# - ESAIM: Mathematical Modelling and Numerical

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- ESAIM: Mathematical Modelling and Numerical

Mathematical Modelling and Numerical Analysis ESAIM: M2AN Vol. 35, No 2, 2001, pp. 239–269 Modélisation Mathématique et Analyse Numérique CONVERGENCE OF FINITE DIFFERENCE SCHEMES FOR VISCOUS AND INVISCID CONSERVATION LAWS WITH ROUGH COEFFICIENTS Kenneth Hvistendahl Karlsen 1 and Nils Henrik Risebro 2 Abstract. We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a “rough” coefficient function k(x). We show that the Engquist-Osher (and hence all monotone) finite difference approximations converge to the unique entropy solution of the governing equation if, among other demands, k0 is in BV , thereby providing alternative (new) existence proofs for entropy solutions of degenerate convectiondiffusion equations as well as new convergence results for their finite difference approximations. In the inviscid case, we also provide a rate of convergence. Our convergence proofs are based on deriving a series of a priori estimates and using a general Lp compactness criterion. Mathematics Subject Classification. 65M06, 35L65, 35L45, 35K65. Received: September 6, 2000. 1. Introduction The main subject of this paper is finite difference schemes for computing the entropy solution of scalar viscous and inviscid conservation laws where the transport term depends explicitly on the spatial location. Such equations are of the form ut + divf (k, u) = ∆A(u), u(x, 0) = u0 (x), (x, t) ∈ ΠT = Rd × (0, T ), (1.1) where the flux function f (k, u) = (f1 (k 1 , u), . . . , fd (k d , u)) depends on the spatial location through the coefficient k = k(x), k(x) = (k 1 (x), . . . , k d (x)). For the initial value problem (1.1) to be well-posed, we must require that the nonlinear elliptic operator u 7→ ∆A(u) satisfies the degenerate ellipticity condition A(·) nondecreasing with A(0) = 0. (1.2) Keywords and phrases. Conservation law, degenerate convection-diffusion equation, entropy solution, finite difference scheme, convergence, error estimate. 1 Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, 5008 Bergen, Norway. e-mail: [email protected]; URL: http://www.mi.uib.no/~kennethk/ 2 Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway. e-mail: [email protected]; URL: http://www.math.uio.no/~nilshr/ c EDP Sciences, SMAI 2001 240 K.H. KARLSEN AND N.H. RISEBRO Note that (1.2) implies that many well known nonlinear partial differential equations are special cases of (1.1). In particular (1.2) includes as special cases the inviscid conservation law, the heat equation, one-point degenerate porous medium type equations [43], two-point degenerate oil reservoir flow equations [15], and strongly degenerate convection-diffusion equations of the type arising in the theory of sedimentation-consolidation processes [4]. We recall that if (1.1) is allowed to degenerate at certain points, that is, A0 (s) = 0 for some values of s, solutions are not necessarily smooth (but typically continuous) and weak solutions must be sought. On the other hand, if A0 (s) is zero on an interval [α, β], (weak) solutions may be discontinuous and they are not uniquely determined by their initial data. Consequently, an entropy condition must be imposed to single out the physically correct solution. Roughly speaking, we call a function u ∈ L1 ∩ L∞ an entropy solution of (1.1) if ∂t |u − c| + div [sign (u − c) (f (k, u) − f (k, c))] − ∆ |A(u) − A(c)| (i) (1.3) +sign (u − c) divf (k, c) ≤ 0 in D0 ∀c ∈ R, 2 (ii) ∇A(u) belongs to L . We refer to Section 2.1 for a more precise statement of the definition of an entropy solution as well as precise conditions on u0 , f, k, A ensuring that this definition makes sense. Relevant mathematical (existence and uniqueness) theory for entropy solutions can be found in [3, 5, 28, 42]. For the hyperbolic equation, the convergence analysis of numerical schemes has very long traditions and goes back to the 1950s. Being extremely selective, we mention only a few references related to finite difference and finite volume approximations. The case of finite difference schemes have been treated by Oleı̆nik [40], Harten et al. [24], Kuznetsov [36], Crandall and Majda [12], Sanders [44], Lucier [37], Osher and Tadmor [41], Cockburn and Gremaud [10], and many others. The study of finite volume methods is more recent and have been conducted by Champier et al. [7], Vila [48], Cockburn et al. [8, 9], Kröner and Rokyta [32], Kröner et al. [31], Noelle [38], Eymard et al. [21], and Chainais-Hillairet [6], as well as many others. Among the cited papers, only [6, 7, 21, 40] treat equations where the nonlinearity f depends on the spatial position x (and time t). Although there seems to be an increasing interest in the (analysis of) numerical approximation of entropy solutions of degenerate parabolic equations, the amount of literature on the subject is at the moment modest. The (very recent) literature include papers by Evje and Karlsen [18], Holden et al. [25], and Holden et al. [26] on operator splitting methods (see also the lecture notes by Espedal and Karlsen [15]); Evje and Karlsen [16, 17, 19, 20] on upwind difference schemes; Kurganov and Tadmor [35] on central difference schemes; Bouchut et al. [2] on kinetic BGK schemes; Afif and Amaziane [1] and Ohlberger [39] on finite volume methods; and Cockburn and Shu [11] on the local discontinuous Galerkin method. Strictly speaking, the authors of [1, 11, 35] do not analyze their numerical methods within an entropy solution framework. It is somewhat surprising that there have been few attempts up to very recently (confer the list of references given above) to develop a systematic treatment of mixed hyperbolic-parabolic partial differential equations within a unified mathematical (entropy solution) framework. In fact, the construction and analysis of numerical methods for first order hyperbolic and second order parabolic equations are usually considered as separate subject areas. In this work we demonstrate that it is possible to give a coherent treatment of numerical methods for such large class of nonlinear partial differential equations. Our main long-term goal is indeed to develop a consistent (mathematical/numerical) framework which is the same whether we are working with the hyperbolic case (A0 ≡ 0), the parabolic case (A0 > 0), or with the mixed hyperbolic-parabolic case (A0 ≥ 0). In the present paper (see also [16,17,19,20]), we are concerned with finite difference schemes and their convergence analysis. For related work on other numerical methods for strongly degenerate parabolic equations, see the list of references given above. To illustrate the results of this paper, we now state them in the one dimensional case (i.e., d = 1). For the general results, we refer to Sections 3 and 4. As a model difference scheme for (1.1), we consider the generalized upwind (Engquist-Osher) scheme f EO ki+1/2 , uni , uni+1 − f EO ki−1/2 , uni−1 , uni A uni+1 − 2A (uni ) + A uni−1 − uni un+1 i + = , ∆t h h2 DIFFERENCE SCHEMES WITH ROUGH COEFFICIENTS 241 where the so-called Engquist-Osher numerical flux [14] takes the form Z f EO (k, u1 , u2 ) = u1 Z fu (k, s) ∨ 0 ds + 0 u2 fu (k, s) ∧ 0 ds + f (k, 0). 