ ) = X j ≥ J hq ( n )+ ǫ hq ( n ) u n +1 j θ n +1 j h − X j ≥ J hq ( n )+ ǫ hq ( n ) u nj θ nj h = X j ≥ J hq ( n − ǫ hq ( n − u nj θ nj h − X j ≥ J hq ( n )+ ǫ hq ( n ) u nj θ nj h, so that I h ( θ ) = − X e hq ( n )= − u nJ hq ( n )+ ǫ hq ( n ) θ nJ hq ( n )+ ǫ hq ( n ) h + X e hq ( n )=1 u nJ hq ( n )+ ǫ hq ( n ) − θ nJ hq ( n )+ ǫ hq ( n ) − h. Observe that, in the second sum above, J hq ( n ) + ǫ hq ( n ) − J hq ( n −
1) + ǫ hq ( n − I h ( θ ) = I h , ( θ ) + I h , ( θ ) with I h , ( θ ) = X e hq ( n )=1 (cid:0) u nJ hq ( n )+ ǫ hq ( n ) − θ nJ hq ( n )+ ǫ hq ( n ) − − u nJ hq ( n )+ ǫ hq ( n ) θ nJ hq ( n )+ ǫ hq ( n ) (cid:1) h and I h , ( θ ) = X e hq ( n )=1 u nJ hq ( n )+ ǫ hq ( n ) θ nJ hq ( n )+ ǫ hq ( n ) h − X e hq ( n )= − u nJ hq ( n )+ ǫ hq ( n ) θ nJ hq ( n )+ ǫ hq ( n ) h. ENERALIZED MONOTONE SCHEMES 19
On one hand, we have I h , ( θ ) ≤ X e hq ( n )=1 (cid:12)(cid:12) u nJ hq ( n )+ ǫ hq ( n ) − θ nJ hq ( n )+ ǫ hq ( n ) − − u nJ hq ( n )+ ǫ hq ( n ) θ nJ hq ( n )+ ǫ hq ( n ) (cid:12)(cid:12) h ≤ O (1) X e hq ( n )=1 (cid:12)(cid:12) u nJ hq ( n )+ ǫ hq ( n ) − − u nJ hq ( n )+ ǫ hq ( n ) (cid:12)(cid:12) h + O (1) X e hq ( n )=1 (cid:12)(cid:12) θ nJ hq ( n )+ ǫ hq ( n ) − − θ nJ hq ( n )+ ǫ hq ( n ) (cid:12)(cid:12) h, and, in view of (3.17) and the smoothness property of θ , I h , ( θ ) ≤ O (1) h / (cid:0) X n | u nJ hq ( n ) − − u nJ hq ( n ) | (cid:1) / + O (1) h ≤ O (1) (cid:0) h + h / (cid:1) ≤ O (1) h / , which implies I h , ( θ ) → h → . (3.31)The expression for I h , ( θ ) can be simplified, namely I h , ( θ ) = − X n e hq ( n ) u nJ hq ( n )+ ǫ hq ( n ) θ nJ hq ( n )+ ǫ hq ( n ) h. Using (3.29), we find that I h , ( θ ) → I ( θ ) as h → . (3.32)In view of (3.25), (3.27), and (3.28), we conclude that I ( θ ) + I ( θ ) + I ( θ ) = 0 . (3.33)Finally, using in (3.33) a sequence of test-functions θ , whose supports shrink andconcentrate on the curve ψ q , the desired Rankine-Hugoniot relation (3.20) with the“+” sign follows at the limit. This completes the proof of Lemma 3.8. (cid:3) Proof of Lemma 3.9.
