# 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

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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 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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