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 . 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, is in , 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 compactness criterion.
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: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {239--269},
year = {2001},
publisher = {EDP Sciences},
volume = {35},
number = {2},
mrnumber = {1825698},
zbl = {1032.76048},
language = {en},
url = {https://www.numdam.org/item/M2AN_2001__35_2_239_0/}
}
TY - JOUR AU - Karlsen, Kenneth Hvistendahl AU - Risebro, Nils Henrik TI - Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2001 SP - 239 EP - 269 VL - 35 IS - 2 PB - EDP Sciences UR - https://www.numdam.org/item/M2AN_2001__35_2_239_0/ LA - en ID - M2AN_2001__35_2_239_0 ER -
%0 Journal Article %A Karlsen, Kenneth Hvistendahl %A Risebro, Nils Henrik %T Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients %J ESAIM: Modélisation mathématique et analyse numérique %D 2001 %P 239-269 %V 35 %N 2 %I EDP Sciences %U https://www.numdam.org/item/M2AN_2001__35_2_239_0/ %G en %F 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: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 2, pp. 239-269. https://www.numdam.org/item/M2AN_2001__35_2_239_0/
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