ESAIM: Mathematical Modelling and Numerical Analysis

Research Article

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

Karlsen, Kenneth Hvistendahla1 and Risebro, Nils Henrika2

a1 Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, 5008 Bergen, Norway. (kennethk@math.uib.no); URL:

a2 Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway. (nilshr@math.uio.no); URL:

Abstract

We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a "rough"coefficient function k(x). We show that the Engquist-Osher (and hence all monotone) finite difference approximations converge to the unique entropy solution of the governing equation if, among other demands, 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.

(Received September 6 2000)

(Online publication April 15 2002)

Key Words:

  • Conservation law;
  • degenerate convection-diffusion equation;
  • entropy solution;
  • finite difference scheme;
  • convergence;
  • error estimate.

Mathematics Subject Classification:

  • 65M06;
  • 35L65;
  • 35L45;
  • 35K65
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