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Upwind finite volume schemes for weakly coupled hyperbolic systems of conservation laws in 2D. (English) Zbl 0918.35086

A weakly coupled system of nonlinear hyperbolic conservation laws in two space dimensions is considered. The coupling of the equations is realized only due to source terms. Systems of this type are widely used in a modelling of combustion processes, hydrological problems, relaxation schemes or mathematical biology. For numerical treatment of the above problem the author derives a class of explicit and implicit upwind finite volume methods on unstructed grids. The convergence of these numerical schemes is proved under the assumption of the existence of the unique exact entropy solution. The proof relies on vanishing viscosity methods and on an extension of DiPerna’s results concerning measure valued solutions to the case of weakly coupled hyperbolic systems.

MSC:

35L60 First-order nonlinear hyperbolic equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
76N15 Gas dynamics (general theory)
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