For definiteness, we assume that u nJ hq ( n ) is a maximumvalue and that: min (cid:0) u nJ hq ( n ) − u nJ hq ( n ) ± (cid:1) = u nJ hq ( n ) − u nJ hq ( n ) − . The other cases are treated similarly. To simplify the notation, set j ∗ = J hq ( n ). Bythe uniform decay property (2.12), we have u nj ∗ − u n +1 j ∗ ≥ α ( u nj ∗ − u nj ∗ − ) . (3.34) By the strong maximum principle (2.11), we have u n +1 j ∗ − − u nj ∗ − ≤
12 ( u nj ∗ − u nj ∗ − ) . (3.35)From (3.35), we deduce u n +1 j ∗ − ≤ u nj ∗ − + 12 ( u nj ∗ − u nj ∗ − )= u nj ∗ −
12 ( u nj ∗ − u nj ∗ − ) ≤ u nj ∗ −
12 min (cid:0) ( u nj ∗ − u nj ∗ − ) , ( u nj ∗ − u nj ∗ − ) (cid:1) (3.36)Note that u nj ∗ − u nj ∗ − might be either ≤ ≥
1. Then we get u n +1 j ∗ − ≤ u nj ∗ − β min (cid:0) ( u nj ∗ − u nj ∗ − ) , ( u nj ∗ − u nj ∗ − ) (cid:1) . (3.37)On the other hand, (3.34) can be written in the form u n +1 j ∗ ≤ u nj ∗ − α ( u nj ∗ − u nj ∗ − ) . (3.38)Moreover, using (2.11) again, we obtain: u n +1 j ∗ +1 ≤ u nj ∗ +1 + 12 (cid:0) u nj ∗ − u nj ∗ +1 (cid:1) ≤ u nj ∗ + 12 (cid:0) u nj ∗ − u nj ∗ − (cid:1) . Hence, in view of (3.36): u n +1 j ∗ +1 ≤ u nj ∗ − β min (cid:0) ( u nj ∗ − u nj ∗ − ) , ( u nj ∗ − u nj ∗ − ) (cid:1) . (3.39)It follows from (3.37)–(3.39) thatmax (cid:0) u n +1 j ∗ − , u n +1 j ∗ , u n +1 j ∗ +1 (cid:1) ≤ u nj ∗ − β min (cid:0) ( u nj ∗ − u nj ∗ − ) , ( u nj ∗ − u nj ∗ − ) (cid:1) , (3.40)and thus u n +1 J hq ( n +1) ≤ u nj ∗ − β min (cid:0) ( u nj ∗ − u nj ∗ − ) , ( u nj ∗ − u nj ∗ − ) (cid:1) , (3.41)since u n +1 J hq ( n +1) by definition achieves the maximum in (3.20). Finally, we haveproved u nJ hq ( n ) − u n +1 J hq ( n +1) ≥ β min (cid:0) ( u nj ∗ − u nj ∗ − ) , ( u nj ∗ − u nj ∗ − ) (cid:1) . (3.42)By assumption, u nj ∗ − u nj ∗ − ≤ u nj ∗ − u nj ∗ . Thus (3.42) gives (3.17) after summationw.r.t. n . (cid:3) ENERALIZED MONOTONE SCHEMES 21
Lemma 3.10.
We have v ( x − , t ) ≥ v ( x + , t ) in the set Ω ( v ) . (3.43) (cid:3) The proof is based on the fact that one of the entropy criteria is satisfied oneach side of a path: the Lax shock inequality in the non-increasing side, and thecell entropy inequality in the non-decreasing one.
Proof.
We claim that, along any path of extremum ψ q , v ( ψ q ( t ) − , t ) ≥ v ( ψ q ( t )+ , t ) for a.e. t ∈ (0 , T q ) . (3.44)We use the notation introduced in the proof of Lemma 3.8. A new difficulty arises:several paths may accumulate in a region. Lemma 3.8 was concerned with thediscrete conservation laws (2.1) which holds in both the non-increasing and non-decreasing regions. For the entropy consistency, we do not use the same criterion,and this complicates the proof.To begin with, consider a path ψ q and a point ( ψ q ( t ) , t ), that is supposed tobe an “isolated” point of change of monotonicity, in the sense that: ψ q − ( t ) <ψ q ( t ) < ψ q +1 ( t ). By continuity, these inequalities then hold with t replaced byany t lying in a small neighborhood of t . For definiteness, we also suppose that ψ q is a path of minimum values. We later analyze the case that two or more pathsof extrema accumulate in a neighborhood of ( ψ q ( t ) , t ).Let θ be a non-negative test-function of the two variables ( x, t ) having its supportincluded in a small neighborhood of ( ψ q ( t ) , t ). We can always assume that u h is non-increasing on the left side of the curve ψ hq , and non-decreasing on the rightside. Using the notation introduced in the proof of Lemma 3.8, we aim at passingto the limit in I h ( θ ) = − X ( j,n ) ∈P hq (cid:0) U n +1 j − U nj + λ ( G nj +1 / − G nj − / ) (cid:1) θ nj h ≥ . Note that I h ( θ ) is non-positive according to the cell entropy inequality (2.9) andsince the u h ’s are non-decreasing on the right side.Integrating by parts in I h ( θ ) gives I h ( θ ) = − X ( j,n ) ∈P hq (cid:0) U n +1 j θ n +1 j − U nj θ nj (cid:1) h + X n G nJ hq ( n )+ ǫ hq ( n ) − / θ nJ hq ( n )+ ǫ hq ( n ) − / τ + X ( j,n ) ∈P hq U n +1 j ( θ n +1 j − θ nj ) h + G nj +1 / ( θ nj +1 − θ nj ) τ = I h ( θ ) + I h ( θ ) + I h ( θ ) . (3.45) The passage to the limit in the term I h ( θ ) is a classical matter. The treatment of I h ( θ ) and I h ( θ ) is similar to what was done to prove (3.27) and (3.28), respectively.Therefore we have I h ( θ ) → I ( θ ) = − Z RI + dψ q dt ( t ) U ( w q ( t )) θ ( ψ q ( t ) , t ) dt. (3.46) I h ( θ ) → I ( θ ) = Z RI + F ( w q ( t )) θ ( ψ q ( t ) , t ) dt, (3.47) I h ( θ ) → I ( θ ) = Z Z(cid:8) x ≥ ψ q ( t ) (cid:9) (cid:0) U ( v ) ∂ t θ + F ( v ) ∂ x θ (cid:1) dxdt. (3.48)It follows from (3.46)–(3.48) that I ( θ ) + I ( θ ) + I ( θ ) ≥ . (3.49)Finally, using a sequence of test-functions whose supports shrink and concentrateon the curve ψ q , we deduce from (3.49) that the entropy dissipation is non-positivealong the path, i.e. − dψ q dt ( t ) (cid:0) U (cid:0) v ( ψ q ( t )+ , t ) (cid:1) − U (cid:0) w q ( t ) (cid:1)(cid:1) + F ( v ( ψ q ( t )+ , t )) − F ( w q ( t )) ≤ . Combined with (3.20), this inequality is equivalent to v ( ψ q ( t )+ , t )) ≤ w q ( t ) , which yields the desired inequality (3.44).Consider next the case that several paths accumulate in the neighborhood of( ψ q ( t ) , t ). In view of (1.4), a finite number of paths only can accumulate at agiven point. For definiteness we suppose that– the point ( ψ q ( t ) , t ) is a point of minimum values for u h ;– the curves ψ q +1 and ψ q +2 coincide with ψ q in a neighborhood of t ;– and we have ψ q − < ψ q and ψ q +2 < ψ q +3 in a neighborhood of t .Suppose that, for instance, w q ( t ) ≤ w q +2 ( t ). The other cases are treated similarly.In this situation, we are going to prove that v ( ψ q ( t ) − , t ) ≥ w q ( t ) = w q +1 ( t ) ≥ w q +2 ( t ) = v ( ψ q ( t )+ , t ) , (3.50)at those points t near t where ψ q ( t ) = ψ q +1 ( t ) = ψ q +2 ( t ). Of course, (3.41) is amuch stronger statement than (3.34). By definition of the paths of extrema, wehave w q ( t ) ≤ w q +1 ( t ) and w q +1 ( t ) ≥ w q +2 ( t ) ,v ( ψ q ( t ) − , t ) ≥ w q ( t ) and w q +2 ( t ) ≤ v ( ψ q ( t )+ , t ) . ENERALIZED MONOTONE SCHEMES 23
Lemma 3.8 shows that w q ( t ) , w q +1 ( t ) , and w q +1 ( t ) ∈ (cid:8) v ( ψ q ( t ) − , t ) , v ( ψ q ( t )+ , t ) (cid:9) . Thus, in order to get (3.50), it is sufficient to check the following two inequalities w q +2 ( t ) ≥ v ( ψ q ( t )+ , t ) , (3.51) w q ( t ) ≥ w q +1 ( t ) . (3.52)On one hand, the argument used in the first part of the present proof appliesdirectly to the path ψ q +2 ( t ) and the region located to the right of this curve,since ψ q +2 ( t ) < ψ q +3 ( t ) in a neighborhood of t . As a consequence, we obtain w q +2 ( t ) ≥ v ( ψ q +2 ( t )+ , t ), which is exactly (3.51), since ψ q +2 = ψ q .On the other hand, to prove (3.52), let P hq,q +1 be the (small) region limited bythe curves ψ hq and ψ hq +1 for t belonging to a small neighborhood of t . Specifically, P hq,q +1 is a set of indices of the form ( j, n ) defined along the lines of the proof ofLemma 3.8. In particular, both paths are modified according to estimate (3.17), asexplained before. Consider I h ( θ ) = − X ( j,n ) ∈P hq,q +1 (cid:0) U n +1 j − U nj + λ ( G nj +1 / − G nj − / ) (cid:1) θ nj h ≥ . (3.53)Note that I h ( θ ) is non-negative and I h ( θ ) = − X ( j,n ) ∈P hq,q +1 (cid:0) U n +1 j θ n +1 j − U nj θ nj (cid:1) h + X n G nJ hq ( n )+ ǫ hq ( n ) − / θ nJ hq ( n )+ ǫ hq ( n ) − / τ − X n G nJ hq +1 ( n )+ ǫ hq +1 ( n ) − / θ nJ hq +1 ( n )+ ǫ hq +1 ( n ) − / τ + X ( j,n ) ∈P hq,q +1 U n +1 j ( θ n +1 j − θ nj ) h + G nj +1 / ( θ nj +1 − θ nj ) τ = I h ( θ ) + I h ,q ( θ ) − I h ,q +1 ( θ ) + I h ( θ ) . Using the technique developed for the proof of Lemma 3.8, we get I h ( θ ) → I ( θ ) = − Z RI + dψ q dt ( t ) U ( w q ( t )) θ ( ψ q ( t ) , t ) dt + Z RI + dψ q +1 dt ( t ) U ( w q +1 ( t )) θ ( ψ q +1 ( t ) , t ) dt,I h ,q ( θ ) → I ,q ( θ ) = Z RI + F ( w q ( t )) θ ( ψ q ( t ) , t ) dt,I h ,q +1 ( θ ) → I ,q +1 ( θ ) = − Z RI + F ( w q +1 ( t )) θ ( ψ q +1 ( t ) , t ) dt,I h ( θ ) → . It follows that I ( θ ) + I ,q ( θ ) + I ,q +1 ( θ ) ≥ , (3.54)which, since ψ q = ψ q +1 near t , is equivalent to the jump condition − dψ q dt ( t ) (cid:0) U ( w q +1 ( t )) − U ( w q ( t )) (cid:1) + F ( w q +1 ( t )) − F ( w q ( t )) ≤ , which gives (3.52). This completes the proof of Lemma 3.10. (cid:3) It is a classical matter to check that the initial condition (1.2) is satisfied byusing (2.2) and the uniform BV bound. Since the function v ( t ) has bounded totalvariation, it admits left and right traces at each point and (1.5) and (1.6) are knownto be equivalent at a point of discontinuity. Therefore the following result followsfrom Volpert’s proof in [37]. Lemma 3.11.
