UPWIND DIFFERENCE APPROXIMATIONS FOR DEGENERATE PARABOLIC CONVECTION-DIFFUSION EQUATIONS WITH A DISCONTINUOUS COEFFICIENT K. H. KARLSEN, N. H. RISEBRO, AND J. D. TOWERS Abstract. We analyze approximate solutions generated by an upwind difference scheme (of Engquist-Osher type) for nonlinear degenerate parabolic convection-diffusion equations where the nonlinear convective flux function has a discontinuous coefficient γ(x) and the diffusion function A(u) is allowed to be strongly degenerate (the pure hyperbolic case is included in our setup). The main problem is obtaining a uniform bound on the total variation of the difference approximation u∆ , which is a manifestation of resonance. To circumvent this analytical problem, we construct a singular mapping Ψ(γ, ·) such that the total variation of the transformed variable z ∆ = Ψ(γ ∆ , u∆ ) can be bounded uniformly in ∆. This establishes strong L1 compactness of z ∆ and, since Ψ(γ, ·) is invertible, also u∆ . Our singular mapping is novel in that it incorporates a contribution from the diffusion function A(u). We then show that the limit of a converging sequence of difference approximations is a weak solution as well as satisfying a Kružkov-type entropy inequality. We prove that the diffusion function A(u) is Hölder continuous, implying that the constructed weak solution u is continuous in those regions where the diffusion is nondegenerate. Finally, some numerical experiments are presented and discussed. 1. Introduction and statement of results We are interested in upwind finite difference approximations for nonlinear degenerate parabolic convection-diffusion initial value problems of the type ( ut + f (γ(x), u)x = A(u)xx , (x, t) ∈ ΠT = R × (0, T ), (1.1) u(x, 0) = u0 (x), x ∈ R, where T > 0 is fixed, u(x, t) is the scalar unknown function that is sought, and f, γ, A, u0 are given functions to be detailed later. The special feature of the problem studied herein, which makes mathematical and numerical analysis more complicated, is the combination of a convection part that depends explicitly on the spatial location through a coefficient γ(x) that may be discontinuous and a diffusion part that strongly degenerates in the sense that A0 (·) ≥ 0. In fact, included in our setup are hyperbolic conservation laws with a discontinuous coefficient: (1.2) ut + f (γ(x), u)x = 0. To facilitate the analysis, (1.2) is often written as a 2×2 nonstrictly hyperbolic system of equations: (1.3) γt = 0, ut + f (γ, u)x = 0. Problems of the type (1.1) occur in several applications. Biased by our own interests, we mention here only flow in porous media (see, e.g., [8, 14]) and sedimentation-consolidation processes [4, 5]. The purely convective version of (1.1) (A0 (u) ≡ 0) provides a simple model of traffic flow on a highway [49, 25], the spatially varying coefficient γ corresponding to varying road conditions. Scalar 1991 Mathematics Subject Classification. 35K65, 35L45, 35L65, 65M06, 65M12. Key words and phrases. degenerate convection-diffusion equation, discontinuous coefficient, weak solution, finite difference scheme, convergence, entropy condition. This work was done while K. H. Karlsen was visiting the Department of Mathematics and the Institute for Pure and Applied Mathematics (IPAM) at the University of California Los Angeles (UCLA). K. H. Karlsen thanks IPAM for their financial support and hospitality. The authors thank the project nonlinear partial differential equations of evolution type - theory and numerics, which is part of the BeMatA program of The Research Council of Norway, for financial support. Finally, we thank the referees for providing valuable comments that have resulted in a number of improvements in this paper. 1 2 KARLSEN, RISEBRO, AND TOWERS conservation laws with discontinuities in the flux also arise in radar shape-from-shading problems [41] and as building blocks in dimensional splitting methods for multi-dimensional Hamilton-Jacobi equations [30]. Before continuing, let us detail the assumptions that we need to impose on the “data” of the problem (1.1). For the coefficient γ, we assume that γ(x) ∈ [γ, γ] ∀x ∈ R; γ ∈ BV (R). In particular, γ is allowed to be discontinuous. For the convective flux function f , we assume that (1.4) f (γ, 0) = f0 ∈ R for all γ and f (γ, 1) = f1 ∈ R for all γ. The purpose of this assumption is to guarantee that a solution initially in the interval [0, 1] remains in [0, 1] for all subsequent times. Furthermore, we assume that f ∈ Lip([γ, γ] × [0, 1]). With this assumption the partial derivatives fγ and fu exist almost everywhere, and kfγ k∞ and kfu k∞ are Lipschitz constants of f with respect to γ and u. In what follows, we will often use the notational shorthands kfu k, kfγ k instead of kfu k∞ , kfγ k∞ , respectively. Let fu+ (γ, u) = max(0, fu (γ, u)), fu− (γ, u) = min(0, fu (γ, u)). We will also require the technical assumption that fu is Lipschitz continuous as a function of γ, with Lipschitz constant Luγ . It follows that fu− and fu+ are also Lipschitz continuous in γ with the same Lipschitz constant. For example, if f (γ, u) = γ f˜(u), where f˜ ∈ Lip([0, 1]), this Lipschitz assumption will hold with Luγ = kf˜0 k. We assume that for each γ ∈ [γ, γ], there is a unique maximum u∗ (γ) ∈ [0, 1] such that f (γ, ·) is strictly increasing for u < u∗ (γ) and strictly decreasing for u > u∗ (γ), and that for each fixed γ ∈ [γ, γ], |fu (γ, u)| > 0 for almost all u ∈ [0, 1]. Finally, we also assume that u∗ is Lipschitz continuous as a function of γ. Regarding the diffusion function A, we assume that it belongs to Lip([0, 1]) with Lipschitz constant kA0 k and that the following degenerate parabolicity condition holds: (1.5) A(·) is nondecreasing with A(0) = 0. Actually we shall be a bit more precise than (1.5). We assume that A degenerates (i.e., is constant) on a finite set of disjoint intervals, that is, A0 (w) = 0, ∀w ∈ M [ [αi , βi ] := Ω, i=1 where αi < βi , i = 1, . . . , M , M ≥ 1. On these intervals, (1.1) acts as a pure hyperbolic problem. We assume that A is non-degenerate (i.e., strictly increasing) off these intervals, so that (1.1) acts as a parabolic problem on [0, 1]\Ω. It is assumed that the maximum u∗ (γ) either lies in Ω for all γ, or lies in the closure of [0, 1]\Ω for all γ. This is trivially satisfied, for example, if f (γ, u) = γ f˜(u), since then u∗ is constant. In view of (1.5), one often refers to (1.1) as a mixed hyperbolic-parabolic problem. In what follows, we assume (since the pure hyperbolic case has already been treated in [44, 45]) max A0 (w) > 0. w∈[0,1] Finally, we assume that the initial function u0 satisfies T 1 u0 ∈ L (R) BV (R); u0 (x) ∈ [0, 1] ∀x ∈ R; (1.6) A(u0 ) is absolutely continuous on R; A(u0 )x ∈ BV (R). Our assumption that A(u0 ) is absolutely continuous requires that any jump in u0 must be contained within one of the intervals [αi , βi ] where A is constant. Independently of the smoothness of γ, 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 and weak solutions must be sought. A weak solution is here defined as follows: Definition 1.1. A measurable function u(x, t) is a weak solution of the initial value problem (1.1) if it satisfies the following conditions: DEGENERATE EQUATIONS WITH A DISCONTINUOUS COEFFICIENT 3 T T (D.1) u ∈ L1 (ΠT ) L∞ (ΠT ) C(0, T ; L1 (R)) and A(u) ∈ L2 (0, T ; H 1 (R)). (D.2) For all test functions φ ∈ D(ΠT ) such that φ|t=T = 0, Z ZZ uφt + (f (γ(x), u) − A(u)x )φx dt dx + u0 (x)φ(x, 0) = 0. (1.7) R ΠT Solutions behave even more dramatically if A0 (s) is zero on a whole interval [α, β]. Then (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. If γ is “smooth”, a weak solution u satisfies the entropy condition if for all convex C 2 functions η : R → R, (1.8) η(u)t + (q(γ(x), u))x + r(u)xx + γ 0 (x) η 0 (u)fγ (γ(x), u) − qγ (γ(x), u) ≤ 0 in D0 (ΠT ), where q, r : R → R are defined by qu (γ(x), u) = η 0 (u)fu (γ(x), u), r0 (u) = η 0 (u)A0 (u). By a standard limiting argument, (1.8) implies that the Kružkov-type entropy condition (1.9) |u − c|t + [sign(u − c)(f (γ(x), u) − f (γ(x), c))]x + |A(u) − A(c)|xx + γ 0 (x)sign(u − c)fγ (γ(x), c) ≤ 0 holds in D0 (ΠT ) for all c ∈ R. The entropy condition (1.9) goes back to Kružkov [37], Vol’pert [46], and Vol’pert and Hudjaev [48]. Existence, uniqueness and stability results for entropy solutions of strongly degenerate parabolic equations with smooth coefficients can be found in [2, 6, 9, 28, 50, 48, 47]. For example, when the coefficients are sufficiently smooth and the initial function satisfies (1.6), there exists a unique entropy solution of (1.1) that belongs to BV (ΠT ) (i.e., ux and ut are 1 finite measures on ΠT ) and A(u) belongs to the Hölder space C 1, 2 (ΠT ). The entropy solution theory breaks down when γ is discontinuous. In Karlsen, Risebro, and Towers [31], we took a first step towards analyzing degenerate parabolic equations with a discontinuous coefficient. More precisely, we proved existence of a weak solution by passing to the limit in a problem where we had smoothed out the coefficient and added artificial viscosity. In contrast to the present paper, the convergence proof in [31] used the compensated compactness theory. In this paper, we are interested in constructing a “simple” numerical scheme for (1.1) and proving its strong convergence towards a weak solution. When γ is constant or at least smooth, several numerical schemes have been proposed and analyzed already in the literature. Let us mention the operator splitting methods in [16, 24], the finite difference schemes in [18, 15, 17], the finite volume schemes in [1, 40, 20], and the kinetic BGK schemes in [3]. For a partial overview of mathematical and numerical theory for degenerate parabolic equations based on “hyperbolic” techniques, see [14]. We now present the numerical scheme that we propose for (1.1) when γ is possibly discontinuous. Let ∆x > 0 and ∆t > 0 denote the spatial and temporal discretization parameters respectively. We then let Ujn denote the finite difference approximation of u(j∆x, n∆t). The difference scheme, which uses the Engquist-Osher numerical flux [13] for the convection part and centered differencing for the parabolic part, takes the following (conservation) form (1.10) n n h(γj+ 12 , Uj+1 , Ujn ) − h(γj− 12 , Ujn , Uj−1 ) Ujn+1 − Ujn + ∆t ∆x n n A(Uj+1 ) − 2A(Ujn ) + A(Uj−1 ) = . (∆x)2 Here the numerical flux h is the Engquist-Osher generalized upwind flux [13] (see Section 3 for precise statements). The scheme (1.10) is the one-dimensional version of the multidimensional algorithm presented in [29], where convergence was established for a “rough” but continuous coefficient γ. It is also closely related to the algorithm presented in [44] and [45], where a staggered mesh EO scheme was investigated for a purely hyperbolic problem with a discontinuous coefficient. 