Geodesic
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

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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 L p compactness criterion.

Classification : 65M06, 35L65, 35L45, 35K65
Keywords: conservation law, degenerate convection-diffusion equation, entropy solution, finite difference scheme, convergence, error estimate 
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     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: Mathematical Modelling and Numerical Analysis },
     pages = {239--269},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {2},
     year = {2001},
     mrnumber = {1825698},
     zbl = {1032.76048},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/M2AN_2001__35_2_239_0/}
}
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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/