A simple well-balanced and positive numerical scheme for the shallow-water system

Abstract

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 entropypreserving 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.