Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001) no. 2, pp. 239-269.

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, k ' is in BV, thereby providing alternative (new) existence proofs for entropy solutions of degenerate convection-diffusion 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 L p compactness criterion.

Classification : 65M06,  35L65,  35L45,  35K65
Mots clés : conservation law, degenerate convection-diffusion equation, entropy solution, finite difference scheme, convergence, error estimate
     author = {Karlsen, Kenneth Hvistendahl and Risebro, Nils Henrik},
     title = {Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     pages = {239--269},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {2},
     year = {2001},
     zbl = {1032.76048},
     mrnumber = {1825698},
     language = {en},
     url = {}
Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik. Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001) no. 2, pp. 239-269.

[1] M. Afif and B. Amaziane, Convergence of finite volume schemes for a degenerate convection-diffusion equation arising in two-phase flow in porous media. Preprint (1999). | MR 2062160 | Zbl 1012.76057

[2] F. Bouchut, F.R. Guarguaglini and R. Natalini, Diffusive BGK approximations for nonlinear multidimensional parabolic equations. Indiana Univ. Math. J. 49 (2000) 723-749. | Zbl 0964.35011

[3] R. Bürger, S. Evje and K.H. Karlsen, On strongly degenerate convection-diffusion problems modeling sedimentation-consolidation processes. J. Math. Anal. Appl. 247 (2000) 517-556. | Zbl 0961.35078

[4] M.C. Bustos, F. Concha, R. Bürger and E.M. Tory, Sedimentation and thickening: Phenomenological foundation and mathematical theory. Kluwer Academic Publishers, Dordrecht (1999). | MR 1747460 | Zbl 0936.76001

[5] J. Carrillo, Entropy solutions for nonlinear degenerate problems. Arch. Rational Mech. Anal. 147 (1999) 269-361. | Zbl 0935.35056

[6] C. Chainais-Hillairet, Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate. RAIRO-Modél. Math. Anal. Numér. 33 (1999) 129-156. | Numdam | Zbl 0921.65071

[7] S. Champier, T. Gallouët and R. Herbin, Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh. Numer. Math. 66 (1993) 139-157. | EuDML 133756 | Zbl 0801.65089

[8] B. Cockburn, F. Coquel and P. Le Floch, An error estimate for finite volume methods for multidimensional conservation laws. Math. Comp. 63 (1994) 77-103. | Zbl 0855.65103

[9] B. Cockburn, F. Coquel and P.G. Lefloch, Convergence of the finite volume method for multidimensional conservation laws. SIAM J. Numer. Anal. 32 (1995) 687-705. | Zbl 0845.65051

[10] B. Cockburn and P.-A. Gremaud, A priori error estimates for numerical methods for scalar conservation laws. I. The general approach. Math. Comp. 65 (1996) 533-573. | Zbl 0848.65067

[11] B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440-2463 (electronic). | Zbl 0927.65118

[12] M.G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws. Math. Comp. 34 (1980) 1-21. | Zbl 0423.65052

[13] M.G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings. Proc. Amer. Math. Soc. 78 (1980) 385-390. | Zbl 0449.47059

[14] B. Engquist and S. Osher, One-sided difference approximations for nonlinear conservation laws. Math. Comp. 36 (1981) 321-351. | Zbl 0469.65067

[15] M.S. Espedal and K.H. Karlsen, Numerical solution of reservoir flow models based on large time step operator splitting algorithms, in Filtration in Porous media and industrial applications. Lect. Notes Math. 1734, Springer, Berlin (2000) 9-77. | Zbl 1077.76546

[16] S. Evje and K.H. Karlsen, Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations. Numer. Math. 86 (2000) 377-417. | Zbl 0963.65094

[17] S. Evje and K.H. Karlsen, Degenerate convection-diffusion equations and implicit monotone difference schemes, in Hyperbolic problems: Theory, numerics, applications, Vol. I (Zürich, 1998). Birkhäuser, Basel (1999) 285-294. | Zbl 0931.65094

[18] S. Evje and K.H. Karlsen, Viscous splitting approximation of mixed hyperbolic-parabolic convection-diffusion equations. Numer. Math. 83 (1999) 107-137. | Zbl 0961.65084

[19] S. Evje and K.H. Karlsen, Monotone difference approximations of BV solutions to degenerate convection-diffusion equations. SIAM J. Numer. Anal. 37 (2000) 1838-1860 (electronic). | Zbl 0985.65100

[20] S. Evje and K.H. Karlsen, Second order difference schemes for degenerate convection-diffusion equations. Preprint (in preparation).

