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A well-balanced positivity preserving “second-order” scheme for shallow water flows on unstructured meshes. (English) Zbl 1087.76072
Summary: We consider the solution of the Saint-Venant equations with topographic source terms on 2D unstructured meshes by a finite volume approach. We first present a stable and positivity preserving homogeneous solver issued from a kinetic representation of the hyperbolic conservation laws system. This water depth positivity property is important when dealing with wet-dry interfaces. Then, we introduce a local hydrostatic reconstruction that preserves the positivity properties of the homogeneous solver and leads to a well-balanced scheme satisfying the steady-state condition of still water. Finally, a formally second-order extension based on limited reconstructed values on both sides of each interface and on an enriched interpretation of the source terms satisfies the same properties and gives a noticeable accuracy improvement. Numerical examples on academic and real problems are presented.

76M12 Finite volume methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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