Supose v is a function of bounded variation and a weak solution tothe conservation law (1 . and satisfies the initial condition (1 . , and the inequal-ities (3 . , (3 . , and (3 . where Ω ( v ) ∪ Ω ( v ) ∪ Ω ( v ) = RI × RI + . Then v isthe unique entropy solution to (1 . - (1 . . (cid:3) The proof of Theorem 2.3 is now complete.The limiting paths ψ q associated with the scheme determine a decompositionof the plane into non-increasing/non-decreasing regions for the exact solution u .Such a decomposition is not unique, in general. Consider the decomposition foundin the Appendix for the function v and the corresponding paths ϕ p . When v isnot constant in any neighborhood of an extremum path ϕ p , the path is unique andmust coincide with one of the path ψ q . When v is constant in the neighborhood ofa path ϕ p , then the path may be arbitrarily modified and it may happen that nolimiting path ψ q coincide with ϕ p .
4. Application to the MUSCL Scheme
The purpose of this section is to apply Theorem 2.3 to van Leer’s MUSCL scheme(for Monotone Upstream Scheme for Conservation Laws); cf. [23, 24]. This sectionalso provides a proof of estimate (2.12) stated in Proposition 2.1, a new propertyof the Godunov scheme which does also hold for the MUSCL scheme.It is convenient to formulate (2.1) in terms of the incremental coefficients C ± ,nj +1 / defined by C + ,nj +1 / = − λ g nj +1 / − f ( u nj ) u nj +1 − u nj , C − ,nj − / = λ f ( u nj ) − g nj − / u nj − u nj − , (4.1)so that u n +1 j = u nj + C + ,nj +1 / (cid:0) u nj +1 − u nj (cid:1) + C − ,nj − / (cid:0) u nj − − u nj ) . (4.2)The numerical viscosity coefficient (Cf. Tadmor [35]) being defined by Q nj +1 / = C + ,nj +1 / + C − ,nj +1 / , (4.3)the viscous form of the scheme is u n +1 j = u nj − λ (cid:0) f ( u nj +1 ) − f ( u nj − ) (cid:1) + 12 Q nj +1 / (cid:0) u nj +1 − u nj (cid:1) − Q nj − / (cid:0) u nj − − u nj ) . ENERALIZED MONOTONE SCHEMES 25
Proposition 4.1.
The scheme (2 . satisfies the local maximum principle (2 . provided C + ,nj +1 / ≥ , C − ,nj − / ≥ , and C + ,nj +1 / + C − ,nj − / ≤ . (4.4) (cid:3) . A sufficient condition for (4.4) to hold is C ± ,nj +1 / ≥ Q nj +1 / ≤ / . (4.5)Namely, if (4.5) holds, then 0 ≤ C + ,nj +1 / ≤ / ≤ C − ,nj +1 / ≤ /
4, so that (4.4)is satisfied. In particular, the Godunov and Engquist-Osher schemes satisfy (4.4)under the CFL condition (2.4). When the numerical flux in independent of λ , thesecond inequality in (4.5) is always satisfied provided λ is small enough.The Lax-Friedrichs type scheme have a constant numerical viscosity Q nj +1 / ≡ Q .For the original Lax-Friedrichs scheme Q = 1. Proposition 4.1 applies provided Q ≤ /
4. Observe that the monotonicity property does fail when Q ∈ (2 / , f ≡ u nj ≡ j = 0 but u n >
0. This initial data hasone maximum point, and at the next time step u n +1 j = Q u nj +1 + (1 − Q ) u nj + Q u nj − . admits two maximum points j = − j = 1. A related observation was madeby Tadmor in [35]: for Q ≤ /
2, better properties can be obtained for the Lax-Friedrichs scheme.
