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, Volume 35 (2001) no. 2, p. 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 $k\left(x\right)$. 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, ${k}^{\text{'}}$ 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
Keywords: conservation law, degenerate convection-diffusion equation, entropy solution, finite difference scheme, convergence, error estimate
@article{M2AN_2001__35_2_239_0,
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},
publisher = {EDP-Sciences},
volume = {35},
number = {2},
year = {2001},
pages = {239-269},
zbl = {1032.76048},
mrnumber = {1825698},
language = {en},
url = {http://www.numdam.org/item/M2AN_2001__35_2_239_0}
}

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, Volume 35 (2001) no. 2, pp. 239-269. http://www.numdam.org/item/M2AN_2001__35_2_239_0/

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