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

Kenneth Hvistendahl Karlsen; Nils Henrik Risebro

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

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topKarlsen, 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 - Modélisation Mathématique et Analyse Numérique 35.2 (2001): 239-269. <http://eudml.org/doc/194049>.

@article{Karlsen2001,

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, $k^\{\prime \}$ 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.},

author = {Karlsen, Kenneth Hvistendahl, Risebro, Nils Henrik},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

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},

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/194049},

volume = {35},

year = {2001},

}

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 - Modélisation Mathématique et Analyse Numérique

PY - 2001

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, $k^{\prime }$ 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.

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/194049

ER -

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