0 Here and in the following, a ∨ b = max {a, b} and a ∧ b = min {a, b}. Note that k and u are discretized on grids that are staggered with respect to each other. Concretely, we set u∆t (x, t) = X χ[xi−1/2 ,xi+1/2 i×[tn ,tn+1 i uni , k∆t = i,n X χ[xi ,xi+1 i ki+1/2 , i where ki+1/2 = k xi+1/2 and xi = ih. We have chosen to analyse the above scheme because of its simplicity. One can, however, adopt the method of proof developed in this paper and obtain similar results for other schemes (e.g., all monotone schemes) as well as more general equations. For example, under the assumption that k ≡ 1, Evje and Karlsen [20] have studied high order difference schemes (based on the MUSCL idea) for degenerate parabolic equations with source terms. One can easily combine the ideas in the present paper with those in [20] and obtain high order difference schemes for degenerate parabolic equations with source terms and k = k(x) non-constant. Moreover, one can easily treat the case where k possesses also a temporal dependence, i.e., k = k(x, t). Although we consider only explicit schemes in this paper, one can adopt the techniques used herein to analyse semi-implicit and implicit schemes (the details will be presented elsewhere). Assuming that u0 , k and k 0 are in BV , we are able to show that the approximate solutions {u∆t}∆t>0 generated by our scheme converge strongly in L1loc as ∆t ↓ 0 to the unique entropy solution. Furthermore, in the hyperbolic case, we show that this convergence has a rate. More precisely, we prove Theorem 1.1. Let u∆t denote the function generated by the Engquist-Osher scheme. We assume that the time step ∆t is related to the spatial step h through an appropriate CFL condition. If u0 , k and k 0 are in BV ∩ L1 ∩ L∞ , then u = lim u∆t ∆t↓0 is the unique entropy solution to (1.1). Furthermore, if A0 ≡ 0, then ku(·, t) − u∆t (·, t)kL1 (R) = O √ ∆t . We remark that Theorem 1.1 provides an existence result for entropy solutions of strongly degenerate parabolic equations which complements those in [5, 50]. We also remark that the question of a convergence rate for the difference approximations to degenerate parabolic equations will be addressed elsewhere. We now relate our results to the ones obtained by Evje and Karlsen [17,19], who analyse monotone difference approximations of (1.1) in the special case k ≡ 1. In this case, the authors gave a fairly complete analysis for the one-dimensional equation under certain smoothness assumptions on the initial function u0 , in which case it actually holds that A(u) belongs to the Hölder space C 1,1/2 (R × [0, T ]), and not merely L2 (0, T ; H 1 (R)) as follows from our analysis. We mention also the work [16] which generalizes the analysis in [17, 19] to the more difficult case of doubly nonlinear degenerate parabolic equations. In the present paper, we dispense with most of the smoothness assumptions on u0 used in [17, 19]. Moreover, in the multidimensional case, the authors of [19] do not prove that the limit u of their monotone difference approximations satisfies (ii) in (1.3), a result that can be easily established by adopting the techniques developed in the present paper. We continue with a few words about the proof of Theorem 1.1. The proof of the first part of Theorem 1.1 is based on deriving uniform L∞ , L1 , and BV bounds on the approximate solution u∆t . Equipped with the BV bound, we use the difference scheme itself and Kružkov’s interpolation lemma [33] to show that u∆t is uniformly L1 continuous in time. Kolmogorov’s compactness criterion then immediately gives L1loc convergence 242 K.H. KARLSEN AND N.H. RISEBRO (along a subsequence) of {u∆t }∆t>0 to a function u ∈ L1 ∩ L∞ . Uniqueness of the entropy solution [5, 28] (see also Th. 2.1 herein) will imply that the whole sequence {u∆t}∆t>0 converges and not just a subsequence. To ensure that the limit u is the (unique) entropy solution in the sense of (1.3), we first prove that the difference scheme satisfies a so-called discrete (or cell) entropy inequality and hence it follows, by arguments analogous to the ones used to prove the Lax-Wendroff theorem, that the entropy condition (i) in (1.3) holds true for the limit u. In passing, we mention that the BV regularity and the cell entropy inequalities are used to derive the error estimate in the hyperbolic case. In doing so, we follow Kuznetsov [36] and Kružkov [34]. Finally, we show that the limit u satisfies (ii) in (1.3). The arguments needed to prove (ii) are rather involved and based on deriving a space estimate that is resemblant of the so-called weak BV estimates employed by Champier et al. [7] and Eymard et al. [21] to prove convergence of finite volume methods on unstructured grids for the hyperbolic equation, see also [1, 22] for the diffusion equation. Equipped with our weak BV estimate and an appropriate time estimate, Kolmogorov’s compactness criterion implies strong L2loc convergence (along a subsequence) of {A(u∆t )}∆t>0 to A(u) and ∇A(u) ∈ L2 . Throughout this paper the coefficient k(x) is not allowed to be discontinuous. In the one-dimensional hyperbolic case (A0 ≡ 0) with k(x) depending discontinuously on x, the equation (1.1) is often written as the following 2 × 2 system: ut + f (k, u)x = 0, kt = 0. (1.4) If ∂f /∂u changes sign, then this system is non-strictly hyperbolic. This complicates the analysis, and in order to prove compactness of approximated solutions a singular transformation Ψ(k, u) has been used by several authors [23, 29, 30, 45]. In these works convergence of the Glimm scheme and of front tracking was established in the case where k may be discontinuous. Under weaker conditions on k, e.g., k 0 ∈ BV , and for f convex in u, convergence of the one-dimensional Godunov method for (1.4) (not for (1.1)) was shown by Isaacson and Temple in [27]. Recently, convergence of the one-dimensional Engquist-Osher method for (1.1) was shown by Towers [46,47] in the case where k is piecewise continuous. In this case, the Kružkov entropy condition (1.3) no longer applies, and in [30] a wave entropy condition analogous to the Oleinik entropy condition introduced in [40] was used to obtain uniqueness. We intend to study the degenerate parabolic case (1.1) with a discontinuous k(x) in future work. The rest of this paper is organized as follows: in the next section we introduce (precisely) the notion of an entropy solution, and state the theorem regarding uniqueness and the L1 contraction property of the solution operator to (1.1). We then proceed to show convergence and convergence rates of difference schemes for the hyperbolic equation. In the last section we show convergence of difference schemes for the degenerate parabolic equation. Throughout this paper we denote by C a generic constant, not depending on our discretization parameter ∆t. Note that the actual value of C may change from one line to the next during a calculation. 