4 KARLSEN, RISEBRO, AND TOWERS In particular, the discretization of γ is staggered with respect to that of the conserved variable u. This results in a significant reduction in complexity compared with the alternative of aligning the two discretizations. In the latter case, a more complicated 2 × 2 Riemann problem has to be solved (exactly or approximately) [38, 39, 21, 35, 34]. Staggering the discretizations also greatly simplifies the analysis, making it possible to apply, with some allowances for the parabolic terms, some of the analytical techniques developed for monotone difference schemes for purely hyperbolic problems. Another important feature of our scheme is its conservation form, i.e., it is a shock capturing algorithm in the purely hyperbolic regime where A0 = 0. For the case of constant γ, [18] provides numerical evidence that differencing the PDE (1.1) directly (i.e., not in conservation form) results in wrong solutions, specifically shocks may move with the wrong speed. Finally, our algorithm is a so-called upwind scheme, meaning that the differencing of the convective flux is biased in the direction of incoming waves, making it possible to resolve shocks without excessive smearing. Let u∆ (x, t) be the piecewise constant approximate solution generated by (1.10), and define σ(w) = w/|w| if w 6= 0 and σ(0) = 0. Roughly speaking, our main results can be stated as follows Theorem 1.1. We have that u∆ converges along a subsequence in L1loc (ΠT ) to a weak solution u of the initial value problem(1.1) in the sense of Definition 1.1. Furthermore, if γ has finitely many discontinuities located at ξ1 , . . . , ξM 0 , the limit satisfies the following entropy condition in D0 (ΠT ) for all c ∈ R: (1.11) |u − c|t + (σ(u − c)(f (γ(x), u) − f (γ(x), c)))x + |A(u) − A(c)|xx − |f (γ(x), c)x | ≤ 0. The convergence proof consists of establishing bounds on the solution and its L1 space and time translates, measured with respect to a transformed variable z. Specifically, we prove that the scalar upwind difference scheme converges (along a subsequence) to a weak solution of (1.1) by constructing a singular mapping Ψ : (γ, u) 7→ (γ, z) such that strong compactness of z ∆ = Ψ(γ ∆ , u∆ ) can be obtained. As in other problems concerning resonance phenomena, it is necessary to measure the space translates with respect to a nonlinear transformation Ψ, since there is generally no spatial variation bound for the conserved variable u itself. The singular mapping approach has been used for at least twenty years in the purely hyperbolic setting. However, the presence of the parabolic term in (1.1) requires a novel, and somewhat more complicated, singular mapping. Specifically, the singular mapping (3.1) includes a contribution from the diffusion term A(u), which make the subsequent analysis a bit more intricate than in the purely hyperbolic case. We prove compactness for two separate parts of the singular mapping. One part, F(γ, u), is associated with the convective portion of the problem, and the other, A(u), is associated with the diffusive portion of the problem. In the process of establishing compactness for the diffusive portion, we also prove that the limit u satisfies A(u) ∈ L2 (0, T ; H 1 (R)). We then combine the two portions to recover the original singular mapping F(γ, u) + A(u), and conclude that since the mapping is strictly increasing as a function of the conserved variable u, convergence of the transformed variable implies convergence of u. We also establish regularity of the diffusion function 1 A(u), specifically that A(u) ∈ C 1, 2 (ΠT ), proving that the solution u itself is continuous in the regions where there is nonzero diffusion. For the purely hyperbolic problem, the singular mapping approach can be traced back to Temple [43], who originated the technique in order to establish convergence of the Glimm scheme for a 2 × 2 resonant system of conservation laws modeling the displacement of oil in a reservoir by water and polymer. In addition to the Glimm scheme, convergence has been established for the 2 × 2 Godunov method by Lin, Temple, and Wang [38, 39]. Specifically, they applied the 2 × 2 Godunov method to the system (1.3) and used a version of the singular mapping to establish compactness (see also Hong [26] for an “improved” singular mapping). The front tracking method, which is based on the work of Dafermos [11] and Holden, Holden, and Høegh-Krohn [23], has been applied to a number of hyperbolic problems with discontinuous coefficients. Gimse and Risebro [21] used the front tracking method to study the two phase flow equation, proving compactness of the sequence of approximations via a bound on the spatial variation, measured with respect to the singular mapping. For the scalar conservation law with a concave flux, Klingenberg and DEGENERATE EQUATIONS WITH A DISCONTINUOUS COEFFICIENT 5 Risebro [35] used the front tracking technique to establish existence, uniqueness, and asymptotic behavior for the Cauchy problem (1.3). The front tracking method has also been applied to the situation where the flux f is neither concave nor convex [34]. A version of the singular mapping was used in both [35] and [34], see also [33]. The singular mapping has also been used to establish convergence of difference schemes for scalar conservation laws having a discontinuous coefficient, see Towers [44, 45]. In [44], it was also proved that limits of the difference approximations satisfied a Kružkov-type entropy condition. Moreover, uniqueness was established in the class of piecewise smooth weak solutions satisfying this entropy condition. This latter result is generalized to (1.1) in [32]. The following example, due to Lin, Temple, and Wang [38], helps to understand the impact of resonance in the purely hyperbolic setting. Assuming that the flux is smooth, they linearize the conservation law. Focusing on the case (for the sake of simplicity) where the original equation is ut + (γ(x)f (u))x = 0, the version that results from linearizing about u∗ (where f 0 (u∗ ) = 0) is ut + f (u∗ )γx = 0, which has the solution u(x, t) = −f (u∗ )γx t + u0 (x). If γ belongs to C 1 , this solution, along with its variation, grows linearly with time, i.e., neither the solution nor its variation is bounded. Furthermore, if γ is allowed to have jumps, then the solution only makes sense as a measure. If one instead linearizes about a point where f 0 (u) 6= 0 and the initial data u0 is bounded, both the solution and its variation will remain bounded; this follows from [45] and Proposition 2.1 herein. The present paper provides the groundwork for future work in several directions. The parabolic term forces a very small time step on our explicit scheme, and so we intend to investigate an implicit version, which will allow for a more efficient algorithm. We also plan to present a second order version of the scheme based on using flux limiters in a novel way that keeps the total variation bounded, as measured via the singular function. We will generalize to the situation where the diffusion term varies spatially, and incorporate more general invariant regions, making it possible to relax the condition (1.4), and allow for singular source terms. Additionally, we will prove uniqueness for piecewise smooth solutions of the initial value problem (1.1) satisfying the Kružkov-type entropy inequality (1.11). One more avenue of investigation is to relax the condition that the flux have a single maximum, allowing for any finite number of critical points. The rest of this paper is organized as follows: Section 2 provides preliminary material concerning the definition of our algorithm and the resulting approximate solutions. In Section 3 we state and prove our main result, convergence of the scheme (2.4) to a weak solution of the initial value 1 problem (1.1). Section 4 establishes that the diffusion function A(u) is continuous of class C 1, 2 . In Section 5 we demonstrate that our scheme satisfies a cell entropy inequality, and that as a consequence, piecewise smooth limits of our algorithm satisfy a Kružkov-type entropy inequality. Section 6 provides the results of some numerical experiments. 2. Definition of approximate solutions Let ∆x > 0 and ∆t > 0 be the spatial and temporal discretization parameters. The spatial domain R is discretized into cells Ij = [xj− 21 , xj+ 12 ), where xk = k∆x for k = 0, ± 12 , ±1, ± 32 , . . . . Similarly, the time interval [0, T ] is discretized via tn = n∆t for n = 0, . . . , N , where the integer N is chosen such that N ∆t = T , resulting in the time strips I n = [tn , tn+1 ). n We let χj (x) and χ (t) be the characteristic functions for the intervals Ij and I n , respectively. We let χnj (x, t) = χj (x)χn (t) be the characteristic function for the rectangle Rjn = Ij × I n . n Also, we let Rj+ 1 = Ij+ 1 × [tn , tn+1 ) with Ij+ 1 = [xj , xj+1 ). To simplify the presentation, we 2 2 2 use ∆+ and ∆− to designate the difference operators in the x direction, e.g., n n ∆+ f (γj , Ujn ) = f (γj+1 , Uj+1 ) − f (γj , Ujn ) = ∆− f (γj+1 , Uj+1 ). 6 KARLSEN, RISEBRO, AND TOWERS Furthermore, ∆u+ and ∆u− are spatial difference operators with respect to u only, keeping γ fixed, e.g., n ∆u+ f (γj , Ujn ) = f (γj , Uj+1 ) − f (γj , Ujn ). Before we can state the finite difference scheme, we need to introduce the Engquist-Osher (EO henceforth) numerical flux [13]: Z 1 v 1 h(γ, v, u) = (f (γ, u) + f (γ, v)) − |fu (γ, w)| dw. 2 2 u The EO numerical flux is consistent with the actual flux in the sense that h(γ, u, u) = f (γ, u). In addition, for fixed γ, h(γ, v, u) is a two-point monotone flux, meaning that it is nonincreasing with respect to v, and nondecreasing with respect to u. Due to the regularity assumptions about the flux f , the numerical flux h is Lipschitz continuous with respect to each of its arguments, and in fact satisfies (2.1) fu− (γ, v) = hv (γ, v, u) ≤ 0 ≤ hu (γ, v, u) = fu+ (γ, u). Thus, if the flux f is C 1 smooth, the numerical flux is also C 1 smooth as a function of the conserved variables u and v. From formula (2.1) it is clear that kfu k is a Lipschitz constant for the conserved variables u and v. It is not hard to check that kfγ k + 21 Luγ is a Lipschitz constant for h with respect to the variable γ. We also recall the decomposition (2.2) n ∆u− h(γj+ 12 , Uj+1 , Ujn ) = ∆u+ h− (γj+ 12 , Ujn ) + ∆u− h+ (γj+ 12 , Ujn ), where h− , h+ are defined in (2.3) via Z ξ (2.3) h− (γj+ 12 , ξ) = fu− (γj+ 21 , w)dw, h+ (γj+ 12 , ξ) = 0 Z 0 ξ fu+ (γj+ 12 , w)dw. It is not hard to check that h− , h+ ∈ Lip([γ, γ] × [0, 1]) with Lipschitz constants kfu k and Luγ . The difference scheme that is analyzed in this paper can then be stated as follows: (2.4) n Ujn+1 = Ujn − λ∆− h(γj+ 21 , Uj+1 , Ujn ) + µ∆− ∆+ A(Ujn ), for j ∈ Z, n = 0, . . . , N − 1, and λ, µ denoting the numbers ∆t , ∆x The iteration (2.4) is started by setting λ= µ= ∆t λ = . 2 ∆x ∆x 1 (u0 (xj −) + u0 (xj +)) , 2 and the discretization of γ is staggered with respect to that of u: 1 (2.6) γj+ 21 = γ(xj+ 12 −) + γ(xj+ 12 +) . 2 (2.5) Uj0 = Observe that the definitions of Uj0 , γj+ 12 are meaningful since u0 , γ ∈ BV (R). It can be shown (see Lemma 3.2) that the difference scheme (2.4) is monotone and Ujn ∈ [0, 1] with the CFL condition (2.7) 2λkfu k + 2µkA0 k ≤ 1. In the case where kA0 k > 0, we will have λ = O (∆x). In fact, then we can assume that there is a constant 0 < b ≤ 1 with b∆x b λ= ≤ ∆x. 2kfu k∆x + 2kA0 k 2kA0 k In the totally degenerate case (pure convection), kA0 k = 0, and then we can allow a less restrictive CFL condition of the form λ = O(1). For the remainder of this paper we will assume that kA0 k > 0. DEGENERATE EQUATIONS WITH A DISCONTINUOUS COEFFICIENT 7 + Let N = bT /∆tc, and ZN = {1, . . . , N }. The difference solution {Ujn } constructed via the scheme (2.4) is extended to all of ΠT by defining X X (2.8) u∆ (x, t) = χnj (x, t)Ujn , (x, t) ∈ ΠT , j∈Z n∈Z+ N where ∆ = (∆x, ∆t). Similarly, the discrete coefficient {γj+ 12 } is extended to all of R by defining X γ ∆ (x) = χj+ 12 (x)γj+ 12 , x ∈ R, j∈Z where χj+ 12 is the characteristic function for the interval Ij+ 12 = [xj , xj+1 ). Before proceeding to the proof of the main result of the paper, we provide the following as motivation for the singular mapping approach. In the situation considered in the following lemma, we can actually bound the spatial variation of Ujn directly. Note however that we are actually bounding the variation of f (Ujn ), and because f 0 > 0, this leads to a variation bound on Ujn . Proposition 2.1. In addition to the assumptions on the data described in Section 1, assume that 0 f is smooth, f (u) → 0 as u → 0, and f 0 is bounded away from zero, f 0 (u) > fmin > 0. Also assume that γ > 0. If the scheme n (2.9) Ujn+1 = Ujn − λ γj+ 12 f (Ujn ) − γj− 12 f (Uj−1 ) . is used to generate approximations {Ujn } to the conservation law (1.2), and the CFL condition (2.7) is satisfied, the following spatial variation bounds hold uniformly in ∆: X X n n γj+1/2 f (Ujn ) − γj−1/2 f (Uj−1 Uj+1 (2.10) ) ≤ C1 , − Ujn ≤ C2 . j∈Z j∈Z Proof. To prove the first estimate in (2.10), 1 X X U n+1 − Ujn ≤ C, ∆− γj+1/2 f (Ujn ) = j λ j∈Z j∈Z by Lemma 3.3. For the second estimate, X 1 X n n f (Ujn ) − f (Uj−1 γj+1/2 f (Ujn ) − γj+1/2 f (Uj−1 ) ) ≤ γ j∈Z j∈Z X 1 n n n γj+1/2 f (Ujn ) − γj−1/2 f (Uj−1 ≤ ) + γj−1/2 f (Uj−1 ) − γj+1/2 f (Uj−1 ) γ j∈Z 1 X 1 X ∆− γj+1/2 , ≤ ∆− γj+1/2 f (Ujn ) + kf k γ γ j∈Z j∈Z and this is uniformly bounded, using the first estimate in (2.10) and the fact that γ ∈ BV . Now, by the mean value theorem, X X 0 n n n f (Ujn ) − f (Uj−1 f (θj )kUjn − Uj−1 ) = , j∈Z for some θjn between Ujn j∈Z n Uj−1 , and from which it follows that X X n 0 n f (Ujn ) − f (Uj−1 Ujn − Uj−1 , C≥ ) ≥ fmin j∈Z j∈Z completing the proof. Remark 2.1. For example, the flux in Proposition 2.1 could be linear, f (u) = u. 8 KARLSEN, RISEBRO, AND TOWERS For the conservation law (1.2), since f 0 > 0, the singular mapping Ψ defined by (3.1) in the next section reduces to Ψ(γ, u) = γf (u), and so the first estimate in (2.10) gives a spatial variation bound for the transformed variable, zjn = Ψ(γj , Ujn ). That bound is then used to derive a variation bound on the conserved quantity itself. Clearly, when f 0 changes sign the argument proving that T V (u) < ∞ breaks down, and it is possible to construct examples where T V (u) actually blows up. This variation blow-up can be viewed as resulting from resonance. On the other hand, the bound on T V (z) remains valid even if f 0 vanishes. This leads to the idea of proving compactness for the transformed variable z, and then using the fact that the singular mapping z = Ψ(γ, u) is strictly increasing as a function of u to recover the limiting value of the conserved quantity. Of course when f 0 changes sign, the version of the singular mapping presented above is not monotone. To see how to circumvent this problem, suppose that the flux f (u) has a single maximum at u∗ , and notice that the entropy flux F (u) = sign(u − u∗ )(f (u) − f (u∗ )) is strictly monotone, making it possible to use γF (u) as the singular mapping, at least in the purely hyperbolic case. When a degenerate diffusion term is present, a somewhat more complicated, but closely related mapping is required, as will be seen in the next section. 3. Compactness of approximate solutions In this section, the goal is to prove strong compactness of our approximate solution u∆ and that any limit of a converging subsequence of u∆ is a weak solution of (1.1). In what follows we will be studying approximate solutions as the mesh size ∆ = (∆x, ∆t) decreases. We will always assume that the mesh refinement parameter ∆ is decreasing with λ constant if kA0 k = 0 (the purely hyperbolic case), or µ constant if kA0 k = 6 0, the constant in each case determined by an appropriate CFL condition. As previously mentioned, our approach is to prove a uniform variation bound with respect to a transformed quantity z = Ψ(γ, u). The singular mapping Ψ(γ, u) is designed to be Lipschitz continuous and strictly increasing as a function of u. Due to the presence of the diffusion term A(u), it is necessary to modify the singular mapping somewhat from the purely hyperbolic setting, where the singular mapping would simply be Z u Ψhyp (γ, u) = |fu (γ, w)| dw. 0 We add the diffusion term, and at the same time, zero out the contribution of the convective flux wherever A(u) is nondegenerate. This allows us to analyze the convective portion F(γ, u) and the S diffusive portion A(u) separately. Let S be the characteristic function for i [αi , βi ]. The singular mapping is then Z u (3.1) Ψ(γ, u) = S(w)|fu (γ, w)| dw + A(u) =: F(γ, u) + A(u). 