[21] R. Eymard, T. Gallouët, M. Ghilani and R. Herbin, Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes. IMA J. Numer. Anal. 18 (1998) 563-594. | Zbl 0973.65078

[22] R. Eymard, T. Gallouët, D. Hilhorst and Y. Naït Slimane, Finite volumes and nonlinear diffusion equations. RAIRO-Modél. Math. Anal. Numér. 32 (1998) 747-761. | Numdam | Zbl 0914.65101

[23] T. Gimse and N.H. Risebro, Solution of the Cauchy problem for a conservation law with a discontinuous flux function. SIAM J. Math. Anal. 23 (1992) 635-648. | Zbl 0776.35034

[24] A. Harten, J.M. Hyman and P.D. Lax, On finite-difference approximations and entropy conditions for shocks. Comm. Pure Appl. Math. XXIX (1976) 297-322. | Zbl 0351.76070

[25] H. Holden, K.H. Karlsen and K.-A. Lie, Operator splitting methods for degenerate convection-diffusion equations I: Convergence and entropy estimates, in Stochastic processes, physics and geometry: New interplays. A volume in honor of Sergio Albeverio. Amer. Math. Soc. (to appear). | MR 1803424 | Zbl 0974.35065

[26] H. Holden, K.H. Karlsen, K.-A. Lie and N.H. Risebro, Operator splitting for nonlinear partial differential equations: An L 1 convergence theory. Preprint (in preparation).

[27] E. Isaacson and B. Temple, Convergence of the 2×2 Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math. 55 (1995) 625-640. | Zbl 0838.35075

[28] K.H. Karlsen and N.H. Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Preprint, Department of Mathematics, University of Bergen (2000). | MR 1974417 | Zbl 1027.35057

[29] C. Klingenberg and N.H. Risebro, Stability of a resonant system of conservation laws modeling polymer flow with gravitation. J. Differential Equations March (2000). | MR 1815188 | Zbl 0977.35083

[30] C. Klingenberg and N.H. Risebro, Convex conservation laws with discontinuous coefficients. Existence, uniqueness and asymptotic behavior. Comm. Partial Differential Equations 20 (1995) 1959-1990. | Zbl 0836.35090

[31] D. Kröner, S. Noelle and M. Rokyta, Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions. Numer. Math. 71 (1995) 527-560. | Zbl 0841.65079

[32] D. Kröner and M. Rokyta, Convergence of upwind finite volume schemes for scalar conservation laws in two dimensions. SIAM J. Numer. Anal. 31 (1994) 324-343. | Zbl 0856.65104

[33] S.N. Kružkov, Results on the nature of the continuity of solutions of parabolic equations, and certain applications thereof. Mat. Zametki 6 (1969) 97-108. | Zbl 0189.10602

[34] S.N. Kružkov, First order quasi-linear equations in several independent variables. Math. USSR Sbornik 10 (1970) 217-243.

[35] A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 241-282. | Zbl 0987.65085

[36] N.N. Kuznetsov, Accuracy of some approximative methods for computing the weak solutions of a first-order quasi-linear equation. USSR Comput. Math. Math. Phys. Dokl. 16 (1976) 105-119. | Zbl 0381.35015

[37] B.J. Lucier, Error bounds for the methods of Glimm, Godunov and LeVeque. SIAM J. Numer. Anal. 22 (1985) 1074-1081. | Zbl 0584.65059

[38] S. Noelle, Convergence of higher order finite volume schemes on irregular grids. Adv. Comput. Math. 3 (1995) 197-218. | Zbl 0834.65088

[39] M. Ohlberger, A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations. Preprint, Mathematische Fakultät, Albert-Ludwigs-Universität Freiburg (2000). | MR 1825703

[40] O.A. Oleĭnik, Discontinuous solutions of non-linear differential equations. Amer. Math. Soc Transl. Ser. 2 26 (1963) 95-172. | Zbl 0131.31803

[41] S. Osher and E. Tadmor, On the convergence of difference approximations to scalar conservation laws. Math. Comp. 50 (1988) 19-51. | Zbl 0637.65091

[42] É. Rouvre and G. Gagneux, Solution forte entropique de lois scalaires hyperboliques-paraboliques dégénérées. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 599-602. | Zbl 0935.35085

[43] A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov and A.P. Mikhailov, Blow-up in quasilinear parabolic equations. Walter de Gruyter & Co., Berlin (1995). Translated from the 1987 Russian original by Michael Grinfeld and revised by the authors. | MR 1330922 | Zbl 1020.35001

[44] R. Sanders, On convergence of monotone finite difference schemes with variable spatial differencing. Math. Comp. 40 (1983) 91-106. | Zbl 0533.65061

[45] B. Temple, Global solution of the Cauchy problem for a class of 2×2 nonstrictly hyperbolic conservation laws. Adv. in Appl. Math. 3 (1982) 335-375. | Zbl 0508.76107

[46] J. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux. Preprint, Available at the URL | MR 1770068 | Zbl 0972.65060

[47] J. Towers, A difference scheme for conservation laws with a discontinuous flux - the nonconvex case. Preprint, Available at the URL | MR 1870839 | Zbl 1055.65104

[48] J.-P. Vila, Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws. I. Explicit monotone schemes. RAIRO-Modél. Math. Anal. Numér. 28 (1994) 267-295. | Numdam | Zbl 0823.65087

[49] A.I. Vol'Pert, The spaces BV and quasi-linear equations. Math. USSR Sbornik 2 (1967) 225-267. | Zbl 0168.07402

[50] A.I. Vol'Pert and S.I. Hudjaev, Cauchy's problem for degenerate second order quasilinear parabolic equations. Math. USSR Sbornik 7 (1969) 365-387. | Zbl 0191.11603