2. Preliminaries In this section we first give a precise definition of an entropy solution, and then present some technical tools that we shall use. 2.1. Definition of the entropy solution Throughout this paper we let fi (k i , u) be smooth functions R×R → R, and set f (k, u) = f1 (k 1 , u), . . . , fd (k d , u) . We assume that A : R → R is a function that satisfies A ∈ Liploc (R) and A(·) is nondecreasing with A(0) = 0. (2.1) Concerning the flux function f : Rd × R → Rd , we assume that f ∈ C 3 Rd × R; Rd and that fi , ∂u fi , ∂k fi , ∂uk fi , ∂ukk fi ∈ Lip(R × R; R), for i = 1, . . . , d. (2.2) DIFFERENCE SCHEMES WITH ROUGH COEFFICIENTS 243 Furthermore, we assume that the relevant Lipschitz constants are bounded by |∂u fi | ≤ Lu , |∂k fi | ≤ Lk , |∂uk fi | ≤ Luk , and so on, for all i and for some constants Lu , Lk , Luk . Without explicitly mentioning this any more, we will always assume in this paper that f (k, 0) = 0 for all k. Regarding the coefficient k we assume that (2.3) k i ∈ C Rd ∩ BV Rd , ∂xj k i ∈ L∞ Rd ∩ BV Rd for all i and j. Under the above assumptions we shall study difference approximations to (1.1). Following [28] (see also Carrillo [5]), an entropy solution is defined as follows: Definition 2.1. An entropy solution of (1.1) is a measurable function u = u(x, t) satisfying: D.1 u ∈ L1 (ΠT ) ∩ L∞ (ΠT ) ∩ C(0, T ; L1 (Rd )). D.2 For all c ∈ R, and all non-negative test functions ϕ in C0∞ (ΠT ) the following entropy inequality holds: ZZ |u − c|ϕt + sign (u − c) f (k(x), u) − f (k(x), c) · ∇ϕ + |A(u) − A(c)|∆ϕ ΠT − sign (u − c) divf (k(x), c)ϕ dt dx ≥ 0. D.3 A(u) ∈ L2 (0, T ; H 1 (Rd )). D.4 Essentially as t → 0+, Z Rd (2.4) u(x, t) − u0 (x) dx → 0. Remark 2.1. (i) Observe that when A0 ≡ 0, (2.4) reduces to the well known entropy inequality for scalar conservation laws introduced by Kružkov [34] and Vol’pert [49]. (ii) Condition (D.4), i.e., that the initial datum u0 should be taken by continuity, motivates the requirement of continuity with respect to t in condition (D.1). The following theorem from [28] shows that the initial value problem (1.1) is well posed: Theorem 2.1. Assume that (2.1, 2.2) and (2.3) hold. Let v, u ∈ L∞ (0, T ; BV (Rd )) be entropy solutions of (1.1) with initial data v0 , u0 ∈ L1 (Rd ) ∩ L∞ (Rd ) ∩ BV (Rd ), respectively. Then for almost all t ∈ (0, T ), kv(·, t) − u(·, t)kL1 (Rd ) ≤ kv0 − u0 kL1 (Rd ) . (2.5) In particular, there exists at most one entropy solution of the initial value problem (1.1). Remark 2.2. At the expense of loosing (2.5), the assumption that v, u ∈ L∞ (0, T ; BV (Rd )) can be removed and uniqueness still holds, see [28]. 2.2. Some mathematical tools In this section we present some mathematical tools that we shall use in the analysis. Let u : Rd ×(0, T ) → R be a function such that u(·, t) ∈ L1 Rd for all t ∈ (0, T ). By a modulus of continuity, we mean a nondecreasing continuous function ν : R0+ → R0+ such that ν(0) = 0. We say that u has a spatial modulus of continuity if Z sup |u(x + ε, t) − u(x, t)| dx ≤ ν(y; u), (2.6) |ε|≤y Rd 244 K.H. KARLSEN AND N.H. RISEBRO (where ν may depend on t). We also say that u has a temporal modulus of continuity if there is a modulus of continuity ω(·; u) such that for each τ ∈ (0, T ), Z sup 0≤ε≤τ Rd |u(x, t + ε) − u(x, t)| dx ≤ ω(τ ; u), ∀t ∈ (0, T − τ ). (2.7) Let θ(r) be a smooth non-negative function of a single variable r such that Z θ(r) = θ(−r), θ(r) = 0, for |r| ≥ 1, and θ(r) dr = 1. R Let δε (x) = (1/εd)θ(|x|/ε), and, with a slight abuse of notation, δε (t) = (1/ε)θ(t/ε). Now define a test function ϕ(x, y, t, s) by ϕ(x, y, t, s) = δε (x − y)δε (t − s). For a function u = u(x, t), set ZZ λ(u, c) = − |u − c| ϕt + sign (u − c) (f (k, u) − f (k, c))ϕx − sign (u − c) f (k, c)x ϕ dt dx ΠT Z + (2.8) t=T |u − c| ϕ t=0 dx. R For two functions u and v we define the functional Λε (u, v) as ZZ Λε (u, v) = λ(u, v(y, s)) ds dy, (2.9) ΠT where u = u(x, t) and v = v(y, s). In passing, we note that if A0 ≡ 0, and u is an entropy solution of (1.1), then Λε (u, v) ≤ 0. (2.10) For two arbitrary functions u and v we have the following result: Lemma 2.1 (Kuznetsov’s lemma). Assume that k = (k 1 , . . . , k d ) is in C(Rd ) and k 0 is in L∞ (Rd ) and that both k and kxi have moduli of continuity for all i = 1, . . . , d. If u and v are in L1 (ΠT ) and have moduli of continuity in space and time, then ku(·, T ) − v(·, T )kL1 (Rd ) ≤ ku(·, 0) − v(·, 0)kL1 (Rd ) + Λε (u, v) + Λε (v, u) 1 [ν(u(·, 0); ε) + ν(v(·, 0); ε) + ν(u(·, T ); ε) + ν(v(·, T ); ε)] 2 1 + [ω(u(·, T ); ε) + ω(v(·, T ); ε) + ω(u(·, 0); ε) + ω(v(·, 0); ε)] 2 + kdivkkL∞ Lu T sup (ν(u(·, t); ε) + ω(u(·, t); ε)) + (2.11) 0≤t≤T + CT kdivkkL∞ ν(k; ε) + max ν (kxi ; ε)) , i where ω(·; ·) denotes a temporal modulus of continuity, and ν(·; ·) denotes a spatial modulus. The constant C depends on fk and fkk . DIFFERENCE SCHEMES WITH ROUGH COEFFICIENTS 245 Proof. We shall prove this lemma for d = 1, the general proof is completely analogous. Remember that ϕx = −ϕy and ϕt = −ϕs . By adding Λε (u, v) and Λε (v, u) we find that ZZ ZZ Λε (u, v) + Λε (v, u) = − sign (u − v) (f (k(x), u) − f (k(x), v)) ϕx − f (k(x), v)x ϕ (2.12) ΠT ΠT − (f (k(y), u) − f (k(y), v)) ϕx + f (k(y), u)y ϕ dt dx ds dy ZZ Z ϕ(x, y, T, s) |u(x, T ) − v(y, s)| dx ds dy + ΠT R ΠT R ZZ Z ϕ(x, y, t, T ) |u(x, t) − v(y, T )| dy dt dx + ZZ Z ϕ(x, y, 0, s) |u(x, 0) − v(y, s)| dx ds dy − ΠT R ZZ Z − ϕ(x, y, t, 0) |u(x, t) − v(y, 0)| dy dt dx. ΠT R Using standard arguments, the four last terms will give the ku − vkL1 terms and the terms starting with 12 [. . . in (2.11). Regarding the remaining term, we follow Kružkov [34]. Let m(x, y, w) = fk (k(x), w)k 0 (x) − fk (k(y), w)k 0 (y). Then we rewrite the square brackets in (2.12) as (f (k(x), u) − f (k(x), v)) ϕx − f (k(x), v)x ϕ − (f (k(y), u) − f (k(y), v)) ϕx + f (k(y), u)y ϕ = [(f (k(y), u) − f (k(x), u)) ϕy + f (k(y), u)y ϕ] + [(f (k(y), v) − f (k(x), v)) ϕx − f (k(x), v)x ϕ] = fk (k(y), u)k 0 (y) [(y − x)ϕy + ϕ] + m(ξ1 , y, u)(x − y)ϕy + fk (k(y), v)k 0 (y) [(y − x)ϕx − ϕ] + m(y, x, v)ϕ + m(ξ2 , y, v)(y − x)ϕx = −fk (k(y), u)k 0 (y) [(y − x)ϕ]x + fk (k(y), v)k 0 (y) [(y − x)ϕ]x + (m(ξ1 , y, u) + m(ξ2 , y, v)) ϕy + m(y, x, v)ϕ = k 0 (y) (fk (k(y), v) − fk (k(y), u)) [(x − y)ϕ]x + (m(ξ1 , y, u) + m(ξ2 , y, v)) (x − y)ϕy + m(y, x, v)ϕ, where |ξi − y| ≤ |x − y| for i = 1, 2. Let now F (u, v) = sign (u − v) k 0 (y) (fk (k(y), v) − fk (k(y), u)) , and note that F is Lipschitz continuous, with Lipschitz constant given by kk 0 kL∞ Lu , in both arguments. Next, we obviously have that ZZ ZZ F (u(y, s), v(y, s)) [(x − y)ϕ]x dt dx ds dy = 0. ΠT ΠT 246 K.H. KARLSEN AND N.H. RISEBRO Thus, our troublesome term reads ZZ ZZ [F (u(x, t), v(y, s)) − F (u(y, s), v(y, s))] [(x − y)ϕ]x dt dx ds dy ΠT ΠT ZZ ZZ + ΠT sign (u − v) (m(ξ1 , y, u) + m(ξ2 , y, v)) (x − y)ϕy + m(y, x, v)ϕ dt dx ds dy. (2.13) (2.14) ΠT Now |m(x, y, u)| ≤ Lk |k 0 (x) − k 0 (y)| + kk 0 kL∞ Luk |k(x) − k(y)| . Hence (2.14) is bounded by integrals of the form ZZ ZZ ΠT |l(ξ) − l(y)| |(x − y)δε0 (x − y)| δε (t − s) dt dx ds dy (2.15) ΠT and ZZ ZZ |l(ξ) − l(y)| δε (x − y)δε (t − s) dt dx ds dy, ΠT (2.16) ΠT where ξ = x, ξ1 or ξ2 , and l = k or k 0 . Since |x − y| ≤ ε we have that |ξ − y| ≤ ε in (2.15) and (2.16), so they are both easily seen to be bounded by ν(l; ε). The rest of (2.13) is bounded as follows ZZ ZZ F (u(x, t), v(y, s)) − F (u(y, s), v(y, s)) |((x − y)ϕ)x | dt dx ds dy ΠT ΠT ZZ ZZ ≤ kk 0 kL∞ Lu |u(x, t) − u(y, s)| |((x − y)ϕ)x | dt dx ds dy kk 0 kL∞ Lu ≤ ε ZZ Z + Z T ZZ |u(x, t) − u(y, t)| dx dy dt 0 |x−y|≤ε |u(y, t) − u(y, s)| dy dt ds |t−s|≤ε ≤ kk 0 kL∞ Lu T sup (ν(u(·, t); ε) + ω(u(·, t); ε)) . 