Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-12T19:45:36.592Z Has data issue: false hasContentIssue false

Convergence of finite difference schemes for viscousand inviscid conservation laws with rough coefficients

Published online by Cambridge University Press:  15 April 2002

Kenneth Hvistendahl Karlsen
Affiliation:
Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, 5008 Bergen, Norway. ([email protected]); URL:
Nils Henrik Risebro
Affiliation:
Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway. ([email protected]); URL:
Get access

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 convergeto 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 Lp compactness criterion.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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).
Bouchut, F., Guarguaglini, F.R. and Natalini, R., Diffusive BGK approximations for nonlinear multidimensional parabolic equations. Indiana Univ. Math. J. 49 (2000) 723-749. CrossRef
Bürger, R., Evje, S. and Karlsen, K.H., On strongly degenerate convection-diffusion problems modeling sedimentation-consolidation processes. J. Math. Anal. Appl. 247 (2000) 517-556. CrossRef
M.C. Bustos, F. Concha, R. Bürger and E.M. Tory, Sedimentation and thickening: Phenomenological foundation and mathematical theory. Kluwer Academic Publishers, Dordrecht (1999).
Carrillo, J., Entropy solutions for nonlinear degenerate problems. Arch. Rational Mech. Anal. 147 (1999) 269-361. CrossRef
Chainais-Hillairet, C., 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. CrossRef
Champier, S., Gallouët, T. and Herbin, R., Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh. Numer. Math. 66 (1993) 139-157. CrossRef
Cockburn, B., Coquel, F. and Le Floch, P., An error estimate for finite volume methods for multidimensional conservation laws. Math. Comp. 63 (1994) 77-103. CrossRef
Cockburn, B., Coquel, F. and LeFloch, P.G., Convergence of the finite volume method for multidimensional conservation laws. SIAM J. Numer. Anal. 32 (1995) 687-705. CrossRef
Cockburn, B. and Gremaud, P.-A., A priori error estimates for numerical methods for scalar conservation laws. I. The general approach. Math. Comp. 65 (1996) 533-573. CrossRef
Cockburn, B. and Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440-2463 (electronic). CrossRef
Crandall, M.G. and Majda, A., Monotone difference approximations for scalar conservation laws. Math. Comp. 34 (1980) 1-21. CrossRef
Crandall, M.G. and Tartar, L., Some relations between nonexpansive and order preserving mappings. Proc. Amer. Math. Soc. 78 (1980) 385-390. CrossRef
Engquist, B. and Osher, S., One-sided difference approximations for nonlinear conservation laws. Math. Comp. 36 (1981) 321-351. CrossRef
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.
Evje, S. and Karlsen, K.H., Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations. Numer. Math. 86 (2000) 377-417. CrossRef
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.
Evje, S. and Karlsen, K.H., Viscous splitting approximation of mixed hyperbolic-parabolic convection-diffusion equations. Numer. Math. 83 (1999) 107-137. CrossRef
Evje, S. and Karlsen, K.H., Monotone difference approximations of BV solutions to degenerate convection-diffusion equations. SIAM J. Numer. Anal. 37 (2000) 1838-1860 (electronic). CrossRef
S. Evje and K.H. Karlsen, Second order difference schemes for degenerate convection-diffusion equations. Preprint (in preparation).
Eymard, R., Gallouët, T., Ghilani, M. and Herbin, R., Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes. IMA J. Numer. Anal. 18 (1998) 563-594. CrossRef
Eymard, R., Gallouët, T., Hilhorst, D. and Naït Slimane, Y., Finite volumes and nonlinear diffusion equations. RAIRO-Modél. Math. Anal. Numér. 32 (1998) 747-761. CrossRef
Gimse, T. and Risebro, N.H., Solution of the Cauchy problem for a conservation law with a discontinuous flux function. SIAM J. Math. Anal. 23 (1992) 635-648. CrossRef
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.
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).
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).
Isaacson, E. and Temple, B., Convergence of the 2 x 2 Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math. 55 (1995) 625-640. CrossRef
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).
C. Klingenberg and N.H. Risebro, Stability of a resonant system of conservation laws modeling polymer flow with gravitation. J. Differential Equations March (2000).
Klingenberg, C. and Risebro, N.H., Convex conservation laws with discontinuous coefficients. Existence, uniqueness and asymptotic behavior. Comm. Partial Differential Equations 20 (1995) 1959-1990. CrossRef
Kröner, D., Noelle, S. and Rokyta, M., 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.
Kröner, D. and Rokyta, M., Convergence of upwind finite volume schemes for scalar conservation laws in two dimensions. SIAM J. Numer. Anal. 31 (1994) 324-343. CrossRef
Kruzkov, S.N., Results on the nature of the continuity of solutions of parabolic equations, and certain applications thereof. Mat. Zametki 6 (1969) 97-108.
Kruzkov, S.N., First order quasi-linear equations in several independent variables. Math. USSR Sbornik 10 (1970) 217-243. CrossRef
Kurganov, A. and Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 241-282. CrossRef
Kuznetsov, N.N., 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. CrossRef
Lucier, B.J., Error bounds for the methods of Glimm, Godunov and LeVeque. SIAM J. Numer. Anal. 22 (1985) 1074-1081. CrossRef
Noelle, S., Convergence of higher order finite volume schemes on irregular grids. Adv. Comput. Math. 3 (1995) 197-218. CrossRef
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).
Oleĭnik, O.A., Discontinuous solutions of non-linear differential equations. Amer. Math. Soc Transl. Ser. 2 26 (1963) 95-172.
Osher, S. and Tadmor, E., On the convergence of difference approximations to scalar conservation laws. Math. Comp. 50 (1988) 19-51. CrossRef
Rouvre, É. and Gagneux, G., Solution forte entropique de lois scalaires hyperboliques-paraboliques dégénérées. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 599-602. CrossRef
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.
Sanders, R., On convergence of monotone finite difference schemes with variable spatial differencing. Math. Comp. 40 (1983) 91-106. CrossRef
Temple, B., Global solution of the Cauchy problem for a class of 2 x 2 nonstrictly hyperbolic conservation laws. Adv. in Appl. Math. 3 (1982) 335-375. CrossRef
J. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux. Preprint, Available at the URL http://www.math.ntnu.no/conservation/
J. Towers, A difference scheme for conservation laws with a discontinuous flux - the nonconvex case. Preprint, Available at the URL http://www.math.ntnu.no/conservation/
Convergence, J.-P. Vila 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.
Vol'pert, A.I., The spaces BV and quasi-linear equations. Math. USSR Sbornik 2 (1967) 225-267. CrossRef
Vol'pert, A.I. and Hudjaev, S.I., Cauchy's problem for degenerate second order quasilinear parabolic equations. Math. USSR Sbornik 7 (1969) 365-387. CrossRef