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Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients. (English) Zbl 1032.76048
Summary: 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.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
76N15 Gas dynamics, general
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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