0 Lemma 3.1. Under the assumptions described in Section 1, the mapping Ψ(γ, u) is strictly increasing as a function of u. Furthermore, both Ψ and F belong to Lip([γ, γ] × [0, 1]). Proof. For Lipschitz continuity of Ψ with respect to u, Z u |Ψ(γ, u) − Ψ(γ, v)| ≤ |fu (γ, w)|dw + |A(u) − A(v)| v ≤ (kfu k + kA0 k) |u − v| , and for Lipschitz continuity as a function of γ, Z u Z u |Ψ(γ1 , u) − Ψ(γ2 , u)| = S(w)|fu (γ1 , w)|dw − S(w)|fu (γ2 , w)|dw 0 0 Z u ≤ |fu (γ1 , w) − fu (γ2 , w)|dw 0 ≤ |u − 0|Luγ |γ1 − γ2 | ≤ Luγ |γ1 − γ2 |. DEGENERATE EQUATIONS WITH A DISCONTINUOUS COEFFICIENT 9 It is clear that essentially the same estimates prove that also F ∈ Lip([γ, γ] × [0, 1]). By definition, Ψu (γ, u) = S(u)|fu (γ, u)| + (1 − S(u))A0 (u) > 0 a.e., which proves strict monotonicity. We will use the notation kΨu k, kΨγ k, kFu k, and kFγ k for the Lipschitz constants provided by the previous lemma. In what follows, Ci will denote positive constants that can depend on the data of the problem but not on ∆. The approach will be to show compactness for the sequence of transformed functions z ∆ (x, t) = Ψ(γ ∆ (x), u∆ (x, t)), where u∆ (x, t) denotes the numerical approximation generated by the scheme. A finite difference scheme such as the scheme (2.4) is monotone [10, 22] if (3.2) Ujn ≤ Vjn ∀j =⇒ Ujn+1 ≤ Vjn+1 ∀j. Lemma 3.2. If the assumptions concerning the data described in Section 1 are satisfied, and the scheme (2.4) is applied with the parameters λ and µ chosen so that the following CFL condition is satisfied (3.3) 2λkfu k + 2µkA0 k ≤ 1. Then the computed solutions remain in the interval [0, 1], the CFL condition (3.3) holds for each succeeding time step, and the scheme (2.4) is monotone. Proof. The formula (2.4) defines Ujn+1 as a function n n Ujn+1 = Gj (Uj+1 , Ujn , Uj−1 , γj+ 12 , γj− 12 ). The partial derivatives with respect to the conserved variables are n n n ∂Ujn+1 /∂Uj+1 = −λfu− (γj+ 12 , Uj+1 ) + µA0 (Uj+1 ) ≥ 0, n n n ∂Ujn+1 /∂Uj−1 = λfu+ (γj− 12 , Uj−1 ) + µA0 (Uj−1 ) ≥ 0, ∂Ujn+1 /∂Ujn = 1 + λfu− (γj+ 12 , Ujn ) − λfu+ (γj− 12 , Ujn ) − 2µA0 (Ujn ). Thus Ujn+1 is a nondecreasing function of the conserved variables at the lower time level if n 1 + λfu− (γj+ 21 , Ujn ) − λfu+ (γj− 12 , Ujn ) − 2µA0 (Uj+1 ) ≥ 0. This will hold if the CFL condition (3.3) is satisfied for the solution at level n. If the CFL condition holds for the initial data, then since each of the functions Gj (·, ·, ·, γj+ 21 , γj− 21 ) is nondecreasing as a function of its first three arguments, and Uj0 ∈ [0, 1] 0 0 0 = Gj (0, 0, 0, γj+ 21 , γj− 12 ) ≤ Gj (Uj+1 , Uj0 , Uj−1 , γj+ 21 , γj− 12 ) = Ujn+1 ≤ Gj (1, 1, 1, γj+ 12 , γj− 12 ) = 1. (3.4) The first and last equalities in this relationship result from the fact that for all γ, f (γ, 0) = f0 and f (γ, 1) = f1 . Proceeding inductively, it is clear that the solution Ujn ∈ [0, 1] for each n ≥ 0, and thus the CFL condition remains satisfied at each succeeding time level. That the scheme is n n monotone is clear from the fact that Ujn+1 is a nondecreasing function of Uj+1 , Ujn , and Uj−1 . The next lemma is of fundamental importance for the subsequent analysis. In addition to providing for L1 time continuity of the numerical approximations, it also plays a key role in our bound on the space translates of the transformed variable. With respect to the spatial variation, the situation is somewhat different here than in the case where γ is constant. Specifically, the variation (measured via the transformed variable) may actually increase from one time step to the next, and so the now classical total variation decreasing (TVD) argument is not available. Instead, we bound the spatial variation in terms of the L1 time translates. 10 KARLSEN, RISEBRO, AND TOWERS Lemma 3.3. Assume that the hypotheses of Lemma 3.2 are satisfied. Then there exists a constant C3 , independent of ∆, such that X X U n+1 − Ujn ≤ ∆x Uj1 − Uj0 ≤ C3 ∆t. ∆x j j∈Z j∈Z Proof. Starting from the marching formula (2.4), the time differences can be expressed as follows: n−1 n Ujn+1 − Ujn = Ujn − Ujn−1 − λ∆− h(γj+ 21 , Uj+1 , Ujn ) − h(γj+ 12 , Uj+1 , Ujn−1 ) + µ∆− ∆+ (A(Ujn ) − A(Ujn−1 )) n− 1 n− 1 n− 1 = 1 − λCj+ 12 + λBj− 12 − 2µDj 2 (Ujn − Ujn−1 ) 2 2 n− 12 n− 12 n−1 n + −λBj+ 1 + µDj+1 (Uj+1 − Uj+1 ) 2 1 1 n− n− n−1 n + λCj− 12 + µDj−12 (Uj−1 − Uj−1 ), 2 where n− 1 Bj+ 12 = 2 n− 1 Cj+ 12 = 2 n− 21 Dj = Z 1 n−1 n fu− (γj+ 12 , θUj+1 + (1 − θ)Uj+1 )dθ ≤ 0, 0 Z 1 0 fu+ (γj+ 12 , θUjn + (1 − θ)Ujn−1 )dθ ≥ 0, A(Ujn ) − A(Ujn−1 ) Ujn − Ujn−1 ≥ 0. Due to the CFL condition (3.3), (3.5) n− 1 n− 1 n− 12 1 − λCj+ 12 + λBj− 12 − 2µDj 2 2 ≥ 0, and so (3.6) n− 1 n− 1 n− 1 |Ujn+1 − Ujn | ≤ 1 − λCj+ 12 + λBj− 12 − 2µDj 2 |Ujn − Ujn−1 | 2 2 n− 12 n− 1 n−1 n + −λBj+ 1 + µDj+12 |Uj+1 − Uj+1 | 2 1 1 n− n− n−1 n + λCj− 12 + µDj−12 |Uj−1 − Uj−1 |. 2 Summing this inequality over j and multiplying by ∆x gives X X ∆x |Ujn+1 − Ujn | ≤ ∆x |Ujn − Ujn−1 |. j∈Z j∈Z Continuing this way by induction yields X X ∆x |Ujn+1 − Ujn | ≤ ∆x |Uj1 − Uj0 |. j∈Z j∈Z Then, using Lipschitz continuity of h and the fact that λ∆x = ∆t, X X 1 0 0 0 ∆− h(γj+ 1 , Uj+1 ∆x |Uj1 − Uj0 | = ∆t , U ) − ∆ ∆ A(U ) + − j j 2 ∆x j∈Z j∈Z X X 1 0 0 0 ≤ ∆t |∆− h(γj+ 12 , Uj+1 , Uj )| + ∆t ∆− ∆x ∆+ A(Uj ) j∈Z j∈Z X 1 1 0 ≤ 2∆t||fu ||T V (u0 ) + ∆t ||fγ || + Luγ T V (γ) + ∆t ∆− ∆+ A(Uj ) . 2 ∆x j∈Z DEGENERATE EQUATIONS WITH A DISCONTINUOUS COEFFICIENT 11 We still have to estimate the last sum in the preceding inequality. To this end, notice first that if u0 has a jump at x = xk for some index k. Then by (1.6), A(Uk0 ) = A(u0 (xk −)) = A(u0 (xk +)). Now using this, it is clear that Z xj+1 1 1 0 ∆ − ∆ A(U ) = ∆ A(u (x)) dx + − 0 x j ∆x ∆x xj 1 Z xj+1 (A(u0 (x))x − A(u0 (x − ∆x))x ) dx = ∆x xj Z xj+1 1 |A(u0 (x))x − A(u0 (x − ∆x))x | dx, ≤ ∆x xj and so summing over j and using standard results concerning total variation (e.g., Lemma 2.4.4 of [27]) yields Z X ∆− 1 ∆+ A(Uj0 ) ≤ 1 |A(u0 (x))x − A(u0 (x − ∆x))x | dx ∆x ∆x R j∈Z ≤ T V (A(u0 )x ), which gives the desired bound and completes the proof. Lemma 3.4. Under the assumptions in Lemma 3.2, the computed solutions u∆ (·, tn ) satisfy a uniform L1 (R) bound for tn ∈ [0, T ]: 1 ku∆ (·, tn )kL1 (R) ≤ C3 T + ku0 kL1 (R) + ∆xT V (u0 ), 2 and if v ∆ is another solution generated using the same discretization of ΠT , the following discrete L1 contraction property holds: (3.7) (3.8) ku∆ (·, tn ) − v ∆ (·, tn )kL1 (R) ≤ ku∆ (·, 0) − v ∆ (·, 0)kL1 (R) . Proof. Using the triangle inequality and the result of Lemma 3.3 yields X X X Ujn ≤ ∆x Ujn − U n−1 + ∆x U n−1 ku∆ (·, tn )kL1 (R) = ∆x j j j∈Z j∈Z j∈Z X X Uj1 − Uj0 + ∆x U n−1 . ≤ ∆x j j∈Z j∈Z Proceeding by induction, ku∆ (·, tn )kL1 (R) ≤ n∆x X X Uj1 − Uj0 + ∆x Uj0 . j∈Z By Lemma 3.3, n∆x 1 j∈Z |Uj P − j∈Z Uj0 | ≤ nC3 ∆t ≤ C3 T . Now applying the estimate Z X XZ u0 (x) − Uj0 dx ∆x |Uj0 | ≤ |u0 (x)| dx + R j∈Z ≤ j∈Z Ij Z 1 |u0 (x)| dx + ∆xT V (u0 ), 2 R the proof of (3.7) is complete. The discrete L1 contraction property (3.8) follows from the CrandallTartar lemma [10], using the fact that the operator which advances the initial approximation to time level n is monotone, conservative, and takes L1 mesh functions into L1 mesh functions. The next three lemmas provide a proof of compactness for the sequence of functions F ∆ defined by F ∆ (x, t) = F(γ ∆ (x), u∆ (x, t)). 12 KARLSEN, RISEBRO, AND TOWERS In what follows, we will use the Kružkov entropy-entropy flux pair indexed by c: V (u) = |u − c|, F (γ, u) = σ(u − c)(f (γ, u) − f (γ, c)). Lemma 3.5. Assume that the hypotheses of Lemma 3.2 are satisfied. For each c ∈ R, n (3.9) V (Ujn+1 ) ≤ V (Ujn ) − λ∆u− H(γj+ 12 , Uj+1 , Ujn ) + µ∆− ∆+ |A(Ujn ) − A(c)| + λC4 |∆+ γj− 12 |, where the EO numerical entropy flux is given by (3.10) 1 1 H(γ, v, u) = (F (γ, u) + F (γ, v)) − 2 2 v Z σ(w − c)|fu (γ, w)| dw. u Proof. Let a ∨ b = max(a, b) and a ∧ b = min(a, b). With n ρn+1 = Ujn − λ∆u− h(γj+ 21 , Uj+1 , Ujn ) + µ∆− ∆+ A(Ujn ), j the following discrete entropy inequality follows from Lemma 3.7 of [18]: V (ρn+1 ) ≤ V (Ujn ) − λ∆u− H(γj+ 21 , Uj+1 , Ujn ) + µ∆− ∆+ |A(Ujn ) − A(c)|, j where H(γ, v, u) = h(γ, v ∨ c, u ∨ c) − h(γ, v ∧ c, u ∧ c). (3.11) For a derivation of the explicit formula (3.10) for the EO numerical entropy flux from (3.11), see [45]. Then n , Ujn ) + µ∆− ∆+ |A(Ujn ) − A(c)| V (Ujn+1 ) ≤ V (Ujn ) − λ∆u− H(γj+ 12 , Uj+1 − V (ρn+1 ) + V (Ujn+1 ). j It remains to estimate V (ρn+1 ) − V (Ujn+1 ): j |V (ρjn+1 ) − V (Ujn+1 )| ≤ |ρn+1 − Ujn+1 | j n n = λ|∆− h(γj+ 21 , Uj+1 , Ujn ) − ∆u− h(γj+ 12 , Uj+1 , Ujn )| 1 ≤ λ kfγ k + Luγ |γj+ 12 − γj− 12 | 2 = λC4 |∆+ γj− 12 |. (3.12) In what follows, χr (w; c) is the characteristic function for the interval [c, +∞), χl (w; c) is the characteristic function for (−∞, c], and χ(α,β] (w) is the characteristic function of the interval (α, β]. The following identities (see [45] for a derivation) will be required in the proof of Lemma 3.6: 1 u n n ∆− H(γj+ 12 , Uj+1 , Ujn ) + ∆u− h(γj+ 12 , Uj+1 , Ujn ) 2 n Z Uj+1 Z Ujn (3.13) = χr (w; c)fu− (γj+ 12 , w) dw + χr (w; c)fu+ (γj+ 12 , w) dw, Ujn (3.14) n Uj−1 1 u n n ∆− H(γj+ 12 , Uj+1 , Ujn ) − ∆u− h(γj+ 12 , Uj+1 , Ujn ) 2 n Z Uj+1 Z Ujn =− χl (w; c)fu− (γj+ 12 , w) dw − χl (w; c)fu+ (γj+ 12 , w) dw. Ujn n Uj−1 Lemma 3.6. Assume that the hypotheses of Lemma 3.2 are satisfied. Let u∗ (γ) denote the unique maximum of f (γ, u) for u ∈ [0, 1]. The following inequality holds for c ≥ u∗ (γj+ 12 ): n Z Uj+1 1 u −∆− χr (w; c)|fu (γj+ 21 , w)| dw + ∆+ (A(Ujn ) − A(c))+ ∆x 0 (3.15) 1 ≤ |Ujn+1 − Ujn | + C5 |∆+ γj− 21 |, λ DEGENERATE EQUATIONS WITH A DISCONTINUOUS COEFFICIENT 13 and the following inequality holds for c ≤ u∗ (γj− 12 ): Z Ujn 1 −∆u− χl (w; c)|fu (γj− 12 , w)| dw − ∆+ (A(Ujn ) − A(c))− ∆x 0 (3.16) 1 ≤ |Ujn+1 − Ujn | + C5 |∆+ γj− 12 |. λ Proof. We start by writing the scheme (2.4) as 1 n ∆u− h(γj+ 12 , Uj+1 , Ujn )− ∆− ∆+ (A(Ujn ) − A(c)) ∆x (3.17) 1 n , Ujn ). = (Ujn − Ujn+1 ) + (∆u− − ∆− )h(γj+ 12 , Uj+1 λ Next, we write the discrete entropy inequality (3.9) as 1 n ∆u− H(γj+ 12 , Uj+1 , Ujn )− ∆− ∆+ |A(Ujn ) − A(c)| ∆x 1 (3.18) ≤ (V (Ujn ) − V (Ujn+1 )) + C4 |∆+ γj− 12 | λ 1 ≤ Ujn − Ujn+1 + C4 |∆+ γj− 12 |. λ Inequality (3.15) is derived by adding (3.17) and (3.18), and then dividing by two. The right hand side of this combination is 1 1 n 1 n (Uj − Ujn+1 ) + (∆u− − ∆− )h(γj+ 12 , Uj+1 , Ujn ) + Ujn − Ujn+1 + C4 |∆+ γj− 12 | 2 λ λ 1 1 1 n = U n − Ujn+1 + + (∆u− − ∆− )h(γj+ 12 , Uj+1 , Ujn ) + C4 |∆+ γj− 12 | λ j 2 2 1 n ≤ U − Ujn+1 + C5 |∆+ γj− 12 |, λ j where we have used the fact that the numerical flux h is Lipschitz continuous as a function of γ. Applying (3.13), the left side of the combination is 1 u n n ∆− h(γj+ 12 , Uj+1 , Ujn ) + ∆u− H(γj+ 12 , Uj+1 , Ujn ) 2 1 1 1 − ∆− ∆+ (A(Ujn ) − A(c)) + ∆− ∆+ A(Ujn ) − A(c) 2 ∆x ∆x n n Uj+1 Z ZUj = χr (w; c)fu− (γj+ 21 , w) dw + χr (w; c)fu+ (γj+ 12 , w) dw Ujn n Uj−1 1 ∆− ∆+ A(Ujn ) − A(c) + . ∆x With c ≥ u∗ (γj+ 12 ), the integral involving χr (w; c)fu+ (γj+ 21 , w) is zero, and − χr (w; c)fu− (γj+ 21 , w) = −χr (w; c)|fu (γj+ 12 , w)|, completing the proof of (3.15). Inequality (3.16) is proven in a similar way by subtracting (3.17) from (3.18) and dividing by two, and then using (3.14). Let cR (γ) = u∗ (γ) ∨ c, (3.19) cL (γ) = u∗ (γ) ∧ c and (3.20) (3.21) Z u 1 ∆+ (A(u) − A(cR (γ)))+ , ∆x 0 Z u 1 L ψ (γ, c, u) = χl (w; cL (γ))|fu (γ, w)| dw − ∆+ (A(u) − A(cL (γ)))− . ∆x 0 ψ R (γ, c, u) = χr (w; cR (γ))|fu (γ, w)| dw + 14 KARLSEN, RISEBRO, AND TOWERS Lemma 3.7. Assume that the hypotheses of Lemma 3.2 are satisfied. For some constant C6 , independent of the mesh size ∆, X ∆u− F(γj− 12 , Ujn ) ≤ C6 . j∈Z Proof. The first step of the proof is to establish a uniform (with respect to ∆) bound for the sequence of divided differences ∆+ A(Ujn )/∆x. We postpone the proof of this result to Lemma 4.1. In addition, it is clear that for any constant κ, the same uniform bound also holds for the divided differences ∆+ (A(Ujn ) − κ)− /∆x and ∆+ (A(Ujn ) − κ)+ /∆x. To continue the proof, fix i ∈ {1, . . . , M }, and take c = αi , c = βi in the formulas (3.19), (3.20), (3.21). Using the fact that A is constant on the interval [αi , βi ], we get Z u ψ L (γ, βi , u) − ψ L (γ, αi , u) = χ(αLi (γ),βiL (γ)] (w)|fu (γ, w)| dw, 0 ψ R (γ, αi , u) − ψ R (γ, βi , u) = u Z 0 R χ(αR (w)|fu (γ, w)| dw, i (γ),βi (γ)] where αiR = u∗ (γ) ∨ αi , βiR = u∗ (γ) ∨ βi , As a consequence, with the help of the identity αiL = u∗ (γ) ∧ αi , βiL = u∗ (γ) ∧ βi . R χ(αLi (γ),βiL (γ)] + χ(αR = χ(αi ,βi ] , i (γ),βi (γ)] we find that Z u Fi (γ, u) := χ(αi ,βi ] (w)|fu (γ, w)| dw Z 0u R (γ)] (w) |fu (γ, w)| dw = χ(αLi (γ),βiL (γ)] (w) + χ(αR (γ),β i i 0 = ψ L (γ, βi , u) − ψ L (γ, αi , u) + ψ R (γ, αi , u) − ψ R (γ, βi , u). Now observing that F has the decomposition F(γ, u) = (3.22) M X Fi (γ, u), i=1 it suffices to show that the sum X |∆u− Fi (γj− 21 , Ujn )| j∈Z is bounded independently of the mesh size ∆, for each i = 1, . . . , M . Recalling Lemma 3.6, for each c ∈ R we have 1 (3.23) −∆u− ψ L (γj− 21 , c, Ujn ) ≤ |Ujn+1 − Ujn | + C5 |∆+ γj− 21 |, λ 1 n+1 n (3.24) −∆u− ψ R (γj− 12 , c, Ujn ) ≤ |Uj−1 − Uj−1 | + C5 |∆− γj− 12 |. λ If the quantities A(cL (γ)), A(cR (γ)) appearing in (3.20), (3.21) are independent of γ, then both ψ L (γ, c, u) and ψ R (γ, c, u) are Lipschitz continuous in γ; this is a result of the Lipschitz continuity (with respect to γ) of fu and of u∗ . When that is the case, by replacing C5 in (3.23), (3.24) by a larger constant C7 , we can replace the ∆u− difference operators with ∆− operators: 1 n+1 |U − Ujn | + C7 |∆+ γj− 21 |, λ j 1 n+1 n (3.26) −∆− ψ R (γj− 12 , c, Ujn ) ≤ |Uj−1 − Uj−1 | + C7 |∆− γj− 12 |. λ Since the right hand sides of (3.25) and (3.26) are nonnegative, we can replace the left sides of (3.25) and (3.26) by the nonnegative quantities −(∆− ψ L (γj− 12 , c, Ujn )− and −(∆− ψ R (γj+ 12 , c, Ujn ))− . Then summing over j ∈ Z, we see that both ψ L (γj− 12 , c, Ujn ) and ψ R (γj− 12 , c, Ujn ) have uniformly (3.25) −∆− ψ L (γj− 12 , c, Ujn ) ≤ DEGENERATE EQUATIONS WITH A DISCONTINUOUS COEFFICIENT 15 bounded negative variation, as a result of Lemma 3.3. Moreover, still assuming that A(cR ) and A(cL ) are independent of γ, both ψ L (γj− 12 , c, Ujn ) and ψ R (γj− 12 , c, Ujn ) are bounded uniformly in ∆. This is a consequence of our bound on the divided differences ∆+ (A(Ujn ) − κ)− /∆x and ∆+ (A(Ujn ) − κ)+ /∆x (with κ = A(cR ), κ = A(cL )), along with the fact that Ujn ∈ [0, 1]. Taken together, the bounds on the negative variation and the amplitude allow us to conclude that the total variations are also uniformly bounded: X X ∆− ψ L (γj− 21 , c, Ujn ) ≤ C8 , ∆− ψ R (γj− 12 , c, Ujn ) ≤ C8 . j∈Z j∈Z Now using Lipschitz continuity (with respect to γ) of ψ R (γ, c, u) and ψ R (γ, c, u) one more time to go back to ∆u− difference operators, we have X X ∆u− ψ R (γj− 12 , c, Ujn ) ≤ C9 . ∆u− ψ L (γj− 21 , c, Ujn ) ≤ C9 , j∈Z j∈Z If we can apply these estimates to the ψ L and ψ R terms making up Fi , then in view of the decomposition (3.22), an application of the triangle inequality will complete the proof. Thus the remainder of the proof consists of checking that none of the quantities A(αiL ), A(αiR ), A(βiL ), A(βiR ) appearing in Fi depend on γ. First take the case where u∗ (γ) ≤ αi ≤ βi . Because of our assumptions about u∗ , if this relationship holds for any value of γ, it holds for all γ. In this case ψ L (γ, βi , u) = ψ L (γ, αi , u) = ψ L (γ, u∗ (γ), u), and thus the ψ L terms cancel, leaving Fi (γ, u) = ψ R (γ, αi , u) − ψ R (γ, βi , u). So, for this case we only have to verify that A(αiR ) and A(βiR ) are independent of γ. With u∗ (γ) ≤ αi ≤ βi , αiR = αi and βiR = βi , i.e., αiR and βiR are independent of γ, and thus so are A(αiR ), A(βiR ). Similarly, in the case where αi ≤ βi ≤ u∗ , Fi (γ, u) = ψ L (γ, βi , u) − ψ L (γ, αi , u), and αiL = αi , βiL = βi , independently of γ. The last case is the one where αi ≤ u∗ (γ) ≤ βi . Here, all four terms making up Fi are present: Fi (γ, u) = ψ L (γ, βi , u) − ψ L (γ, αi , u) + ψ R (γ, αi , u) − ψ R (γ, βi , u), but since A is constant on [αi , βi ], A(αiL ) = A(αiR ) = A(βiL ) = A(βiR ) = A(αi ) = A(βi ) = A(u∗ (γ)), all of which are independent of γ. Lemma 3.8. Assume thatthe hypotheses of Lemma 3.2 are satisfied. There exists a subsequence of F ∆ , also denoted by F ∆ , and a function \ F ∈ L1 (ΠT ) L∞ (ΠT ) such that F ∆ → F in L1loc (ΠT ) and a.e. in ΠT . Furthermore, F(·, t) ∈ L1 (R) for all t ∈ [0, T ]. Proof. The first step of the proof is to establish a uniform variation bound for F ∆ (·, tn ). The step function F ∆ (·, tn ) has jumps at cell centers, due to jumps in γ ∆ , plus jumps at cell boundaries, due to jumps in u∆ . When both types of jumps are accounted for, we arrive at X X T V (F ∆ (·, tn )) = ∆u− F(γj− 21 , Ujn ) + F(γj+ 12 , Ujn ) − F(γj− 21 , Ujn ) j∈Z j∈Z X ≤ ∆u− F(γj− 12 , Ujn ) + ||Fu || T V (γ), j∈Z yielding the desired variation bound as a result of Lemma 3.7. 16 KARLSEN, RISEBRO, AND TOWERS Using the fact that u∆ ∈ [0, 1] and the definition (3.1) of F provides an L∞ bound: kF ∆ k ≤ kfu k. To show that F ∆ (·, t) is in L1 (R) (uniformly) on each of the time slices 0 ≤ t ≤ T , observe that F(γ, 0) = 0, and so, with n chosen so that t ∈ [tn , tn+1 ), Z Z ∆ ∆ F (x, t) dx = F (x, tn ) dx R ZR F(γ ∆ (x), u∆ (x, tn )) dx = ZR F(γ ∆ (x), u∆ (x, tn )) − F(γ ∆ (x), 0) dx = R Z ∆ u (x, tn ) − 0 dx ≤ kfu k R 1 (3.27) ≤ kfu k C3 T + ku0 kL1 (R) + ∆xT V (u0 ) , 2 by Lemma 3.4. For time continuity, Lipschitz continuity of F with respect to its second argument gives Z |F ∆ (x, tn + ∆t) − F ∆ (x, tn )| dx R ∆x X |F(γj+ 12 , Ujn+1 ) − F(γj+ 12 , Ujn )| + |F(γj− 12 , Ujn+1 ) − F(γj− 12 , Ujn )| 2 j∈Z X ≤ ||Fu ||∆x |Ujn+1 − Ujn | ≤ C3 ||Fu ||∆t, = j∈Z by Lemma 3.3. Using this estimate it is easy to check that kF ∆ (·, t + τ ) − F ∆ (·, t)kL1 (R) ≤ C3 ||Fu ||(|τ | + ∆t). (3.28) Let {Xk }∞ results k=1 be a sequence of positive numbers with Xk → ∞. From standard compactness (Lemma 16.8 of [42]), there is a subsequence (which we do not bother to relabel) of F ∆ such that for any fixed Xk , Z Xk ∆ F (x, t) − F(x, t) dx → 0, t ∈ [0, T ], and (3.29) −Xk Z (3.30) 0 T Z Xk |F ∆ (x, t) − F(x, t)| dx dt → 0, −Xk for some measurable function F. By passing to a further subsequence if necessary, we may assume that F ∆ converges a.e. in [−Xk , Xk ] × [0, T ]. By applying a standard diagonal process, we can find a subsequence (which we do not bother to relabel) such that F ∆ converges to F a.e. in R × [0, T ]. Note that (3.29) and (3.30) remain valid for our subsequence. To show that each F(·, t) ∈ L1 (R), an application of the triangle inequality yields, for any Xk , Z Xk Z Xk Z Xk ∆ |F(x, t)| dx ≤ |F(x, t) − F (x, t)| dx + |F ∆ (x, t)| dx −Xk −Xk Xk ≤ −Xk Z 1 |F(x, t) − F ∆ (x, t)| dx + kfu k C3 T + ku0 kL1 (R) + ∆xT V (u0 ) . 2 −Xk Letting first ∆ → 0 and then Xk → ∞, and applying (3.29), we have kF(·, t)kL1 (R) ≤ kfu k C3 T + ku0 kL1 (R) , proving that F(·, t) ∈ L1 (R). It is clear from this estimate that kFkL1 (ΠT ) ≤ T kfu k C3 T + ku0 kL1 (R) , and so F ∈ L1 (ΠT ). That F ∈ L∞ (ΠT ) is clear from the bound on F ∆ . DEGENERATE EQUATIONS WITH A DISCONTINUOUS COEFFICIENT 17 To show strong compactness of A(u∆ ), we shall in the following three lemmas obtain uniform L2 (ΠT ) estimates on the space and time translates of A(u∆ ). Lemma 3.9. Assume that the hypotheses of Lemma 3.2 are satisfied. There exists a constant C11 , independent of ∆, such that 12 X X n 2 (3.31) ∆t ∆x (∆+ A(Uj )) ≤ C11 ∆x. j∈Z n∈Z+ N Proof. The proof is a discrete energy argument similar to the one used in [29] (but the proof here is a bit simpler since we know that the difference approximations are L1 Lipschitz continuous in time). Multiplying (2.4) by ∆t ∆xUjn , summing over n, j, and doing summation by parts in j, we find that X X X X n ∆x Ujn (Ujn+1 − Ujn ) + ∆t Ujn ∆− h(γj+ 12 , Uj+1 , Ujn ) j∈Z n∈Z+ N j∈Z n∈Z+ N + ∆t X X ∆+ Ujm ∆+ A(Ujn ) = 0. ∆x + n∈ZN j∈Z Observe that we can write Ujn (Ujn+1 − Ujn ) = 1 n+1 2 (Uj ) − (Ujn )2 − (Ujn+1 − Ujn )2 . 