0≤t≤T This concludes the proof of the lemma. We need the following general L1 and L2 compactness criteria. Lemma 2.2 (L1loc compactness lemma). Let {zh }h>0 be a sequence of functions defined on Rd × (0, T ) which satisfies: (1) There exists a constant C1 > 0 which is independent of h such that kzh (·, t)kL1 (Rd ) and kzh (·, t)kL∞ (Rd ) ≤ C1 , ∀t ∈ (0, T ). DIFFERENCE SCHEMES WITH ROUGH COEFFICIENTS 247 (2) There exists a spatial modulus of continuity ν which is independent of h such that kzh (· + y, t) − zh (·, t)kL1 (Rd ) ≤ ν(|y|; zh ) as y → 0, ∀t ∈ (0, T ). (3) There exists a temporal modulus of continuity ω which is independent of h such that kzh (·, t + τ ) − zh (·, t)kL1 (Rd ) ≤ ω(τ ; zh ), ∀t ∈ (0, T − τ ) whenever τ ∈ (0, T ). Then {zh }h>0 is compact in the strong topology of L1loc (Rd × (0, T )). Moreover, any limit point of {zh }h>0 belongs to L1 (Rd × (0, T )) ∩ L∞ (Rd × (0, T )) ∩ C(0, T ; L1(Rd )). Lemma 2.3 (L2loc compactness lemma). Let {zh }h>0 be a sequence of functions defined on Rd × (0, T ) which satisfies: (1) There exists a constant C1 > 0 which may depend on T , but not h, such that kzh kL2 (Rd ×(0,T )) ≤ C1 . (2) There exists a constant C2 > 0 which may depend on T but not h such that kzh (· + y, ·) − zh (·, ·)kL2 (Rd ×(0,T )) ≤ C2 (|y| + h) for all y as h ↓ 0. (3) There exists a constant C3 > 0 which may depend on T but not h such that √ kzh (·, · + τ ) − zh (·, ·)kL2 (Rd ×(0,T −τ )) ≤ C3 τ + h for all τ > 0 as h ↓ 0. Then {zh }h>0 is compact in the strong topology of L2loc (Rd × (0, T )). Moreover, any limit point of {zh }h>0 belongs to L2 (0, T ; H 1(Rd )). To prove that the difference approximations possess some L1 time continuity, we shall use the following lemma due to Kružkov [33]. Lemma 2.4 (Kružkov’s interpolation lemma [33]). Let z(x, t) be a bounded measurable function defined on Rd × (0, T ). For t ∈ (0, T ) assume that z possesses a spatial modulus of continuity Z z x + ε, t − z(x, t) dx ≤ ν(|ε| ; z), (2.17) Rd where ν does not depend on t. Suppose that for any φ ∈ C0∞ (Rd ) and any t1 ,t2 ∈ (0, T ), Z X α cα kD φkL∞ (Rd ) · |t2 − t1 | , d (z (x, t2 ) − z (x, t1 )) φ(x) dx ≤ ConstT · R (2.18) |α|≤m where α denotes a multi-index, and cα are constants not depending on φ or t. Then for any t1 , t2 ∈ (0, T ) and all ε > 0 Z X cα |z(x, t2 ) − z(x, t1 )| dx ≤ C · |t2 − t1 | + ν(z; ε) . (2.19) ε|α| Rd |α|≤m Proof. Let δε (x) denote the usual mollifying kernel of radius ε, and let d(x) = z(x, t2 ) − z(x, t1 ). For r > ε, set ( sign (d(x)) for |x| ≤ r − ε, β(x) = 0 otherwise, 248 K.H. KARLSEN AND N.H. RISEBRO and set βε = β ∗ δε . Then βε is in C0∞ (Rn ), has support inside the ball Br , and we have the bound |Dα βε | ≤ Const/ε|α| . Also Z Z Z d(x) dx ≤ |d(x) − βε (x)d(x)| dx + βε (x)d(x) dx Br B Br Z Zr X cα ≤ |d(x) − d(x − y)| δε (y) dx dy + C |t2 − t1 | ε|α| |α|≤m Rd X cα ≤ C ν(ε; z) + |t2 − t1 | · ε|α| |α|≤m Letting r ↑ ∞, we obtain (2.19). We shall also need the technical result: Lemma 2.5 (Crandall and Tartar [13]). Let (Ω, µ) be some measure space and let D be a subset of L1 (Ω). Assume that if u and v are in D, then also u ∨ v is in D. Let T be a map D → D such that Z Z T (u) dµ = Ω u dµ, ∀u ∈ D. Ω Then the following statements, valid for all u and v in D, are equivalent: (i) IfR u ≤ v, then T (u) ≤ T (v).R (ii) RΩ (T (u) − T (v)) ∨ 0 dµR≤ Ω (u − v) ∨ 0 dµ. (iii) Ω |T (u) − T (v)| dµ ≤ Ω |u − v| dµ. 3. Difference approximations: The hyperbolic equation In this section we analyze a difference approximation to the solution of the hyperbolic equation ut + divf (k, u) = 0, u(x, 0) = u0 (x), (x, t) ∈ ΠT , (3.1) where k and f satisfy (2.2) and (2.3) respectively. For simplicity, we shall assume that u0 has compact support, which implies that all subsequent sums over I are finite. To obtain results in the general case, we can use the stability result in Theorem 2.1 (these standard details will not be written out). We have chosen to analyze the hyperbolic equation separately since the analysis parallels the general case but is simpler. In the next section, where we consider the general case, we shall use several of the estimates obtained in this section. Furthermore, we provide a convergence rate in the hyperbolic case. As already mentioned in the introduction, we use the Engquist-Osher scheme to make the analysis more concrete, but our methods can easily be adapted to general monotone schemes. For a scalar flux function fi (k, u), the Engquist-Osher flux fiEO [14] can be written as fiEO (k, u, v) 1 = 2 Z fi (k, u) + fi (k, v) − v |∂u fi (k, s)| ds . (3.2) u To define the scheme, let I be a multi-index I = (i1 , . . . , id ) and set ei to be a multi-index with zeros everywhere except for a 1 at the ith place. Furthermore, we choose a time step ∆t such that N ∆t = T and a spatial DIFFERENCE SCHEMES WITH ROUGH COEFFICIENTS 249 discretization parameter h > 0. Letting λ = ∆t/h, the Engquist-Osher scheme reads (d ≥ 2) = unI − λ un+1 I d h i X EO ¯EO f¯I+e − f I−ei /2 i /2 i=1 = unI − λ d X i=1 h i i n n EO i n n fiEO kJ+e , u , u − f k , u , u , I I+e i I−e I J−ei /2 i i i /2 X 1 2d−1 (3.3) j6=i J=I±ej /2 where EO f¯I+e = i /2 i n n fiEO kJ+e , u , u I I+ei . i /2 X 1 2d−1 (3.4) j6=i J=I±ej /2 For d = 1, the Engquist-Osher scheme reads h i n EO n n EO n n un+1 = u − λ f k , u , u − f k , u , u . i+1/2 i−1/2 i i i+1 i−1 i i The approximate solution u∆t is then defined as u∆t (x, t) = unI , for (x, t) ∈ χI × [tn , tn+1 ), (3.5) where χI denotes the set n o χI = x ∈ Rd xI−ei /2 i ≤ xi < xI+ei /2 i , i = 1, . . . , d and xI = hI. We initialize the scheme by setting u0I = 1 |χI | Z u0 (x) dx. χI Note that fiEO (k, u, v) is not continuously differentiable in the first variable but merely Lipschitz. Therefore we introduce the following auxiliary numerical flux fiEO,ε (k, u, v) 1 = 2 Z fi (k, u) + fi (k, v) − v u |∂u fi (k, s)|ε ds , where | · |ε is a smooth approximation to the absolute value function | · | such that |a| − |a|ε ≤ ε and |a| = |a|ε for |a| > ε. 250 K.H. KARLSEN AND N.H. RISEBRO Note in particular that EO EO,ε (k, u, v) ≤ |u − v| ε, fi (k, u, v) − fi ∀k, u, v. (3.6) This scheme can be analyzed as follows. Set X d i = X 1 2d−1 and ψi (k, u, v) = j6=i J=I±ej /2 ∂fiEO,ε (k, u, v). ∂k Note that |ψi (k, u, v)| ≤ Lk + 1 |u − v| Luk . 