A SIMPLE WELL-BALANCED AND POSITIVE NUMERICAL SCHEME FOR THE SHALLOW-WATER SYSTEM.

Emmanuel Audusse (1, 2) , Christophe Chalons (3) , Philippe Ung (1, 4)

1 ANGE - Numerical Analysis, Geophysics and Ecology
2 LAGA - Laboratoire Analyse, Géométrie et Applications
3 LMV - Laboratoire de Mathématiques de Versailles
4 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

This work considers the numerical approximation of the shallow-water equations. In this context, one faces three important issues related to the well-balanced, positivity and entropy-preserving properties, as well as the ability to consider vacuum states. We propose a Godunov-type method based on the design of a three-wave Approximate Riemann Solver (ARS) which satisfies the first two properties and a weak form of the last one together. Regarding the entropy, the solver satisfies a discrete non-conservative entropy inequality. From a numerical point of view, we also investigate the validity of a conservative entropy inequality.

Mots clés: shallow-water equations; approximate Riemann solver; finite volume method; positivity preserving; well-balanced scheme

Domaines: Mathématiques [math]; Analyse numérique [math.NA]

Emmanuel Audusse, Christophe Chalons, Philippe Ung. A SIMPLE WELL-BALANCED AND POSITIVE NUMERICAL SCHEME FOR THE SHALLOW-WATER SYSTEM.. 2014. ⟨hal-01083364v1⟩