2 Since A0 (·) ≥ 0, we also have 1 (∆+ A(Ujn ))2 ≤ ∆+ Ujn ∆+ A(Ujn ) . 0 kA k Using these observations in (3) as well as (2.2), (3.12) and Lemma 3.3, we get ∆t X X X X Ujn ∆u+ h− (γj+ 21 , Ujn ) + ∆t Ujn ∆u− h+ (γj+ 12 , Ujn ) j∈Z n∈Z+ N j∈Z n∈Z+ N 1 ∆t X X + (∆+ A(Ujn ))2 kA0 k ∆x + n∈ZN j∈Z −∆x X X n+1 2 ≤ (Uj ) − (Ujn )2 − (Ujn+1 − Ujn )2 + C12 T · T V (γ) 2 + n∈ZN j∈Z ≤ ∆x X X n+1 ∆x X 0 2 (Uj − Ujn )2 + (Uj ) + C12 T · T V (γ) 2 2 + n∈ZN j∈Z j∈Z X X X U n+1 − Ujn + ∆x ≤ max Ujn ∆x (Uj0 )2 + C12 T · T V (γ) ≤ C13 , j n,j 2 + n∈ZN j∈Z j∈Z (3.32) for some finite constant C13 that is independent of ∆. To continue our analysis, we need to introduce the functions Z ξ Z H± (γ, ξ) = w ∂w h± (γ, w) dw = ξh± (γ, ξ) − 0 ξ h± (γ, ξ) dξ. 0 Then the following equalities hold − Z Ujn ∆u− h+ (γj+ 21 , Ujn ) = ∆u− H+ (γj+ 12 , Ujn ) + Z Ujn ∆u+ h− (γj+ 12 , Ujn ) = ∆u+ H− (γj+ 12 , Ujn ) Ujn n Uj+1 Ujn n Uj−1 n h− (γj+ 12 , w) − h− (γj+ 12 , Uj+1 ) dw, n h+ (γj+ 12 , w) − h+ (γj+ 12 , Uj−1 ) dw. 18 KARLSEN, RISEBRO, AND TOWERS Hence n X X Ujn ∆u+ h− (γj+ 12 , Ujn ) = − Uj X X Z n h− (γj+ 12 , w) − h− (γj+ 12 , Uj+1 ) dw j∈ZU n n∈Z+ N j+1 j∈Z n∈Z+ N + X X ∆u+ − ∆+ H− (γj+ 12 , Ujn ), j∈Z n∈Z+ N (3.33) n X X Ujn ∆u− h+ (γj+ 12 , Ujn ) = j∈Z n∈Z+ N Uj X X Z n h+ (γj+ 12 , w) − h+ (γj+ 12 , Uj−1 ) dw j∈ZU n n∈Z+ N j−1 + X X ∆u− − ∆− H+ (γj+ 12 , Ujn ), j∈Z n∈Z+ N and X X ∆u− − ∆− H+ (γj+ 12 , Ujn ) ≤ C14 T · T V (γ), ∆t j∈Z n∈Z+ N (3.34) ∆t X X ∆u+ − ∆+ H− (γj− 12 , Ujn ) ≤ C14 T · T V (γ). j∈Z n∈Z+ N To bound the terms involving integrals, we need the following technical result (an easy proof can be found in [19]): Let g : R → R be a monotone Lipschitz continuous function with Lipschitz constant Lg . Then we have Z ξ2 2 1 g(w) − g(ξ1 ) dw ≥ g(ξ2 ) − g(ξ1 ) , ∀ξ1 , ξ2 ∈ R. ξ1 2Lg Applying this to h− , h+ we find that Z Ujn n − h− (γj+ 12 , w) − h− (γj+ 12 , Uj+1 ) dw ≥ (3.35) Z n Uj+1 Ujn n Uj−1 n h+ (γj+ 12 , w) − h+ (γj+ 12 , Uj−1 ) dw ≥ 1 (∆u h− (γj+ 12 , Ujn ))2 , 2kfu k + 1 (∆u h+ (γj+ 12 , Ujn ))2 . 2kfu k − Inserting (3.35) into (3.32) yields via (3.33) the following inequality: ∆t X X u ∆t X X u (∆+ h− (γj+ 12 , Ujn ))2 + (∆− h+ (γj+ 12 , Ujn ))2 2kfu k 2kf k u j∈Z j∈Z n∈Z+ n∈Z+ N N (3.36) 1 ∆t X X + (∆+ A(Ujn ))2 ≤ C1 + C12 T · T V (γ) + 2C14 T · T V (γ). kA0 k ∆x + n∈ZN j∈Z From (3.36), we conclude that (3.31) holds. Remark 3.1. Although we did not need it in the proof of Lemma 3.9, (3.36) also provides us with the estimates (3.37) 12 12 X X X X √ u n 2 n 2 u 1 1 ∆t ∆x (∆+ h− (γj+ 2 , Uj )) , ∆t ∆x (∆− h+ (γj+ 2 , Uj )) ≤ C15 ∆x, j∈Z n∈Z+ N j∈Z n∈Z+ N which imply, for any X > 0, an estimate of the type X X √ (3.38) ∆t ∆x |∆u+ h− (γj+ 12 , Ujn ))| + |∆u− h+ (γj+ 21 , Ujn ))| ≤ C16 (X) ∆x, |j|≤X/∆x n∈Z+ N DEGENERATE EQUATIONS WITH A DISCONTINUOUS COEFFICIENT 19 where C16 (X) is a constant depending on X but not ∆. This is a sort of variation bound and for that reason the estimates in (3.37) are sometimes called weak BV estimates in the literature [7, 19, 36]. We mention that under a stronger CFL condition it is possible to prove Lemma 3.10 without using L1 Lipschitz continuity in time of the approximate solutions as stated in Lemma 3.3, see [29]. Lemma 3.10. Assume that the hypotheses of Lemma 3.2 are satisfied. There exists a constant C17 , independent of ∆, such that p kA(u∆ (· + y, ·)) − A(u∆ (·, ·))kL2 (ΠT ) ≤ C17 |y|(|y| + ∆x), ∀y ∈ R. Proof. Let I(x) ∈ Z be the integer such that x ∈ II(x) . For x, y ∈ R and ` ∈ Z, let χ(x, `) = 1 whenever the line segment from x to x + y intersects both I` , I`+1 if y > 0 or both I` , I`−1 if y < 0, otherwise let χ(x, `) be zero. Observe that X χ(x, `) ≤ |y| + ∆x ∆x `∈Z and Z χ(x, `) dx ≤ |y|. R The last estimate is true since R R χ(x, `) dx is the measure of the set {x ∈ R: the line segment from x to x + y intersects both I` , I`+1 if y > 0 or both I` , I`−1 if y < 0} . Equipped with these estimates on χ(x, `) and Lemma 3.9, we calculate as follows: ZZ 2 A(u∆ (x + y, t)) − A(u∆ (x, t)) dt dx ΠT = ∆t X Z 2 A(u∆ (x + y, tn )) − A(u∆ (x, tn )) dx n∈Z+ N R = ∆t X Z A X Z X n UI(x+y) −A n UI(x) !2 dx n∈Z+ N R = ∆t n∈Z+ N R X Z !2 ∆+ A (U`n ) χ(x, `) dx `∈Z 2 1 X ∆+ A (U`n ) χ(x, `) dx ∆x `∈Z `∈Z n∈Z+ N R Z X X 2 1 n ≤ (|y| + ∆x) ∆t ∆+ A Uj+` χ(x, `) dx, ∆x R + ≤ ∆t ∆x X χ(x, l) n∈ZN `∈Z 2 ≤ C11 |y| (|y| + ∆x) , where C11 is the constant in Lemma 3.9. This concludes the proof of the lemma. Lemma 3.11. Assume that the hypotheses of Lemma 3.2 are satisfied. There exists a constant C18 , independent of ∆, such that √ kA(u∆ (·, · + τ )) − A(u∆ (·, ·))kL2 (ΠT −τ ) ≤ C18 τ + ∆t, ∀τ ∈ (0, T ). Proof. From Lemma 3.3 it follows that Z |u∆ (x, t + τ ) − u∆ (x, t)| dx ≤ C3 (τ + ∆t), R 20 KARLSEN, RISEBRO, AND TOWERS for all t, τ such that t, t + τ ∈ (0, T ). Now “interpolating between L1 and L∞ ”, we obtain the desired result: Z T −τ Z (A(u∆ (x, t + τ )) − A(u∆ (x, t)))2 dx dt 0 R ≤ C19 Z 0 T −τ Z |u∆ (x, t + τ ) − u∆ (x, t)| dx dt ≤ C20 (τ + ∆t), R where the constants C19 and C20 do not depend on ∆. Remark 3.2. Again under a stronger CFL condition, it is possible to prove Lemma 3.11 without using L1 Lipschitz continuity in time of the approximate solutions (Lemma 3.3), see [29]. Lemma 3.12. Assume that the hypotheses of Lemma 3.2 are satisfied. There exists a subsequence of A∆ , also denoted by A∆ , and a function A ∈ L2 (0, T ; H 1 (R)) (3.39) such that A∆ → A in L2loc (ΠT ) and a.e. in ΠT . Furthermore, A = A(u) a.e. in ΠT , where u denotes the L∞ weak-∗ limit of u∆ . Proof. For the compactness part of the lemma, one has to use Lemmas 3.10 and 3.11 and repeat the proof of Kolmogorov’s Lp compactness criterion (see Theorem IV.8.21 in [12]). We omit the tedious but straightforward details. In view of Lemma 3.10 and the strong convergence A∆ → A, we clearly have ZZ 2 2 A(x + y, t) − A(x, t) dt dx ≤ C17 |y|2 , ΠT for any y ∈ R. From this it follows that (3.39) holds, and the L2 (0, T ; H 1 (R)) norm of A is bounded by C17 . The proof of the final part of the lemma can be found in [31]. We are now in a position to prove our main theorem. Theorem 3.1. Assume that the hypotheses concerning the data stated in Section 1 are satisfied. Let u∆ be defined by (2.8) and the scheme (2.4), with the parameters λ and µ chosen so that the CFL condition (3.3) holds. Then there exists a weak u of the initial solution value problem (1.1) in the sense of Definition 1.1, and a subsequence of u∆ , also denoted by u∆ , such that u∆ → u in L1loc (ΠT ) and a.e. in ΠT . Proof. Let z ∆ = Ψ(γ ∆ , u∆ ) = F ∆ + A∆ . Both of the sequences F ∆ and A∆ have subsequences converging boundedly a.e. in ΠT , by Lemma 3.8 and 3.12, Lemma ∆ and therefore in ∆ L1loc (ΠT ). By passing to a further subsequence on which both F and A converge, there T is a subsequence, also denoted by z ∆ , such that for some z ∈ L1loc (ΠT ) L∞ (ΠT ), z ∆ → z in L1loc (ΠT ) and a.e. Let u(x, t) = Ψ−1 (γ(x), z(x, t)), which is well-defined a.e. in ΠT , thanks to the fact that Ψ(γ, w) is strictly increasing as a function of w. The immediate goal is to show that we have pointwise convergence of u∆ a.e. in ΠT . Suppressing the dependence on the point (x, t), Ψ(γ, u∆ ) − Ψ(γ, u) ≤ Ψ(γ, u∆ ) − Ψ(γ ∆ , u∆ ) + Ψ(γ ∆ , u∆ ) − Ψ(γ, u) ≤ kΨγ k γ − γ ∆ + z ∆ − z . Thus, since γ ∆ → γ a.e. and z ∆ → z a.e., Ψ(γ, u∆ ) → Ψ(γ, u) a.e. in ΠT . Since Ψ(γ, ·) is strictly increasing, it follows that u∆ → u boundedly a.e., from which convergence in L1loc (ΠT ) follows. Since each u∆ ∈ [0, 1], it is clear that u ∈ L∞ (ΠT ). Also, it is immediate from Lemma 3.12 that A(u) ∈ L2 (0, T ; H 1 (R)). To prove that u ∈ L1 (ΠT ), fix X > 0, and set ΠX T = [−X, X] × [0, T ]. Then ZZ ZZ ZZ ∆ u(x, t) − u∆ (x, t) dt dx + u (x, t) dt dx |u(x, t)| dt dx ≤ ΠX T ΠX T ΠX T DEGENERATE EQUATIONS WITH A DISCONTINUOUS COEFFICIENT ≤ ZZ ΠX T u(x, t) − u∆ (x, t) dt dx + ZZ ΠT 21 ∆ u (x, t) dt dx. After invoking Lemma 3.4, then letting ∆ → 0 and subsequently X → ∞, we get ZZ |u(x, t)| dt dx ≤T C3 T + ku(·, 0)kL1 (R) , ΠT 1 proving that u ∈ L (ΠT ). As a result of the time continuity estimate of Lemma 3.3, and by passing to a further subsequence if necessary, u∆ (·, t) → u(·, t) in L1 (R) for each t ∈ [0, T ] (see, e.g., the proof of Lemma 16.8 of [42]). To show that u ∈ C(0, T ; L1 (R)), let τ > 0, and apply the triangle inequality: ku(·, t + τ ) − u(·, t)kL1 (R) ≤ ku(·, t + τ ) − u∆ (·, t + τ )kL1 (R) + ku∆ (·, t + τ ) − u∆ (·, t)kL1 (R) + ku∆ (·, t) − u(·, t)kL1 (R) . It is a simple consequence of Lemma 3.3 that ku∆ (·, t + τ ) − u∆ (·, t)kL1 (R) ≤ C3 (τ + ∆t). Using this fact, and letting ∆ → 0 in the preceding inequality gives the desired L1 time continuity estimate for the limit function u, proving that u ∈ C(0, T ; L1 (R)). It remains to show that the limit solution u is a weak solution to the initial value problem (1.1), for which a version of the Lax-Wendroff theorem is required. Let φ ∈ D(ΠT ) with φ|t=T = 0. Fix X > 0 such that φ vanishes for |x| ≥ X. Let φnj = φ(xj , tn ). Multiplying the difference scheme (2.4) by φnj ∆x, then summing by parts results in − ∆x ∆t X X Ujn j∈Z n∈Z+ N − ∆x ∆t X φnj − φjn−1 − ∆x Uj0 φ0j ∆t j∈Z X X j∈Z n∈Z+ N n h(γj+ 21 , Uj+1 , Ujn ) X X 1 1 ∆+ φnj + ∆x ∆t