2 Then we define FI as the right hand side in (3.3), i.e., un+1 = FI (un ) . I (3.7) Assuming the CFL condition λ d X i=1 it is easy to show that U n = maxI |unI |, then ∂FI ∂un J max |∂u fi (k, u)| ≤ 1, (3.8) k,u ≥ 0 for all J. In other words, the Engquist-Osher scheme is monotone. Let n+1 u = |FI (un )| ≤ FI (U n ) I = Un + λ d X h i X d i n i n fi kJ+e , U − f k , U i /2 J−e /2 i i i=1 i d X d 1 X X i d i i n kx ∞ d . ≤ U + Lk ∆t kJ+ei /2 − kJ−ei /2 ≤ U + Lk ∆t i L (R ) i h i=1 i=1 n From this it follows that ku∆t (·, T )kL∞ (Rd ) ≤ ku0 kL∞ (Rd ) + T Lk d max kxi i L∞ (Rd ) . i Next, by the Crandall and Tartar lemma (see Lem. 2.5) and the monotonicity of FI , X I |FI (un ) − FI (0)| ≤ X I |unI | . (3.9) 251 DIFFERENCE SCHEMES WITH ROUGH COEFFICIENTS Hence hd d X X X X X d i i i d un+1 ≤ hd |unI | + hd λ , 0 ≤ h |unI | . f kJ+ei /2 , 0 − f i kJ−e /2 I i I i=1 I i I This means that ku∆t (·, T )kL1 (Rd ) ≤ ku0 kL1 (Rd ) . (3.10) For any quantity XI defined on our grid let D` XI denote the upward difference D` XI = XI+e` − XI . (3.11) To bound the total variation of u∆t we again use the Crandall-Tartar lemma, which in this case gives X FI+e ` X n n u un·+e` − FI+e` (un ) ≤ I+e` − uI . I I Note that we have n un+1 I+e` = FI+e` u·+e` and Xh i X X D` un+1 ≤ |D` unI | + FI+e` (un ) − FI (un ) . I I I I Before we start to estimate the difference on the right-hand side of (3.12), note that 1 ψi (k1 , u, v) − ψi (k2 , u, v) = (∂kk fi (η, u) + ∂kk fi (ν, v)) (k1 − k2 ) 2 Z v − [signε (∂u fi (k1 , s)) − signε (∂u fi (k2 , s))] ∂uk fi (k1 , s) u + signε (∂u fi (k2 , s)) ∂ukk fi (γ, s) ds ≤ max |∂kk fi (k, u)| |k1 − k2 | + max |∂uk fi (k, u)| |u − v| k,u k,u 1 + max |∂ukk fi (k, u)| |u − v| |k1 − k2 | , 2 k,u where signε (·) denotes the derivative of | · |ε . Furthermore, we have |ψi (k, u1 , v) − ψi (k, u2 , v)| ≤ max |∂ku fi (k, u)| |u1 − u2 | , k,u |ψi (k, u, v1 ) − ψi (k, u, v2 )| ≤ max |∂ku fi (k, u)| |v1 − v2 | . k,u (3.12) 252 K.H. KARLSEN AND N.H. RISEBRO i i Using the above estimates on ψi and (3.6), there exist numbers ξJ+e` /2±ei /2 between kJ±e and kJ+e i /2 ` ±ei /2 such that d X X d i n n EO i n n fiEO kJ+e , u , u − f k , u , u FI+e` (u ) − FI (u ) = −λ I I+e i I I+e J+e +e /2 /2 i i i i ` n n − i=1 fiEO i = O(ε) − λ i kJ+e , unI−ei , unI ` −ei /2 − fiEO i n n kJ−ei /2 , uI−ei , uI d X X d i=1 i i ψi ξJ+e` /2+ei /2 , unI , unI+ei D` kJ+e i /2 i − ψi ξJ+e` /2−ei /2 , unI−ei , unI D` kJ−e i /2 = O(ε) − λ d X X d i=1 − i i ψi ξJ+e` /2+ei /2 , unI , unI+ei Di D` kJ+e i /2 Di ψi ξJ+e` /2−ei /2 , unI−ei , unI = O(ε) − λ i D` kJ−e i /2 d X X d i=1 i i ψi ξJ+e` /2+ei /2 , unI , unI+ei Di D` kJ+e i /2 − ψi ξJ+e` /2+ei /2 , unI , unI+ei − ψi ξJ+e` /2−ei /2 , unI , unI+ei + ψi ξJ+e` /2−ei /2 , unI , unI+ei − ψi ξJ+e` /2−ei /2 , unI , unI n n n n i + ψi ξJ+e` /2−ei /2 , uI , uI − ψi ξJ+e` /2−ei /2 , uI−ei , uI D` kJ−ei /2 d X X d i ≤ O(ε) + λ ψi ξJ+e` /2+ei /2 , unI , unI+ei Di D` kJ+e /2 i i=1 i Lukk − Lkk Di ξJ+e` /2−ei /2 + Luk |Di unI | + Di unI Di ξJ+e` /2−ei /2 2 i + Lku |Di unI | + Di unI−ei D` kJ−e . i /2 As the above inequality holds for any ε > 0, we can let ε ↓ 0 and obtain |FI+e` (un ) − FI (un )| ≤ λ d X X d i ψi ξJ+e` /2+ei /2 , unI , unI+ei Di D` kJ+e i /2 i=1 i Lukk − Lkk Di ξJ+e` /2−ei /2 + Luk |Di unI | + Di unI Di ξJ+e` /2−ei /2 2 i + Lku |Di unI | + Di unI−ei D` kJ−e . i /2 253 DIFFERENCE SCHEMES WITH ROUGH COEFFICIENTS Since we have that ψi is bounded, we find that d X X X XX d n d−1 d−2 i D` un+1 hd−1 ≤ D |D u | h + C∆t h D k ` I ` i J+ei /2 I I I I + hd−1 max kxi j i,j L∞ (Rd ) i i=1 d X XX d i i d−1 + h max D` kJ−e kxj i /2 I i=1 i i,j L∞ (Rd ) d XX I |Di unI | , i=1 for some constant C independent of ∆t. The first and second sums inside {· · · } are bounded since we assume that k i and kxi j are in BV Rd . By summing the above over ` = 1, . . . , d, we find that |u∆t (·, tn+1 )|BV (Rd ) = hd−1 d X X D` un+1 ≤ (1 + C∆t) |u∆t (·, tn )| I `=1 I + C∆t |k|BV (Rd ) + max kxi j i,j BV (Rd ) BV (Rd ) . Consequently, |u∆t (·, t)|BV (Rd ) ≤ C |u0 |BV (Rd ) + t , ∀t ∈ (0, T ), (3.13) where C does not depend on ∆t. Next, we shall use the scheme to show that u∆t ∈ C 0, T ; L1(Rd ) uniformly in ∆t. This is done as follows d X XX d X EO i n+1 n d−1 i n n h uI − uI ≤ ∆th kJ+ei /2 , unI , unI+ei − fiEO kJ+e , u , u fi I−e I /2 i i d I i I i=1 EO ∂fi i + max Di kJ−ei /2 ∂k ≤ C∆t |u∆t |BV (Rd ) + |k|BV (Rd ) , from which we obtain ku∆t (·, t + τ ) − u∆t (·, t)kL1 (Rd ) ≤ Cτ, ∀t ∈ [0, T − τ ]. (3.14) By Lemma 2.2, we have that the sequence {u∆t }∆t>0 is compact in L1loc (ΠT ). Moreover, any limit point of this sequence satisfies (D.1) and (D.4). Next, we shall prove two cell entropy inequalities. The first one is based on Kružkov’s entropies and will be used later to prove an error estimate. The second one is based on smooth (C 2 ) convex entropies and will be used to show that the limit of any convergent subsequence of {u∆t }∆t>0 satisfies the entropy condition (D.2). Let wIn = unI ∨ c, vIn = unI ∧ c, wIn+1 = FI (wIn ) , vIn+1 = FI (vIn ) , cn+1 = FI (c), I where FI is defined in (3.7). Then we have that ≥ vIn+1 , wIn+1 ≥ un+1 I This implies that vIn+1 ≤ cn+1 ≤ wIn+1 . I n+1 u − cn+1 ≤ wn+1 − v n+1 . I I I I 254 K.H. KARLSEN AND N.H. RISEBRO Now wIn+1 − vIn+1 = |unI d X X d i n n − c| − λ fiEO kJ+e , u ∨ c, u ∨ c I I+e /2 i i i i=1 i n n EO i n n , u ∧ c, u ∧ c − f k , u ∨ c, u ∨ c −fiEO kJ+e I I+e i I−e I J−ei /2 i i i /2 i n n −fiEO kJ−e , u ∧ c, u ∧ c . I−ei I i /2 Denoting the numerical entropy flux by qiEO (k, u, v) = fiEO (k, u ∨ c, v ∨ c) − fiEO (k, u ∧ c, v ∧ c) and noting that d X h i X n+1 n+1 d n+1 i i u fi kJ+ei /2 , c − fi kJ−ei /2 , c − cI = uI − c + λ I i i=1 ! d X h i X d i i ≥ sign un+1 − c un+1 −c+λ fi kJ+e , c − fi kJ−e ,c I I i /2 i /2 i=1 i d X h i n+1 X d n+1 i i ≥ uI − c + sign uI − c λ fi kJ+e , c − fi kJ−e ,c , i /2 i /2 i=1 i we find that d X X n+1 d EO i n n n EO i n n u − c ≤ |uI − c| − λ qi kJ+ei /2 , uI , uI+ei − qi kJ−ei /2 , uI−ei , uI I i i=1 − sign un+1 I d X X d i −c λ Di fi kJ−e ,c . i /2 i=1 (3.15) i This discrete entropy inequality will be used later to derive an error estimate. Let U ∈ C 2 be a convex entropy and let q̃iEO (k, u, v) denote the associated numerical entropy flux defined by Z q̃iEO (k, u, v) = u Z U 0 (s) ∂u fi (k, s) ∨ 0 ds + 0 v U 0 (s) ∂u fi (k, s) ∧ 0 ds. (3.16) 0 Let us write the scheme (3.3) as un+1 = wIn − λ I d X h i X d i n n EO i n n fiEO kJ+e , u , u − f k , u , u , I I+ei i I I+ei J−ei /2 i /2 i i=1 (3.17) where wIn = unI − λ d X h i X d i i fiEO kJ−e , unI , unI+ei − fiEO kJ−e , unI−ei , unI . i /2 i /2 i=1 i (3.18) 255 DIFFERENCE SCHEMES WITH ROUGH COEFFICIENTS From (3.18), it follows in a standard way using monotonicity (i.e., the CFL condition) that − ∆t U (wIn ) U (unI ) + n n EO i n n d X q̃ EO k i − q̃ k , u , u , u , u X i I I+ei I−ei I J−ei /2 J−ei /2 d i h i i=1 ≤ 0. (3.19) Using (3.17) and convexity of U (·), it follows that ) + U 0 (un+1 )λ U (wIn ) ≥ U (un+1 I I d X h X d i i=1 i i i fiEO kJ+e , unI , unI+ei − fiEO kJ−e , unI , unI+ei . i /2 i /2 Plugging this into (3.19), we get our desired cell entropy inequality − ∆t U (un+1 ) I U (unI ) + n n EO i n n d X q̃ EO k i , u , u − q̃ k , u , u X i i I I+e I−e I J+e J−e /2 /2 i i d i i i=1 i + U 0 (un+1 ) I n n EO i d X f EO k i , unI , unI+ei kJ−e X J+ei /2 , uI , uI+ei − fi d i i /2 i=1 − d X X d i=1 h q̃iEO h i i n n EO i n n , u , u , u , u kJ+e − q̃ k i I I+ei I I+ei J−ei /2 i /2 h i (3.20) ≤ 0. We now multiply (3.20) by a nonnegative test function, do summation by parts, and then subsequently send ∆t → 0. We conclude that a limit point u of {u∆t }∆t>0 satisfies the entropy condition ZZ U (u)ϕt + q(k, u) · ∇ϕ − U 0 (u)fx (k, u) − qx (k, u) ϕ dt dx ≥ 0, (3.21) ΠT where q = (q1 , . . . , qd ) satisfies ∂u q(k, u) = U 0 (u)∂u f (k, u). Furthermore, fx , qx denote the functions obtained by taking the divergence of f (k(·), u), q(k(·), u), respectively. Since (3.21) holds for any convex C 2 entropy U and corresponding entropy flux q, a standard approximation argument applied to | · −c| will reveal that (2.4) (with A0 ≡ 0) holds for all c ∈ R. Hence, by the uniqueness of the entropy solution, the whole sequence {u∆t }∆t>0 converges to the unique entropy solution. Now we shall use the cell entropy inequality (3.15) and Kuznetsov’s lemma (Lem. 2.1) to show that u∆t converges to the unique entropy solution at a rate of ∆t1/2 . For simplicity we shall restrict our proof to the case of one space dimension, i.e., d = 1. However, with some effort, the calculations given below can be generalized to the multi-dimensional case d > 1. Let u(x, t) be the unique entropy solution to ut + f (k, u)x = 0, u(x, 0) = u0 (x) (3.22) where u0 and k 0 is of bounded variation, and let u∆t (x, t) and k∆t be as before. Now Λε (u, u∆t ) ≤ 0, and all the continuity moduli in Kuznetzov’s lemma are linear in ε, therefore (2.11) reads ku(·, T ) − u∆t (·, T )kL1 (R) ≤ ku0 − u∆t (·, 0)kL1 (R) + Λε (u∆t , u) + Cε, (3.23) 256 K.H. KARLSEN AND N.H. RISEBRO where the constant C does not depend on ∆t. We must estimate Λε (u∆t , u). Set η(u) = |u − c| , q(k, u) = sign (u − c) (f (k, u) − f (k, c)). Multiplying the cell entropy inequality (3.15) by positive numbers hϕni with ϕni = 0 for |i| large, and summing over i and n = 0, . . . , N − 1 where N ∆t = T , we find that Xh l(u∆t , c) := ηin+1 − ηin ϕni h + q EO ki+1/2 , uni , uni+1 − q EO ki−1/2 , uni−1 , uni ϕni ∆t i,n n i + sign un+1 − c f k , c − f k , c ϕi ∆t i+1/2 i−1/2 i (3.24) =l1 + l2 + l3 ≤ 0. We also find that λε (u∆t , c) = " X ηin+1 − ηin Z xi+1/2 ϕ(x, tn ) dx xi−1/2 i,n + q ki+1/2 , uni+1 − q ki−1/2 , uni Z (3.25) tn+1 ϕ(xi , t) dt tn Z tn+1 ϕ xi+1/2 , t − ϕ (xi , t) dt tn # Z tn+1 n+1 + sign ui − c f ki+1/2 , c − f ki−1/2 , c ϕ (xi , t) dt + q ki+1/2 , uni+1 − q ki+1/2 , uni (3.26) (3.27) (3.28) tn =: λ1 + λ2,1 + λ2,2 + λ3 . Since l(u∆t , c) ≤ 0, λε (u∆t , u) ≤ |λε (u∆t , u) − l (u∆t , u)| ≤ |λ1 − l1 | + |λ2,1 − l2 | + |λ3 − l3 | + |λ2,2 | . First we note that |λ2,2 | ≤ Lu XZ i,n and xi+1/2 Z xi tn+1 |δε0 (x − y)| δε (t − s) dt dx uni+1 − uni tn ZZ |λ2,2 | ds dy ≤ Lu T h |u∆t |BV (R) . ε ΠT We choose the numbers ϕni as ϕni = 1 ∆th Z xi+1/2 Z tn+1 ϕ(x, t) dt dx. xi−1/2 tn Using this we find that Z tn+1 Z xi+1/2 X n+1 n 1 |λ1 − l1 | ≤ ηi − ηi |ϕ(x, t) − ϕ(x, tn )| dx dt ∆t tn xi−1/2 i,n Z Z X tn+1 xi+1/2 η n+1 − ηin ≤ |ϕt | dx dt. i i,n tn xi−1/2 (3.29) 257 DIFFERENCE SCHEMES WITH ROUGH COEFFICIENTS Therefore, Z X n+1 n |λ1 − l1 | ds dy ≤ ηi − ηi ZZ Z tn i,n ΠT tn+1 xi+1/2 ZZ |δε0 (t − s)| δε (x − y) ds dy dx dt xi−1/2 X η n+1 − ηin C ∆th ≤ i ε i,n ≤C (3.30) ∆t · ε We continue with |λ2,1 − l2 | ≤ " X Z tn+1 ϕ(xi , t) dt × q ki+1/2 , uni+1 − q EO ki+1/2 , uni , uni+1 tn i,n − q ki−1/2 , uni − q EO ki−1/2 , uni−1 , uni # Z tn+1 EO n n n EO n n + ϕ(xi , t) dt − ϕi ∆t q ki+1/2 , ui , ui+1 − q ki−1/2 , ui−1 , ui tn " Z X n EO n n + q ki+1/2 , ui+1 − q ki+1/2 , ui , ui+1 tn i,n Z tn+1 Z xi+1/2 + tn " Z X ≤ Lu i,n tn+1 tn+1 tn+1 |ϕx | dx dt xi # EO |ϕx | dx dt q ki+1/2 , uni , uni+1 − q EO ki−1/2 , uni−1 , uni Z xi+1 xi Z |ϕx | dx dt uni+1 − uni # n n n o n n |ϕx | dx dt ki+1/2 − ki−1/2 + ui − ui−1 + ui+1 − ui · xi+1/2 +C tn xi+1 xi−1/2 tn Z Z xi−1/2 Consequently ZZ h h |λ2,1 − l2 | dy ds ≤ C |u∆t |BV (R) + |k|BV (R) ≤C · ε ε (3.31) ΠT Finally we estimate " Z tn+1 Z xi+1/2 X f ki+1/2 , c − f ki−1/2 , c 1 |λ3 − l3 | ≤ |ϕ(xi , t) − ϕ(x, t) + ϕ(x, t) − ϕ(x, t − ∆t)| dx dt h tn xi−1/2 i,n Z ∆t # X N −1 + f ki+1/2 , c − f ki−1/2 , c ϕ(xi , t) dt + ∆tϕi i " Z X ki+1/2 − ki−1/2 ≤ Lk 0 tn+1 Z tn i,n h |ϕx | + ∆t |ϕt | dx dt xi−1/2 Z X ki+1/2 − ki−1/2 + Lk i xi+1/2 0 ∆t ϕ(xi , t) dt + ∆tϕiN −1 # · 258 K.H. KARLSEN AND N.H. RISEBRO Therefore ZZ 2 X X 2 ki+1/2 − ki−1/2 h ∆t + h∆t + ki+1/2 − ki−1/2 C∆t ε i,n i 2 h + h∆t ≤ C |k|BV (R) + ∆t . ε |λ3 − l3 | ds dy ≤ Lk ΠT (3.32) Collecting the bounds (3.29–3.31), and (3.32), we find that h + ∆t + h2 + h∆t , Λε (u∆t , u) ≤ C ∆t + ε (3.33) for some constant C not depending on ∆t or h. Since we assume that u0 has bounded variation, ku(·, 0) − u∆t (·, 0)kL1 (R) ≤ Ch, and using this in (3.23) as well as h = ∆t/λ we arrive at the inequality ∆t ku(·, T ) − u∆t (·, T )kL1 (R) ≤ C ε + , ε (3.34) √ which is minimized by setting ε = ∆t. The main result of this section is summed up in the following theorem, which is stated for multi-dimensional equations: Theorem 3.1. Assume that f and k satisfy (2.2) and (2.3), respectively. Moreover, assume that u0 is a function in L1 (Rd )∩L∞ (Rd )∩BV (Rd ). Let u be the unique entropy solution of (3.1). If the CFL condition (4.16) holds, then there exists a constant C, depending on k, kxi , u0 , f and T , but not on ∆t, such that √ ku∆t (·, T ) − u(·, T )kL1 (Rd ) ≤ C ∆t, (3.35) where the Engquist-Osher approximate solution u∆t is build from (3.3) and (3.5). Remark 3.1. The assumptions on k = (k 1 , . . . , k d ) in Theorem 3.1 are (slightly) less restrictive than those used in [1, 6, 7, 21] for finite volume methods. 4. Difference approximations: The degenerate parabolic equation In this section we analyse the Engquist-Osher scheme for the degenerate parabolic equation (1.1). Again we shall assume that u0 has compact support so that all subsequent sums over I are finite. To obtain results for the general case, we can use the stability result in Theorem 2.1. Let λ = ∆t/h (as usual) and µ = ∆t/h2 , then this the scheme reads (d ≥ 2) un+1 = unI − λ I d X X d i=1 i d X i n n Di fiEO kJ−e , u , u + µ Di2 A unI−ei I−e I /2 i i i=1 n =: GI (u ) , where Di denotes the usual upwind difference operator, see (3.11). For d = 1, the scheme reads un+1 = uni − λDf EO ki−1/2 , uni−1 , uni + µD2 A(uni−1 ). i (4.1) 259 DIFFERENCE SCHEMES WITH ROUGH COEFFICIENTS Let u∆t be the piecewise constant function defined by (4.1) and (3.5). As a starting point we assume that the following CFL condition holds CFL = λ d X i=1 max |∂u fi (k, u)| + 2µd max A0 (u) ≤ 1. (4.2) u k,u Remark 4.1. The CFL condition (4.2) will be sufficient to establish the convergence of the sequence {u∆t}∆t>0 and moreover that a limit point u of this sequence satisfies (D.1), (D.2), and (D.4). However, to prove that u obeys (D.3), we shall later need a slightly stronger CFL condition (see (4.16) below). I Now it is easy to see that the CFL condition (4.2) implies that ∂G ≥ 0 for all J and the scheme (4.1) is ∂un J monotone. In the same manner as the bounds (3.9, 3.10), and (3.13), we show that ku∆t (·, t)kL∞ (Rd ) ≤ C, ku∆t (·, t)kL1 (Rd ) ≤ C, |u∆t (·, t)|BV (Rd ) ≤ C, ∀t ∈ (0, T ), (4.3) for some constant C not depending on ∆t. To show compactness of the scheme, we must also show that u∆t ∈ C 0, T ; L1(Rd ) uniformly in ∆t. In order to do this, we use the Kružkov interpolation lemma (Lem. 2.4). Let ϕ(x) be a test function and set ϕI = ϕ (xI ). Let Dt XIn = XIn+1 − XIn for any XIn . From (4.1) we find that X I Dt unI ϕI hd = d X d X X X d EO λ fi kJ−ei /2 , unI−ei , unI Di ϕI−ei hd + µ Di A (unI ) Di ϕI−ei hd I i i=1 ≤ C∆t k∇ϕkL∞ (Rd ) i=1 ! d d XX XX d X EO n n d n d−1 fi kJ−ei /2 , uI−ei , uI h + |Di A (uI )| h . I i i=1 I i=1 In view of the uniform L1 and BV bounds in (4.3), an application of Lemma 2.4 gives ku∆t (·, t1 ) − u∆t (·, t2 )kL1 (Rd ) ≤ C p |t1 − t2 |, (4.4) for some constant C not depending on ∆t. By Lemma 2.2, the sequence {u∆t }∆t>0 is compact in L1loc (ΠT ) and any limit point will satisfy (D.1) and (D.4). We next show that the entropy condition (D.2) holds. To this end, a cell entropy inequality for the scheme (4.1) is established in the same way as in the hyperbolic case. A modification of the arguments leading to equation (3.15) yields X q̃i ) − U (unI ) X d U (un+1 I + i ∆t i=1 d EO i i kJ+e , unI , unI+ei − q̃iEO kJ−e , unI−ei , unI i /2 i /2 h n n EO i n n d d X f EO k i n n n , u , u − f k , u , u X X i r(uI+ei ) − 2r(uI ) + r(uI−ei ) I I+ei I I+ei J+ei /2 J−ei /2 d i − + U 0 (un+1 ) I 2 i h h i=1 i=1 EO i n n EO i n n d q̃ k , u , u − q̃ k , u , u XX i i I I+e I I+e J+e J−e /2 /2 i i d i i − ≤ 0, (4.5) i h i=1 260 K.H. KARLSEN AND N.H. RISEBRO for any convex C 2 entropy U and corresponding numerical entropy fluxes q̃iEO defined in (3.16) and r defined by r0 (u) = U 0 (u)A0 (u). Consequently, any limit point of {u∆t }∆t>0 satisfies ZZ U (u)ϕt + q(k, u) · ∇ϕ + r(u)∆ϕ − U 0 (u)fx (k, u) − qx (k, u) ϕ dt dx ≥ 0, (4.6) ΠT where q obeys ∂u q(k, u) = U 0 (u)∂u f (k, u) and fx , qx denote the divergence of f (k(·), u), q(k(·), u), respectively. Since (4.6) holds for any convex C 2 entropy U and corresponding entropy fluxes q, r, the usual approximation argument will show that (2.4) holds ∀c ∈ R. It remains to show that a limit u of {u∆t }∆t>0 satisfies (D.3). This will be done by deriving a weak BV estimate [1, 7, 21, 22]. Multiplying (4.1) by unI hd , and summing over I, we find that d X X X d n EO i i n n unI Dt unI hd + ∆t kJ+ei /2 , unI , unI+ei − fiEO kJ+e hd−1 u I fi , u , u I−e I /2 i i i=1 I +µ i d X d X XX d n EO i Di A unI−ei Di unI−ei = −∆t u I fi kJ+ei /2 , unI−ei , unI i=1 We can write unI Dt unI = Di A i n n d−1 , u , u h . − fiEO kJ−e I−ei I i /2 1 2 2 Dt (unI ) − (Dt unI ) , 2 and we also have that unI−ei i i=1 I Di unI−ei 2 Di A unI−ei ≥ maxu A0 (u) since A0 (u) ≥ 0. Using these observations, we find that XX ∆thd 0 maxu A (u) i=1 d I !2 d X XX Di A unI−ei d n EO i + ∆t u I fi kJ+ei /2 , unI , unI+ei i h I i=1 i −hd Xh 2 2 i −fiEO kJ+e , unI−ei , unI hd−1 ≤ Dt (unI ) − (Dt unI ) i /2 2 I − ∆t d X XX d I i=1 i i i unI fiEO kJ+e , unI−ei , unI − fiEO kJ−e , unI−ei , unI hd−1 . (4.7) i /2 i /2 Before we proceed, we note that fiEO (k, v, w) − fiEO (k, u, v) = fi− (k, w) − fi− (k, v) + fi+ (k, v) − fi+ (k, u) , where fi− (k, u) = Z u 0 ∂u fi (k, s) ∧ 0 ds, fi+ (k, u) = Z u ∂u fi (k, s) ∨ 0 ds. 0 261 DIFFERENCE SCHEMES WITH ROUGH COEFFICIENTS Thus we can write inequality (4.7) as XX ∆thd 0 maxu A (u) i=1 d Di A unI−ei h I !2 + ∆th d−1 + unI ≤ i i unI fi− kJ+e , unI+ei − fi− kJ+e , unI i /2 i /2 i i=1 I " d X XX d (4.8) # + + i n i n fi kJ+ei /2 , uI − fi kJ+ei /2 , uI−ei (4.9) d i XX −hd Xh d−1 2 2 i Dt (unI ) − (Dt unI ) + C∆t Di kI−e h , (4.10) i /2 2 i=1 I I for some constant C. The sum in (4.9) can be analyzed further by introducing the functions Fi± (k, u) Z u = s∂u f ± (k, s) ds. 0 Then an integration by parts reveals that Fi± (k, b) − Fi± (k, a) =b fi± (k, b) − fi± (k, a) Z b − fi± (k, s) − fi± (k, a) ds. a Therefore + + + i n i n i n i n unI fi+ kJ+e , u , u , u , u − f k = F k − F k I I−ei I I−ei J+ei /2 J+ei /2 J+ei /2 i i i i /2 Z un I + un I−e (4.11) + i i n fi+ kJ+e , s − f k , u ds, I−ei J+ei /2 i i /2 i − i n i n i i unI fi− kJ+e , u − f k , u = Fi− kJ+e , unI+ei − Fi− kJ+e , unI I+ei I J+ei /2 i i /2 i /2 i /2 Z − un I un I+e (4.12) − i i n fi− kJ+e , s − f k , u ds. I+ei J+ei /2 i i /2 i Consequently (4.9) can be written d−1 ∆th d X XX d I i=1 " Fi+ i Z + i n i n kJ+ei /2 , uI − Fi kJ+ei /2 , uI−ei + un I un I−e i fi+ kJ+e , s /2 i i − fi+ i i i kJ+e , unI−ei ds + Fi− kJ+e , unI+ei − Fi− kJ+e , unI i /2 i /2 i /2 Z − un I un I+e i fi− i kJ+e ,s i /2 − fi− # i n kJ+ei /2 , uI+ei ds . 262 K.H. KARLSEN AND N.H. RISEBRO We also have that Fi± is locally Lipschitz continuous in k as Z ± F (k1 , u) − F ± (k2 , u) = i i u 0 s∂u fi± (k1 , s) − fi± (k2 , s) ds ≤ max |∂uk fi | |u| |k1 − k2 | . k,u Hence we obtain d X h X X i d ± ± i n i n d−1 Fi kJ+ei /2 , uI − Fi kJ−ei /2 , uI h i I i=1 ≤ Luk max ku∆t (·, t)kL∞ (Rd ) t d XX d−1 i Di kJ−e h , i I (4.13) i=1 which is bounded uniformly in ∆t. To bound the terms involving integrals, we need the following technical lemma (whose easy proof can be found in [22]): Lemma 4.1. Let h : R → R be a monotone, Lipschitz continuous function, with a Lipschitz constant Lh . Then we have Z b 1 2 (h(ξ) − h(a)) dξ ≥ (h(b) − h(a)) , a 2Lh ∀a, b ∈ R. Applying this to fi± we find that Z un I un I−e + i i n fi+ kJ+e , s − f k , u ds I−e J+e /2 /2 i i i i i ≥ 2 1 i i fi+ kJ+e , unI − fi+ kJ+e , unI−ei , i /2 i /2 2 maxu |∂u f (k, u)| Z unI − i i n − fi− kJ+e , s − f k , u ds I+e J+e /2 /2 i i i i un I+e i ≥ 2 1 i i fi− kJ+e , unI+ei − fi− kJ+e , unI . i /2 i /2 2 maxu |∂u f (k, u)| The above and (4.7) imply that !2 d d X XX XX Di A unI−ei ∆thd ∆thd−1 d − i n + f k , u I+ei J+ei /2 i i maxu A0 (u) h 2 maxu,k |∂u f (k, u)| I i=1 I i=1 2 2 + + i n i n i n −fi− kJ+e , u + f k , u − f k , u I I I−ei J+ei /2 J+ei /2 i i i /2 ≤ i −hd Xh 2 2 Dt (unI ) − (Dt unI ) + C∆t, 2 I (4.14) DIFFERENCE SCHEMES WITH ROUGH COEFFICIENTS 263 where the constant C does not depend on ∆t. Furthermore, using the definition of the scheme, (4.1), and the inequality (a + b)2 ≤ 2(a2 + b2 ), we find that d X 2 X 1 d n 2 2 i n n EO i n n (Dt uI ) ≤ 2λ d fiEO kJ+e , u , u − f k , u , u I I+e i I−e I J+e /2 /2 i i i i i 2 i=1 2 i i + fiEO kJ+e , unI−ei , unI − fiEO kJ−e , unI−ei , unI i /2 i /2 + 2µ2 d d h X 2 i 2 (Di A (unI )) + Di A unI−ei i=1 ≤ 4λ2 d 2 d − i i n fi kJ+ei /2 , unI+ei − fi− kJ+e , u I i /2 d X X i=1 i (4.15) 2 + i n i n + fi+ kJ+e , u − f k , u I I−ei J+ei /2 i i /2 + 2λ2 d d X 2 X d i n n EO i n n , u , u , u , u fiEO kJ+e − f k I−ei I i I−ei I J−ei /2 i /2 i=1 + 2µ2 d i d h X 2 i (Di A (unI ))2 + Di A unI−ei . i=1 In what follows, we assume the following strengthened CFL condition CFLε := 8λd max |∂u fi (k, u)| + 4µd max A0 (u) ≤ 1 − ε, u i,k,u (4.16) where ε ∈ (0, 1) is given a real number. Note that if A0 ≡ 0, i.e., in the hyperbolic case, (4.16) implies CFLε ∈ (0, 18 ), which should be compared with the usual CFL ∈ (0, 1), see (4.2). The new CFL condition implies in particular that ∆t ∆t ∆t(1 − ε) = 8λd max |∂u fi (k, u)| ≤ , k,u,i h 2h maxk,u,i |∂u fi (k, u)| 2h maxk,u,i |∂u fi (k, u)| ∆t ∆t ∆t(1 − ε) 2µ2 d = 2µd 2 = 4d max A0 (u) 2 ≤ 2 , u h 2h maxu A0 (u) 2h maxu A0 (u) 4λ2 d = 4λd and therefore d X 2 X 1 ∆t(1 − ε) d + n 2 i n i n (Dt uI ) ≤ fi+ kJ+e , u − f k , u I I−e J+e /2 /2 i i i i i 2 2h maxk,u,i |∂u fi (k, u)| i=1 2 − i n i n + fi− kJ+e , u − f k , u I+e I J+e i /2 /2 i i i 2 XX ∆t(1 − ε) d i + const. Di kJ−e /2 i i 2h maxk,u,i |∂u fi (k, u)| i=1 d Xh 2 i ∆t(1 − ε) (Di A (unI ))2 + Di A unI−ei . 