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

Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik

Geodesic

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

Karlsen, Kenneth Hvistendahl ; Risebro, Nils Henrik

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 35 (2001) no. 2, pp. 239-269.

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Résumé

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 $B V$ , 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.

MR Zbl

Classification : 65M06, 35L65, 35L45, 35K65
Keywords: conservation law, degenerate convection-diffusion equation, entropy solution, finite difference scheme, convergence, error estimate

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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 , Tome 35 (2001) no. 2, pp. 239-269. http://geodesic.mathdoc.fr/item/M2AN_2001__35_2_239_0/

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