A(Ujn ) ∆ ∆ φn = 0, 2 − + j ∆x ∆x + n∈ZN j∈Z where J∆x = X and N = bT /Dtc, and j ∈ {−J, . . . , J}, n ∈ {0, . . . , N }. For (x, t) ∈ Rjn we have that φnj − φn−1 j = φt (x, t) ∆t 1 n ∆+ φj = φx (x, t) + O (∆t + ∆x) , ∆x 1 n ∆ ∆ φ = φ (x, t) − + xx j 2 ∆x and φ0j = φ(x, 0) + O (∆x) . Therefore we have that X X X X X φnj − φn−1 1 j n Ujn ∆x∆t + ∆x∆t A(Ujn ) ∆ ∆ φ + ∆x Uj0 φ0j − + j ∆t ∆x2 + j∈Z + j∈Z j∈Z n∈Z n∈ZN (3.40) Z Z N Z = u∆ φt + A(u∆ )φxx dt dx + u∆ (x, 0)φ(x, 0) dx + O (∆x + ∆t) . ΠT Now defining γj = R 1 2 (γ(xj −) + γ(xj +)) and defining γ̃ ∆ as X γ̃ ∆ (x) = χj (x)γj , j∈Z we compute n n n h(γj+ 21 , Uj+1 , Ujn ) − f (γj , Ujn ) = h(γj+ 12 , Uj+1 , Ujn ) − h(γj , Uj+1 , Ujn ) 22 KARLSEN, RISEBRO, AND TOWERS = Z = Z n + h(γj , Uj+1 , Ujn ) − h(γj , Ujn , Ujn ) n Z Uj+1 γj+ 1 2 n n hγ (w, Uj+1 , Uj ) dw + fu− (γj , w) dw γj γj+ 1 2 Ujn n hγ (w, Uj+1 , Ujn ) dw + ∆u+ h− (γj , Ujn ). γj Consequently, ∆x∆t X X n h(γj+ 12 , Uj+1 , Ujn ) j∈Z n∈Z+ N = ZZ 1 ∆− φnj ∆x f (γ̃ ∆ (x), u∆ )φx (x, t) dt dx + O(∆x) ΠT + ∆xO (T V (γ)) + ∆x∆t X X ∆u+ h− (γj+ 12 , Ujn ) j∈Z n∈Z+ N 1 ∆− φnj . ∆x Now, using (3.38), we find that the last term above can be bounded by √ (3.41) Cφ ∆x, where the constant Cφ depends on φ but not on ∆. Collecting these bounds we find that ZZ Z √ ∆ ∆ ∆ ∆ (3.42) u φt + f (γ̃ (x), u )φx + A(u )φxx dt dx + u∆ (x, 0)φ(x, 0) dx = O ∆x , ΠT R where the O () term on the right depends only on φ. Since A(u) ∈ L2 (0, T ; H 1 (R)) it is possible to integrate by parts in x, and hence ZZ ZZ ZZ A(u∆ )φxx dt dx → A(u)φxx dt dx = − A(u)x φx dt dx. ΠT ΠT ΠT Letting ∆ ↓ 0 in (3.42), we thus find that u is a weak solution, and that we have a “weak convergence rate” of 1/2. Remark 3.3. Note that T V[−X,X] z ∆ (·, t) ≤ C21 , for some constant C21 that is independent of ∆ but dependent on X. Here the R dependence comes from only having an L2 space translation estimate on A(u∆ ). This bound could have been used directly to get strong compactness of z ∆ . 4. Additional regularity In this section we show that A(u), where u is the limit constructed in Theorem 3.1, can be 1 identified a.e. with a C 1, 2 (ΠT ) function. To this end, we will in the next two lemmas obtain ∞ uniform L estimates of the space and time translates of {A(Ujn )}. Lemma 4.1. With the hypotheses of Theorem 3.1, there exists a constant C22 , independent of ∆, such that A(Ujn ) − A(Uin ) ≤ C22 |j − i|∆x, ∀i, j ∈ Z. Proof. Assuming that M is a positive integer such that Ujn = 0 for j ≤ −M +1, from the definition of the scheme (2.4), 1 1 n n n n n n ∆+ A(Uj ) − h(γj+ 12 , Uj+1 , Uj ) ≤ h(γj+ 12 , Uj+1 , Uj ) − ∆+ A(Uj ) ∆x ∆x j X 1 n = ∆− h(γ`+ 12 , U`+1 , U`n ) − ∆+ A(U`n ) ∆x `=−M j 1 X = U`n+1 − U`n λ `=−M DEGENERATE EQUATIONS WITH A DISCONTINUOUS COEFFICIENT ≤ 23 ∆x X n+1 Uj − Ujn ≤ C3 . ∆t j∈Z Since n h(γj+ 21 , Uj+1 , Ujn ) is bounded, ∆+ A(Ujn ) ≤ C22 ∆x, and the lemma follows. Although the above L∞ (ΠT ) space translation estimate was an easy consequence of Lemma 3.3, the L∞ (ΠT ) time translation estimate is a bit trickier to obtain, as shown by the proof of the next lemma. Lemma 4.2. With the hypotheses of Theorem 3.1, there exists a constant C23 , independent of ∆, such that p A(Ujn ) − A(Ujm ) ≤ C23 |n − m|∆t. Proof. The proof is an adaptation of a technique used in [17]. To prove the lemma, we shall need to work with an interpolant of the discrete values {Ujn } that is continuous everywhere and differentiable almost everywhere. For this purpose, define ũn (x) as 1 n ũn (x) = (x − xj−1 ) Ujn + (xj − x) Uj−1 , x ∈ [xj−1 , xj ]. ∆x Then define 1 ũ∆ (x, t) = (t − tn ) ũn+1 (x) + (tn+1 − t) ũn (x) , t ∈ [tn , tn+1 ]. ∆t As before, let I(x) ∈ Z be the integer satisfying x ∈ [xI(x) , xI(x)+1 ), so that I(xj + α) − j =: J for some J ∈ Z. Since ũ∆ is differentiable in time almost everywhere on ΠT , we can proceed as follows: xZ j +α ũ∆ (x, tn ) − ũ∆ (x, tm ) dx xj = xZ j +αZtn xj = ∂t ũ∆ (x, τ ) dτ dx tm I(xj +α)−1 n−1 X X k=j = `=m ZZ ∂t ũ∆ (x, τ ) dτ dx ` Rk−1/2 I(xj +α)−1 n−1 xZk+1 X X k=j = `=m x k I(xj +α)−1 n−1 xZk+1 X X ũ`+1 (x) − ũ` (x) dx k=j = ∆x 2 `=m x k 1 `+1 ` (x − xk ) Uk+1 − Uk+1 + (xk+1 − x) Uk`+1 − Uk` dx ∆x +α)−1 n−1 X I(xjX `=m k=j `+1 ` Uk+1 − Uk+1 + Uk`+1 − Uk` . Hence, using Lemma 3.3, we find that (4.1) xZj +α ∆ ∆ ũ (x, tn ) − ũ (x, tm dx ≤ C3 |n − m| ∆t. xj 24 KARLSEN, RISEBRO, AND TOWERS p Now set α = |n − m|∆t. By the mean value theorem, there exists a number x∗ in [xj , xj + α] such that Z p ∆ ∗ 1 xj +α ∆ ∆ ∗ ∆ (4.2) ũ (x , tn ) − ũ (x , tm ) = ũ (ξ, tn ) − ũ (ξ, tm ) dξ = O |n − m| ∆t , α xj where we have used (4.1). From this we derive the following estimate: E2 = A(ũ∆ (x∗ , tn )) − A(ũ∆ (x∗ , tm )) ≤ kA0 k ũ∆ (x∗ , tn ) − ũ∆ (x∗ , tm ) p (4.3) =O |n − m|∆t . By the triangle inequality, A(Ujn ) − A(Ujm ) = A(ũ∆ (xj , tn )) − A(ũ∆ (xj , tm )) ≤ E1 + E2 + E3 , where E1 = |A(ũ∆ (xj , tn )) − A(ũ∆ (x∗ , tn ))|, E2 = |A(ũ∆ (x∗ , tn )) − A(ũ∆ (x∗ , tm ))|, E3 = |A(ũ∆ (x∗ , tm )) − A(ũ∆ (xj , tm ))|. By Lemma 4.1, E1 + E3 = O (|x∗ − xj |) = O (α) = O |n − m| ∆t , p which finishes the proof of the lemma. Introduce the piecewise bilinear interpolant A∆ (x, t) interpolating the values A(Ujn ) in the same way that ũ∆ interpolates the values Ujn . Then the main theorem of this section can be stated as follows: Theorem 4.1. With the hypotheses of Theorem 3.1, there exists a subsequence of A∆ , also denoted by A∆ , and a function 1 Ā ∈ C 1, 2 (ΠT ) such that A∆ → Ā in L∞ loc (ΠT ). Furthermore, Ā = A(u) a.e. in ΠT , where u is the weak solution constructed in Theorem 3.1. n m Proof. Let (j, n) and (i, m) be integers such that (x, t) ∈ Rj+ . Then 1 and (x + y, t + τ ) ∈ R i+ 12 2 ∆ A (x + y, t + τ ) − A∆ (x, t) ≤ E1 + E2 + E3 , where E1 = A∆ (x + y, t + τ ) − A∆ (xi , tm ) , E2 = A∆ (xi , tm ) − A∆ (xj , tn ) , E3 = A∆ (xj , tn ) − A∆ (x, t) . p n Lemmas 4.1 and 4.2 imply that E2 = O |i − j|∆x + |n − m|∆t . Obviously, for (x, t) ∈ Rj+ 1 2 we have n+1 n min A(Ujn ), A(Uj+1 ), A(Ujn+1 ), A(Uj+1 ) (4.4) n+1 n ≤ A∆ (x, t) ≤ max A(Ujn ), A(Uj+1 ), A(Ujn+1 ), A(Uj+1 ) . √ From this and again Lemmas 4.1 and 4.2, we have E1 + E3 = O ∆x + ∆t . Thus there exists a constant C24 , independent of ∆, such that √ ∆ A (x + y, t + τ ) − A∆ (x, t) ≤ C24 y + τ + ∆x + ∆t . Equipped with this estimate, that we repeat the proofof the Ascoli-Arzela theorem to conclude 1 there is a subsequence of A∆ , still denoted by A∆ , and a limit function Ā ∈ C 1, 2 (ΠT ) such that A∆ → Ā uniformly on compact sets and pointwise on ΠT . DEGENERATE EQUATIONS WITH A DISCONTINUOUS COEFFICIENT 25 Next we show that Ā = A(u) almost everywhere. Without loss of generality, assume for some sequence ∆ → 0 that u∆ → u a.e. and A∆ → Ā pointwise everywhere. Pick an arbitrary but fixed point (x, t) such that u∆ (x, t) → u(x, t). We have A(u(x, t)) − Ā(x, t) ≤ E1 + E2 + E3 , where E1 = A(u(x, t)) − A(u∆ (x, t)) , E2 = A(u∆ (x, t)) − A∆ (x, t) , E3 = A∆ (x, t) − Ā(x, t) . n Obviously, E1 and E3 vanish as ∆ → 0. Let j and n be integers such that (x, t) ∈ Rj+ 1 . Then, 2 √ in view of (4.4) and Lemmas 4.1 and 4.2, E2 = A(Ujn ) − A∆ (x, t) = O ∆x + ∆t , which also tends to zero as ∆ → 0. This concludes the proof of the theorem. 5. Entropy satisfaction Because the diffusion term is strongly degenerate, solutions to the initial value problem (1.1) can develop discontinuities, and so solutions are not a priori unique. In the case where the coefficient γ is continuous, an entropy condition is used to single out the physically relevant solution. In the remainder of this section we establish a cell entropy inequality for solutions of our finite difference algorithm, and then show that limit solutions satisfy a Kružkov-type entropy inequality. This generalizes the corresponding result in [44]. In [32], we use this entropy inequality to establish that there is an L1 contraction principle for weak solutions of the initial value problem (1.1) that are additionally assumed to be piecewise smooth. Lemma 5.1. With the hypotheses of Theorem 3.1, the following cell entropy inequality is satisfied by approximate solutions Ujn generated by the scheme (2.4): (5.1) n V (Ujn+1 ) ≤ V (Ujn ) − ∆− λH(γj+ 21 , Uj+1 , Ujn ) − µ∆+ A(Ujn ) − A(c) + λ ∆+ f (γj− 12 , c) , n where the numerical entropy flux H(γj+ 12 , Uj+1 , Ujn ) is defined by (5.2) n n n H(γj+ 12 , Uj+1 , Ujn ) = f (γj+ 12 , Uj+1 ∨ c, Ujn ∨ c) − f (γj+ 12 , Uj+1 ∧ c, Ujn ∧ c). Proof. The proof is an adaptation of a portion of the proof of Lemma 4.2 of [10]. Let 1 n+1 n n n n n n n ∆+ A(Uj ) , Gj (Uj+1 , Uj , Uj−1 ) = Uj = Uj − λ∆− h(γj+ 21 , Uj+1 , Uj ) − ∆x and observe that Gj (c, c, c) = c − λ∆− f (γj+ 12 , c). The following inequalities are a consequence of monotonicity of the numerical scheme: (5.3) n n n n Gj (c, c, c) ∨ Gj (Uj+1 , Ujn , Uj−1 ) ≤ Gj (c ∨ Uj+1 , c ∨ Ujn , c ∨ Uj−1 ), (5.4) n n n n −Gj (c, c, c) ∧ Gj (Uj+1 , Ujn , Uj−1 ) ≤ −Gj (c ∧ Uj+1 , c ∧ Ujn , c ∧ Uj−1 ). Following [10], (5.3) and (5.4) are added, and the identity a ∨ b − a ∧ b = |a − b| is applied, giving n n Gj (Uj+1 , Ujn , Uj−1 ) − Gj (c, c, c) (5.5) n n n n ≤ Gj (c ∨ Uj+1 , c ∨ Ujn , c ∨ Uj−1 ) − Gj (c ∧ Uj+1 , c ∧ Ujn , c ∧ Uj−1 ). Take the left side of (5.5): n n n n Gj (Uj+1 , Ujn , Uj−1 ) − Gj (c, c, c) = Gj (Uj+1 , Ujn , Uj−1 ) − c + λ∆− f (γj+ 12 , c) (5.6) ≥ Ujn+1 − c − λ ∆− f (γj+ 12 , c) . 