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

Published online by Cambridge University Press:  15 April 2002

Kenneth Hvistendahl Karlsen  and
Nils Henrik Risebro
Kenneth Hvistendahl Karlsen
Affiliation:
Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, 5008 Bergen, Norway. (kennethk@math.uib.no); URL:
Nils Henrik Risebro
Affiliation:
Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway. (nilshr@math.uio.no); URL:
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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.

Keywords

Conservation lawdegenerate convection-diffusion equationentropy solutionfinite difference schemeconvergenceerror estimate.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 2 , March 2001 , pp. 239 - 269
Copyright
© EDP Sciences, SMAI, 2001

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