Proof of Proposition 4.1.
Inequalities (2.11) can be written in terms of theincremental coefficients, namely12 min (cid:0) δ nj +1 , δ nj , δ nj − (cid:1) ≤ C + ,nj +1 / δ nj +1 + (cid:0) − C + ,nj +1 / − C − ,nj − / (cid:1) δ nj + C − ,nj − / δ nj − ≤
12 max (cid:0) δ nj +1 , δ nj , δ nj − (cid:1) (4.6)with δ nj +1 = u nj +1 − u nj , δ nj = 0 , δ nj − = u nj − − u nj . If (4.4) holds, then2 C + ,nj +1 / δ nj +1 + (cid:0) − C + ,nj +1 / − C − ,nj − / (cid:1) δ nj + 2 C − ,nj − / δ nj − )is a convex combination of the δ j ’s. So (4.6) and therefore (2.11) follows. (cid:3) We now introduce the van Leer’s scheme, composed of a reconstruction stepbased on the min-mod limitor and a resolution step based on the Godunov solver.
We use the notation introduced in Section 2. For simplicity in the presentation, wenormalize the flux to satisfy f (0) = f ′ (0) = 0. From the approximation (cid:8) u nj (cid:9) atthe time t = t n , we construct a piecewise affine function˜ u nj ( x ) = u nj + s nj ( x − x j ) /h for x ∈ ( x j − / , x j +1 / ) , (4.7)where the slope s nj is s nj = minmod (cid:0) u j − u j − , ( u j +1 − u j − ) / , u j +1 − u j (cid:1) (4.8)with minmod( a, b, c ) = min( a, b, c ) if a > , b > , and c > , max( a, b, c ) if a < , b < , and c < . u nj +1 / − = u nj + s nj / u nj +1 / = u nj +1 − s nj +1 / . Then the solution is up-dated with (2.1) where the numerical flux is defined de-pending upon the values of the reconstruction at the interfaces.(1) If either 0 ≤ u nj ≤ u nj +1 or 0 ≤ u nj +1 ≥ u nj , then the numerical flux isdefined by using the characteristic line traced backward from the point( x j +1 / , t n +1 / ). Since the latter has a positive slope, we set g nj +1 / = f ( u nj +1 / − ) + f ′ ( u nj +1 / − ) ( v − u nj +1 / − ) (4.10a)with v solving u nj +1 / − = v + λ f ′ ( v ) s nj . (4.10b)(2) If either u nj ≤ u nj +1 ≤
0, or u nj +1 / ≤ u nj ≤
0, then the backward characteris-tic has a negative slope and we set g nj +1 / = f ( u nj +1 / ) + f ′ ( u nj +1 / ) ( v − u nj +1 / ) (4.11a)with v solving u nj +1 / = v + λ f ′ ( v ) s nj +1 . (4.11b)(3) In all other cases we set g nj +1 / = f (0) = 0 . (4.12)Equations (4.10b) and (4.11b) can be solved explicitly for the Burgers equationsince then f ′ ( u ) = u is linear. Observe that the scheme reduces to first order atsonic points and extrema.The main result of this section is: ENERALIZED MONOTONE SCHEMES 27
Theorem 4.2.
For λ small enough, the MUSCL method defined by (4 . – (4 . is a generalized monotone scheme in the sense of Definition . . When (1 . - (1 . hold, the scheme converges in the strong L topology to the unique entropy solutionof (1 . - (1 . . (cid:3) It would be interesting to extend Theorem 4.2 to higher-order methods such asthe Woodward-Collela’s P.P.M. scheme.
Proof of Theorem 4.2.
