An all-regime and well-balanced Lagrange-projection type scheme for the shallow water equations on unstructured meshes

Abstract

In this work, we focus on the numerical approximation of the shallow water equations in two space dimensions. Our aim is to propose a well-balanced, all-regime and positive scheme. By well-balanced, it is meant that the scheme is able to preserve the so-called lake at rest smooth equilibrium solutions. By all-regime, we mean that the scheme is able to deal with all flow regimes, including the low-Froude regime which is known to be challenging when using usual Godunov-type finite volume schemes. At last, the scheme should be positive which means that the water height stays positive for all time. Our approach is based on a Lagrange-projection decomposition which allows to naturally decouple the acoustic and transport terms. Numerical experiments on unstructured meshes illustrate the good behaviour of the scheme.