2 0 2h maxu A (u) i=1 d + (4.17) 264 K.H. KARLSEN AND N.H. RISEBRO Now we multiply the above inequality (4.17) by hd and sum over I and n = 0, . . . , N , and sum (4.14) over n, and add the results to find that ∆tεhd X X maxu A0 (u) d n,I i=1 !2 d X XX Di A unI−ei ∆thd−1 ε d i fi+ kJ+e + , unI /2 i i h 2 maxi,k,u |∂u fi (k, u)| n,I i=1 2 2 + − − i n i n i n −fi kJ+ei /2 , uI−ei + fi kJ+ei /2 , uI+ei − fi kJ+ei /2 , uI ≤ hd X 0 2 1 uI + CT ≤ ku0 kL∞ (Rd ) ku0 kL1 (Rd ) + CT ≤ C, 2 2 (4.18) I for some constant C not depending on ∆t. We thus have the following bound ∆thd 2 d XX Di A (un ) I n,I i=1 h ≤ C, (4.19) for some constant C not depending on ∆t. Next let I(x) be the multi-index such that x ∈ χI . Then we have that I(x + y) − I(x) =: J = (J1 , . . . , Jd ), and |Jh| ≤ |y| + h. Set Ki = I + (J1 , . . . , Ji−1 , 0, . . . , 0). Using this notation we can write Ji d X X A (u∆t (x + y, t)) − A (u∆t (x, t)) = A unI(x+y) − A unI(x) = Di A unKi −(j−1)ei . i=1 j=1 By the Cauchy-Schwartz inequality Ji d X 2 2 X A (u∆t (x + y, t)) − A (u∆t (x, t)) ≤ |J| Dj A unKi −(j−1)ej , i=1 j=1 where |J| d X = |J i | . i=1 Hence using (4.19), ZZ 2 2 X A (u∆t (x + y, t)) − A (u∆t (x, t)) dt dx ≤ ∆thd A unI(xI +y) − A unI(xI ) n,I ΠT ≤ ∆th |J| d Ji d X XX 2 Di A unKi −(j−1)ei n,I i=1 j=1 ≤ |J| ∆thd Ji X d d X 2 XX D` A unKi −(j−1)ei n,I i=1 j=1 `=1 ≤ (|J|)2 d∆thd d XX (Di A (unI ))2 ≤ C (|J| h)2 , n,I i=1 where the constant C does not depend on y or h. Noting that |J| h ≤ C (|y| + h), we find that kA (u∆t (· + y, ·)) − A (u∆t (·, ·))kL2 (ΠT ) ≤ C(|y| + h). (4.20) DIFFERENCE SCHEMES WITH ROUGH COEFFICIENTS 265 Next, we will use the weak space estimate (4.19) and the difference scheme itself to show that A (u∆t ) is also L2 continuous in time. Let n(t) denote the integer such that t ∈ [tn(t) , tn(t)+1 ). Then we have ZZ 2 A (u∆t (x, t + τ )) − A (u∆t (x, t)) dt dx ΠT −τ ≤ max A0 (u) u ZZ A (u∆t (x, t + τ )) − A (u∆t (x, t)) (u∆t (x, t + τ ) − u∆t (x, t)) dt dx ΠT −τ = max A0 (u) u Z T −τ hd 0 X n(t+τ ) n(t) n(t+τ ) n(t) − uI A uI − A uI uI dt. I We denote the above integrand by B(t) and write n(t)+n(τ X n(t+τ ) X )−1 n(t) Dt unI . B(t) = h A uI − A uI d I (4.21) n=n(t) Then each term in the sum over n above equals hd X n(t+τ ) n(t) ( A uI − A uI Dt unI I = −∆th − fi− d−1 d X XX d I i=1 i kJ+e , unI i /2 i " n(t+τ ) n(t) i n A uI − A uI fi− kJ+e , u I+e /2 i i fi+ + i n i n kJ+ei /2 , uI − fi kJ+ei /2 , uI−ei + n(t+τ ) n(t) + A uI − A uI # i i × fiEO kJ+e , unI−ei , unI − fiEO kJ−e , unI−ei , unI i /2 i /2 + ∆thd−2 d X X n(t+τ ) n(t) Di2 A unI−ei . A uI − A uI I (4.22) (4.23) (4.24) i=1 We can do a partial summation in (4.22) to find that d X X d n(t+τ ) n(t) EO i n n |(4.22)| = ∆th Di A uI−ei − A uI−ei fi kJ+ei /2 , uI−ei , uI i i=1 2 2 n(t+τ ) n(t) d X d X A u A u D D i i I−e I−e ∆th i i d EO i + ≤ fi kJ+ei /2 , unI−ei , unI . i 2 i=1 h h d−1 (4.25) 266 K.H. KARLSEN AND N.H. RISEBRO In (4.25), one should recall that we have uniform control over the L2 (ΠT ) norm of the discrete gradient of A(u∆t ), see (4.19). Since k is of bounded variation, |(4.23)| ≤ ∆tC |k|BV (Rd ) , (4.26) where C does not depend on ∆t. Regarding (4.24) we have that d X X n(t+τ ) n(t) |(4.24)| = ∆thd−2 −Di A uI−ei Di A unI−ei + Di A uI−ei A unI−ei I i=1 " # d 2 1 2 XX 2 1 n(t+τ ) n(t) d−2 n . ≤ ∆th Di A uI−ei + Di A uI−ei + Di A uI−ei 2 2 i=1 (4.27) I We can write Z Z 1 1 B1 (t) + B2 (t) + B3 (t) dt, 2 2 0 0 where C does not depend on ∆t and B1 , B2 , B3 are defined via (4.21) and (4.27). Now Z T −τ B(t) dt ≤ (T − τ )C n(τ )∆t + Z T −τ T −τ B1 (t) dt = 0 ∆thd−2 d n(t)+n(τ XX X )−1 0 i=1 I N −n(τ ) = X T −τ ∆thd−2 m=0 2 n(t+τ ) Di A uI dt n=n(t) d n(tm )+n(τ XX X )−1 2 n(t +τ ) Di A uI m ∆t I i=1 ≤ (n(τ ) + 1)∆t∆thd−2 n=n(tm ) d XXX n I (Di A (unI ))2 i=1 ≤ C(τ + ∆t). (4.28) Similarly Z T −τ B2 (t) dt ≤ C (τ + ∆t) . (4.29) 0 Finally Z T −τ N −n(τ ) B3 (t) dt ≤ 0 X ∆th n=m m=0 X d XX I (Di A (unI ))2 ∆t i=1 N −n(τ ) n(τ )−1 = ∆t X m+n(τ )−1 d−2 ∆thd−2 k=0 d X XX n=0 I Di A un+k I 2 i=1 ≤ C(τ + ∆t), where C does not depend on ∆t. Using the bounds (4.28–4.30), we find that √ kA (u∆t (·, · + τ )) − A (u∆t (·, ·))kL2 (Rd ×(0,T −τ )) ≤ C ∆t + τ . In view Lemma 2.3, we conclude that A (u∆t ) → A strongly in L2loc (R × (0, T )) as ∆t ↓ 0 and A ∈ L2 (0, T ; H 1(R)). (4.30) (4.31) DIFFERENCE SCHEMES WITH ROUGH COEFFICIENTS 267 Equipped with the strong convergence u∆t → u a.e., we conclude immediately that A = A(u) and thus (D.3) holds. We sum up our results in the following theorem: Theorem 4.1. Assume that A, f , and k satisfy (2.1, 2.2), and (2.3), respectively. Furthermore, assume that u0 is a function in L1 (Rd ) ∩ L∞ (Rd ) ∩ BV (Rd ). If the CFL condition (4.16) holds, then the piecewise constant approximate solutions (3.5) generated by the Engquist-Osher scheme (4.1) converge to the unique entropy solution of (1.1). In the special case without coefficients, i.e., k = 1, we can use our techniques to prove existence of an entropy solution without necessarily having initial data in BV (Rd ). This can be done as follows. Assuming that u0 ∈ L∞ (Rd ) ∩ L1 (Rd ), we study the problem ut + divf (u) = ∆A(u), u(x, 0) = u0 (x), (4.32) where f and A are as before. We obtain the two first bounds in (4.3) as before. Fixing ε ∈ Rd , we have ku∆t (· + ε, t) − u∆t (·, t)kL1 (Rd ) ≤ ku0 (· + ε) − u0 (·)kL1 (Rd ) ≤ ν(|ε| ; u0 ), since any function in L∞ (Rd )∩L1 (Rd ) has some modulus of continuity and the scheme is now translation invariant. Then we use Kružkov’s interpolation lemma (Lem. 2.4) to show that u∆t also has a modulus of continuity in time. Next, we use the L1 compactness lemma (Lem. 2.2) to show that {u∆t }∆t>0 has a subsequence that converges strongly in L1 to a function that satisfies (D.1), (D.4), and the entropy condition (D.3). To finally show that the limit satisfies (D.3), note that to obtain the crucial estimates (4.20) and (4.31) we did not use a BV bound on u∆t . Thus we have shown the following theorem: Theorem 4.2. Assume that the function u0 is in L∞ (Rd ) ∩ L1 (Rd ). If the CFL condition (4.16) holds, then then the piecewise constant approximate solutions (3.5) generated by the Engquist-Osher scheme (4.1) converge to the unique entropy solution of (4.32). References [1] M. 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