We have to check that the scheme satisfies the threeconditions in Definition 2.2. We always assume that λ is, at least, less or equal to1 / Step 1:
Local maximum principle.Estimate (2.11) is easily obtained by applying Proposition 4.1 and relying on theconvexity of the flux function f . We omit the details. Step 2:
Cell entropy inequality.Consider a region where the sequence (cid:8) u nj (cid:9) is non-decreasing. We will use theentropy pair ( U, F ) with U ( u ) = u / F ′ ( u ) = uf ′ ( u ). Define the numericalentropy flux by G nj +1 / = F ( u nj +1 / − ) + U ′ ( u nj +1 / − ) ( f ( v ) − f ( u nj +1 / − )) , (4.13i) G nj +1 / = F ( u nj +1 / ) + U ′ ( u nj +1 / ) ( f ( v ) − f ( u nj +1 / )) , (4.13ii)and G nj +1 / = 0 (4.13iii)in Cases (1), (2), and (3), respectively. Inequality (2.9) is checked by direct calcu-lation, for λ small enough. Observe that Case (3) is obvious since our scheme thenreduces to a first order, entropy consistent scheme.For definiteness we treat Case (1), i.e. f ′ > u under consid-eration. We view the left hand side of (2.9) as a function of w = u j − / − , u = u nj , v = u j +1 / − and the value ˜ w defined as w = ˜ w + λ f ′ ( ˜ w ) t, where t stands for the slope in the cell j −
1. Introduce also ˜ v by v = ˜ v + λ f ′ (˜ v ) s with s = 2( v − u ). Since the approximate solution is non-increasing, we have˜ w ≤ w ≤ u − v ≤ u ≤ v ≤ ˜ v . SetΩ( ˜ w, w, u, v ; λ )= U (¯ u ) − U ( u ) + λ (cid:2) F ( v ) + U ′ ( v )( f (˜ v ) − f ( v )) − F ( w ) − U ′ ( w )( f ( ˜ w ) − f ( w )) (cid:3) , and ¯ u = u − λ (cid:2) f ( v ) + f ′ ( v )(˜ v − v ) − f ( w ) − f ′ ( w )( ˜ w − w ) (cid:3) . Observe that ∂ ˜ w Ω = U ′ (¯ u ) λf ′ ( w ) − λU ′ ( w ) f ′ ( ˜ w ) , and ∂ w Ω = U ′′ (¯ u ) λ f ′ ( w ) − λU ′ ( w ) f ′′ ( ˜ w ) ≤ − Cλ | w | for w > λ small enough. Therefore Ω is a concave function in ˜ w andΩ( ˜ w, w, u, v ; λ ) ≤ Ω( w, w, u, v ; λ ) − ( w − ˜ w ) ∂ ˜ w Ω( w, w, u, v ; λ ) − Cλ | w | | ˜ w − w | . But ∂ ˜ w Ω( w, w, u, v ; λ ) = λf ′ ( w ) (cid:0) U ′ (¯ u ) − U ′ ( w ) (cid:1) = U ′′ ( ξ ) λf ′ ( w ) (cid:2) u − w − λ (cid:0) f ( v ) + f ′ ( v )(˜ v − v ) − f ( w ) (cid:1)(cid:3) ≤ Cλ | u − w | for some ξ > λ small enough. This proves thatΩ( ˜ w, w, u, v ; λ ) ≤ Ω( w, w, u, v ; λ ) − Cλ (cid:0) | ˜ w − w || u − w | + | w | | ˜ w − w | (cid:1) ≤ Ω( w, w, u, v ; λ ) , and we now simply use the notation Ω( w, u, v ; λ ).Taylor expanding Ω with respect to λ shows that the dominant term is the firstorder coefficient in λ given byˆΩ ( w, u, v ) ≡ − U ′ ( u ) (cid:0) f ( v ) − f ( w ) (cid:1) + F ( v ) − F ( w ) , in which w ≤ u − v ≤ u ≤ v . Since ∂ w ˆΩ ( w, u, v ) = ( U ′ ( u ) − U ′ ( w ) (cid:1) f ′ ( w ) ≥ C | w | | u − w | , we have ˆΩ ( w, u, v ) ≤ ˆΩ (2 u − v, u, v ) − C | w | Z u − vw ( u − z ) dz, so ˆΩ ( w, u, v ) ≤ ˆΩ (2 u − v, u, v ) − C ′ | w | | u − v − w | | u − w | . It remains to study ˆΩ (2 u − v, u, v ) = ˜Ω ( u, v ) with u ≤ v . We find ∂ v ˜Ω ( u, v ) = (cid:0) U ′ ( v ) − U ′ ( u ) (cid:1) f ′ ( v ) + (cid:0) U ′ (2 u − v ) − U ′ ( u ) (cid:1) f ′ (2 u − v )= ( v − u ) (cid:0) f ′ ( v ) − f ′ (2 u − v ) (cid:1) ≤ − C | u − v | . It follows that ˜Ω( u, v ) is a non-increasing function of v for all v ≥ u , and since itvanishes for v = u , ˜Ω ( u, v ) ≤ − C ′ | u − v | . ENERALIZED MONOTONE SCHEMES 29
This proves that the first order term in λ in the expansion of the function Ω isnegative.The same arguments are now applied to the function Ω( λ ) directly. We have ∂ w Ω ( w, u, v ; λ ) = ( U ′ (¯ u ) − U ′ ( w )) λf ′ ( w )= λf ′ ( w ) U ′′ ( ξ )( u − w − λ ( f ( v ) + f ′ ( v )(˜ v − v ) − f ( w )))= λf ′ ( w ) U ′′ ( ξ )( u − w − λO (1)( u − w )) ≥ λC | w | | u − w | . Therefore,Ω( w, u, v ; λ ) ≤ Ω(2 u − v, u, v ; λ ) − λC ′ | w | | u − v − w | | u − w | . Denote ˆΩ( u, v ; λ ) = Ω(2 u − v, u, v ; λ )ˆΩ( u, v ; λ ) = U (¯ u ) − U ( u ) + λ (cid:2) F ( v ) + U ′ ( v )( f (˜ v ) − f ( v )) − F (2 u − v ) (cid:3) , where ¯ u = u − λ (cid:2) f ( v ) + f ′ ( v )(˜ v − v ) − f (2 u − v ) (cid:3) . We easily compute that ∂ v ˆΩ( u, v ; λ ) λ = f ′ ( v ) (cid:0) U ′ ( v ) − U ′ (¯ u ) (cid:1) + f ′ (2 u − v ) (cid:0) U ′ (2 u − v ) − U ′ (¯ u ) (cid:1) + λA ( u, v ; λ )with | A ( u, v ; λ ) | ≤ C | u − v | . This establishes the desired conclusion for λ small enough. Step 3:
Quadratic decay property.Near a local extremum, the MUSCL scheme essentially reduces to the Godunovscheme. So it is enough to check the quadratic decay property (2.12) for the Go-dunov scheme. This can be done from the explicit formula (2.6).The simplest situation is obtained with the Godunov scheme and when f ′ has asign, say is positive. Assume u nj is a local maximum. We have u n +1 j = u nj − λ (cid:0) f ( u nj ) − f ( u nj − ) (cid:1) , thus u nj − u n +1 j = λ (cid:0) f ( u nj ) − f ( u nj − ) (cid:1) ≥ λf ′ ( u nj − ) (cid:0) u nj − u nj − (cid:1) + λ (cid:0) inf f ′′ / (cid:1) (cid:0) u nj − u nj − (cid:1) ≥ λ (cid:0) inf f ′′ / (cid:1) min ± (cid:0) u nj − u nj ± (cid:1) . This establishes (2.11) when f ′ > It remains to treat the sonic case where f ′ has no definite sign. We will rely onthe following technical remark. Given two points such that u − < < u + , f ( u − ) = f ( u + ) , there exist c , c > u ± ) such that c | u − | ≤ | u + | ≤ c | u − | . Consider the case u nj − < < u nj , and use Osher’s formula for the Riemannproblem, we have u nj − u n +1 j = λ ( max ( u nj +1 ,u nj ) f − min ( u nj − ,u nj ) f )= λ (cid:0) max ( u nj +1 ,u nj ) f − f (0))= λ (cid:0) inf f ′′ / (cid:1) ( max | u nj +1 | , | u nj | f ) ≥ c | u nj +1 − u nj | . Consider next the case 0 < u nj − < u nj , then u nj − u n +1 j = λ (cid:0) max ( u nj +1 ,u nj ) f − f ( u nj )) ≥ λ (cid:0) inf f ′′ / (cid:1) min( | u nj +1 − u nj | , | u nj − u nj − | )This completes the proof of Theorem 4.2. (cid:3) Acknowledgments.
Most of this work was done in 1992 while P.G.L. and J.G.L.were Courant instructors at the Courant Institute of Mathematical Sciences, NewYork University. The authors are very grateful to Peter D. Lax for helpful remarkson a first draft of this paper.
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Appendix: Monotonicity Property.
In this appendix we brieffly discuss the monotonicity property together with morebasic properties of entropy solutions to conservation laws, which go back to Kruzkov[19] and Volpert [37]. In the paper by Keyfitz, [18], somewhat simpler proofs areavailable for piecewise Lipschitz continuous solutions. We are interested in thelocal versions of the properties, i.e. formulated in domains limited by characteristiccurves. To cope with discontinuous solutions, we use the concept of generalizedcharacteristic curves introduced for ordinary differential equations by Filippov [9]and developed in the context of conservation laws by Dafermos; see e.g. [7] andthe references therein. We recall that, through any point ( x , t ), there exists afunnel of forward and backward generalized characteristic curves, which fill up adomain (cid:8) ξ m ( t ) ≤ x ≤ ξ M ( t ) (cid:9) . Here ξ m (respectively ξ M ) is called the minimal ENERALIZED MONOTONE SCHEMES 33 (resp. maximal) characteristic curve originating at ( x , t ). It is known [9, 7] thata characteristic, say ξ , is Lipschitz continuous and for almost every time t > dξdt ( t ) = ( f ′ ( u ( ξ ( t ) , t )) if u − ( t ) = u + ( t ) = u ( ξ ( t ) , t ) , f ( u + ) − f ( u − ) u + − u − if u − = u + , (A.1)where u ± = u (cid:0) ξ ( t ) ± , t (cid:1) . For our purposes, f is strictly convex and there is aunique forward characteristic issued from ( x ) , t and there is no need to distinguishbetween the minimal characteristic and the maximal one, with the exception ofthose points where t = 0 and u has an increasing jump at x ; cf. [7].Solutions u to (1.1) are Lipschitz continuous in time with values in L and, forall times t , u ( t ) has bounded total variation in x .The following properties follows from [37, 19] and the technique of generalizedcharacteristic in [9, 7]. Proposition A.1.
