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

Kenneth Hvistendahl Karlsen; Nils Henrik Risebro

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 35, Issue: 2, page 239-269
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Karlsen, Kenneth Hvistendahl, and Risebro, Nils Henrik. "Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients." ESAIM: Mathematical Modelling and Numerical Analysis 35.2 (2010): 239-269. <http://eudml.org/doc/197398>.

@article{Karlsen2010,
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. },
author = {Karlsen, Kenneth Hvistendahl, Risebro, Nils Henrik},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Conservation law; degenerate convection-diffusion equation; entropy solution; finite difference scheme; convergence; error estimate.; initial value problem; degenerate scalar conservation laws; entropy solutions; Engquist-Osher finite difference approximations; flux function; degenerate convection-diffusion equations; rate of convergence; a priori estimates; compactness criterion},
language = {eng},
month = {3},
number = {2},
pages = {239-269},
publisher = {EDP Sciences},
title = {Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients},
url = {http://eudml.org/doc/197398},
volume = {35},
year = {2010},
}

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: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 2
SP - 239
EP - 269
AB - 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.
LA - eng
KW - Conservation law; degenerate convection-diffusion equation; entropy solution; finite difference scheme; convergence; error estimate.; initial value problem; degenerate scalar conservation laws; entropy solutions; Engquist-Osher finite difference approximations; flux function; degenerate convection-diffusion equations; rate of convergence; a priori estimates; compactness criterion
UR - http://eudml.org/doc/197398
ER -

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