26 KARLSEN, RISEBRO, AND TOWERS Now take the right side of (5.5): n n n n Gj (c ∨ Uj+1 , c ∨ Ujn , c ∨ Uj−1 ) − Gj (c ∧ Uj+1 , c ∧ Ujn , c ∧ Uj−1 ) = c ∨ Ujn − c ∧ Ujn n n , c ∨ Ujn ) − h(γj+ 12 , c ∧ Uj+1 , c ∧ Ujn ) − λ∆− h(γj+ 12 , c ∨ Uj+1 1 λ∆− ∆+ (A(Ujn ∨ c) − A(Ujn ∧ c)) ∆x 1 n (5.7) = Ujn − c − λ∆− H(γj+ 12 , Uj+1 , Ujn ) − ∆+ A(Ujn ) − A(c) . ∆x n n The last step in (5.7) used the fact that A(Uj ∨ c) − A(Uj ∧ c) = |A(Ujn ) − A(c)|, which results from the fact that A is nondecreasing. The proof is completed by comparing (5.6) with (5.7). + Theorem 5.1. Suppose that the hypotheses of Theorem 3.1 hold. Let u be a limit point of the sequence u∆ generated by the scheme (2.4) (see Theorem 3.1). The following entropy inequality holds for all nonnegative test functions φ ∈ D0 (ΠT ) such that φ|t=0 = φ|t=T = 0: ZZ |u − c|φt + σ(u − c)(f (γ(x), u) − f (γ(x), c))φx +|A(u) − A(c)|φxx dt dx (5.8) ΠT + ZZ |f (γ(x), c)x |φ dtdx ≥ 0. ΠT Remark 5.1. Note that in the last integral in (5.8), since γ is not continuous, but only of bounded variation, the term |f (γ(x), c)x | must be interpreted as a measure. If we label this measure ν, then for any set E ⊆ R, Z ν(E) = T.V. E (f (γ(·), c)) = |f (γ(x), c)x | dx. E Proof. Let V (u) = |u − c| and F (γ, u) = σ(u − c)(f (γ, u) − f (γ, c)). Following the proof of the Lax-Wendroff theorem, the discrete entropy inequality (5.1) is multiplied by φnj ∆x, then summed over j and n. Here φnj = φ(xj , tn ) and φ is a test function of type described in the statement of the theorem. This yields (5.9) ∆t ∆x X X φnj j∈Z n∈Z+ N V (Ujn+1 ) − V (Ujn ) 1 n + ∆− H(γj+ 12 , Uj+1 , Ujn ) ∆t ∆x (5.10) − X X 1 1 n n ≤ 0. 1 ∆ ∆ A(U ) − A(c) − ∆t∆x φ ∆ f (γ , c) − + + j− 2 j j 2 ∆x ∆x + n∈ZN j∈Z Summing by parts and letting ∆ → 0 gives, by the bounded convergence theorem, X X V (Ujn+1 ) − V (Ujn ) 1 n (5.11) ∆t ∆x φnj + ∆− H(γj+ 21 , Uj+1 , Ujn ) ∆t ∆x j∈Z n∈Z+ N ZZ (5.12) →− V (u)φt + F (γ(x), u)φx dt dx, ΠT as in the proof of the Lax-Wendroff theorem. For the term containing |A(Ujn ) − A(c)|, summing by parts and letting ∆ → 0 gives ZZ X X n 1 n (5.13) ∆t ∆x φj ∆− ∆+ A(Uj ) − A(c) → |A(u) − A(c)|φxx dtdx. ∆x2 + n∈ZN j∈Z ΠT DEGENERATE EQUATIONS WITH A DISCONTINUOUS COEFFICIENT 27 For the remaining term, (5.14) X X ∆t ∆x j∈Z n∈Z+ N → ZZ ΠT \{ξm } φnj 1 ∆x ∆+ f (γj− 12 , c) |f (γ(x), c)x | φ(x, t) dtdx + Z 0 M0 T X m=1 + − f (γ(ξm ), c) − f (γ(ξm ), c) φ (ξm , t) dt, which follows by breaking the spatial portion of the sum into sums over intervals where γ is differentiable and isolating the finite number of cells where the jumps in γ are located. The proof is complete once (5.11), (5.13), and (5.14) are combined. 6. Numerical examples This section discusses some numerical examples for the equation ut + (γ(x)f (u))x = A(u)xx , each using the convective flux f (u) = u(1 − u). We will focus on Riemann problems, with initial data denoted by (uL , uR ), meaning that u0 (x) = uL for x < 0, u0 (x) = uR for x > 0. Similarly the coefficient has a single jump at the origin, which we denote by (γL , γR ). The data satisfy all of the hypotheses of the paper, with the exception that we do not have u0 ∈ L1 (R). This is not a serious violation, since we can imagine that u0 eventually vanishes for large |x| (thus satisfying u0 ∈ L1 (R)), and that the waves generated by this variation in u0 never reach the boundary of our computational domain. By choosing our output time small enough that no waves from either the interior or exterior of the computational domain ever reach the boundary, we can handle the boundary conditions by simply setting the boundary data to their initial values, and keeping them fixed throughout the computation. Example 1. Figure 1 shows the result of two runs of the scheme (2.4) with the Riemann problem having constant initial data (uL , uR ) = (0.6, 0.6), and the coefficient given by (γL , γR ) = (0.05, 0.1). The scheme was run for 500 time steps, with ∆x = .02 and ∆t = .04. In (a), the diffusion term was A(u) = 0, i.e., the purely hyperbolic problem. In (b), the diffusion term was (6.1) A(u) = .0025 uχ[0,.45) (u) + .45χ[.45,.55) (u) + (u − 0.1)χ[.55,1] (u) , which is degenerate (A0 (u) = 0) in the interval (.45, .55), and linear with A0 (u) = 1/400 elsewhere. In (a), the constant state on the left uL = 0.6 is connected to a steady jump at x = 0 by a rarefaction, which is connected to a constant state of approximately u = 0.15. This constant state is connected to uR = 0.6 by a shock moving to the right. All of these waves are created by the jump in the coefficient γ. In (b), the shock moving to the right is smaller, due to the diffusion, and the steady jump at x = 0 has been replaced by a shock moving to the left. Both shocks have end states at approximately u = 0.45 and u = 0.55, providing numerical evidence that discontinuities can only occur in regions of state space where the diffusion A(u) degenerates to a constant. Example 2. The initial data u0 and the coefficient γ are the same as in the previous example. The diffusion term is now given by (6.2) A(u) = .0025(u − 0.3)χ[.3,1] (u), which is degenerate for u < .3, and allows for a stationary jump in the solution u at x = 0. Figure 2 (a) shows the solution for the purely hyperbolic problem, and (b) shows the effect of adding the diffusion term. The scheme was run for 500 time steps, with ∆x = .02 and ∆t = .04, as in the previous example. The diffusion term has the effect of changing the upper end states for both the steady jump and the shock moving to the right, lowering them to approximately u = 0.3, providing more numerical evidence that the solution is continuous wherever A(u) is nondegenerate. 28 KARLSEN, RISEBRO, AND TOWERS (a) (b) 1 1 .............................................. .... ....................... . .55 .45 ......................... 0 −1 0 1 0 ..................................... .......................... . ..... .. ... .. .. . .. ..... ............. −1 0 1 Figure 1. Example 1. (a) Purely hyperbolic problem; A(u) = 0. (b) With degenerate diffusion term (6.1). (a) (b) 1 1 .............................................. .... ....................... . ......................... 0 −1 0 1 ............................ ...................... ........ .... ...... ... .... . ... .. . . .3 ................ 0 −1 0 1 Figure 2. Example 2. (a) Purely hyperbolic problem; A(u) = 0. (b) With the degenerate diffusion term (6.2). Example 3. Figure 3 shows the result of two runs of the scheme (2.4) with the Riemann problem having initial data (uL , uR ) = (0.8, 0.2), and the coefficient given by (γL , γR ) = (0.05, 0.1). Both plots in Figure 3 show the purely hyperbolic case. In (a), ∆x = 0.02, ∆t = 0.04, with 100 time steps. In (b), ∆x = 0.01, ∆t = 0.01, with 400 time steps. Both plots show a spurious bump that starts out as a kink, and then moves to the right at the edge of the rarefaction. Refining the mesh causes the bump to decrease in amplitude and width, as can be seen by comparing Figure 3 (a) with (b). We have found that these spurious bumps turn up in certain (but not all) Riemann problems. As predicted by our convergence theory, they shrink as the mesh size diminishes. For a Riemann problem the discretizations (2.5) and (2.6) result in an intermediate state uM = (uL + uR )/2 and a sharp jump from γL to γR . We have found from experience that the bump can be removed by moving the jump in γ one mesh width in the direction of the bump. All of our convergence theory remains valid with a change of this type, since we still have γ ∆ → γ in L1loc and boundedly a.e. Figure 4 (a) is the same as Figure 3 (a) with the exception that the jump in u0 has been moved one mesh width to the right, with the result that the spurious bump does not appear. Figure 4 (b) is the same as Figure 4 (a), except that the degenerate diffusion term (6.3) A(u) = .0025 uχ[0,.2] (u) + .2χ(.2,1] (u) is incorporated. DEGENERATE EQUATIONS WITH A DISCONTINUOUS COEFFICIENT (a) (b) 1 1 ............................................ .. .. .. ............................................. .. .. . ...................................... ........... 0 29 −1 0 1 ..................................... ............. 0 −1 0 1 Figure 3. Example 3. (a) Purely hyperbolic problem, showing spurious bump; A(u) = 0. (b) Purely hyperbolic problem. Mesh size reduced to show that the spurious bump reduces in width and amplitude. (a) (b) 1 1 ............................................ .. .. .. . ............................................ .. .. .. . ............................................. ... ................................................ 0 −1 0 1 0 −1 0 1 Figure 4. Numerical example 3. (a) Purely hyperbolic problem, with jump in γ moved by one mesh width to the right to get rid of spurious bump. (b) With degenerate diffusion term (6.3). The effect of the diffusion is to smear out the corners where the rarefaction meets the constant states. Although not shown in this plot, when the mesh size is reduced sufficiently, the small jump between the minimum point on the graph and u = .2 fills in, so that the solution is continuous in the region where A0 (u) > 0. Example 4. In this example we study the convergence rate for the problem in the previous example, with the diffusion term included. 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Box 1053, Blindern N–0316 Oslo, Norway E-mail address: [email protected] URL: www.math.uio.no/~nilshr (Towers) MiraCosta College 3333 Manchester Avenue Cardiff-by-the-Sea, CA 92007-1516, USA E-mail address: [email protected]

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