Let u be the entropy solution to (1 . - (1 . . Given x and x with x < x , consider the maximal forward characteristic ξ u ( t ) issued from (0 , x ) and the minimal forward characteristic ξ u ( t ) from (0 , x ) . For all times t ≥ , u satisfies (1) the local maximum principle for all t ≤ s and y ∈ ( ξ u ( s ) , ξ u ( s )) : inf ξ u ( t )
Proof of Proposition A.2.
First of all, the points ϕ q (0) and the set E ( u ) aredefined from the initial condition u in an obvious way so that the conditions (A.7)–(A.10) hold true at time t = 0. Let us define ϕ p ( t ) to be the maximal forwardcharacteristic issued from ϕ p (0). Similarly, let ϕ p +1 ( t ) be the minimal forwardcharacteristics issued from ϕ p +1 (0). Indeed one need to distinguish between mini-mal and maximal characteristics only in the case of an initially increasing jump. Itmay happen that both a path of minimum and a path of maximum may originatefrom such a point of increasing jump.Property (A.7) is an immediate consequence of the uniqueness property of theforward characteristic. Condition (A.8) follows from the property of propagationwith finite speed satisfied by solutions to (1.1) and the fact that the initial data hasa locally finite number of local extremum. Indeed (A.1) yields a uniform bound forthe slopes of the characteristics.In order to establish (A.9) and (A.10), we first suppose that u does not admitincreasing jumps. Consider an interval of the form ( ϕ p ( t ) , ϕ p +1 ( t )) for thosevalues of t when this interval is not empty. Note first that, taking ξ u = ϕ p and ξ u = ϕ p +1 , the local maximum principle (A.2) implies in particular that u (cid:0) ϕ p ( t )+ , t (cid:1) ≥ u (cid:0) ϕ p (0)+ (cid:1) ,u (cid:0) ϕ p +1 ( t ) − , t (cid:1) ≤ u (cid:0) ϕ p +1 (0) − (cid:1) . (A.11)Let w be the entropy solution to (1.1) associated with the initial condition w ( x,
0) = w ( x ) ≡ u (cid:0) ϕ p (0)+ (cid:1) if x < ϕ p (0) ,u ( x ) if ϕ p (0) < x < ϕ p +1 (0) ,u (cid:0) ϕ p +1 (0) − (cid:1) if x > ϕ p +1 (0) . (A.12)The data w is non-decreasing and, in view of (A.6),the solution w is non-decreasing for all times. (A.13)Let ψ p and ψ p +1 be the forward characteristics associated with w and issued ϕ p (0) and ϕ p +1 (0) at time t = 0, respectively. Observe that the maximum forwardand the minimum forward curves coincide since by construction w is continuousat ϕ p (0) and ϕ p +1 (0). Note in passing that the function w satisfies: w ( x, t ) = (cid:26) u (cid:0) ϕ p (0)+ (cid:1) if x < ψ p ( t ) ,u (cid:0) ϕ p +1 (0) − (cid:1) if x > ψ p +1 ( t ) . (A.14)Using (A.11) and (A.14) and the fact that f ′ ( . ) is increasing, one gets dϕ p +1 dt ( t ) ≤ f ′ ( u ( ϕ p +1 ( t ) − , t )) ≤ f ′ ( u ( ϕ p +1 (0) − )) ≤ dψ p +1 dt ( t ) , which implies ϕ p +1 ( t ) ≤ ψ p +1 ( t ) . Similarly ψ p ( t ) ≤ ϕ p ( t ) . Using the L contraction principle (A.3), it follows that u = w for ϕ p ( t ) < x < ϕ p +1 ( t ) , and, in view of (A.13), the function u is non-decreasing and (A.9) holds. Using(A.9) and the local maximum principle (A.2) finally provides (A.10). The proofis complete in the case of an interval of the form ( ϕ p ( t ) , ϕ p +1 ( t )). An interval( ϕ p − ( t ) , ϕ p ( t )) can be treated in a similar fashion.It remains to consider increasing jumps in u . That situation can be treated byusing the following property. Suppose u has an increasing jump at a point x andlet ϕ m ( t ) and ϕ M ( t ) be the minimal and maximal forward curves from x . It isknown that at least for small times the function u ( t ) coincides with the rarefactionwave connecting the values u ( x ± ) in the interval (cid:0) ϕ m ( t ) , ϕ M ( t ) (cid:1) .This completes the proof of Proposition A.2..This completes the proof of Proposition A.2.