A 2d Well-balanced Positivity Preserving Second Order Scheme for Shallow Water Flows on Unstructured Meshes
Résumé
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 this 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 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.