On the well-balanced numerical discretization of shallow water equations on unstructured meshes

We consider in this work a finite volume numerical approximation of weak solutions of the shallow water equations with varying topography, on unstructured meshes. Relying on an alternative formulation of the shallow water equations that involves the free surface as a conservative variable, instead of the water height, we introduce a simple discretization of the bed slope source term, together with some suitable conservative variables reconstructions. The resulting scheme is automatically consistent and well-balanced, for any given consistent numerical flux for the homogeneous system. We obtain a very simple formulation, which do not need to be modified when second order accuracy MUSCL reconstructions are adopted. Additionally, the positivity of the water height is preserved under a relevant stability condition, as soon as the numerical flux for the associated homogeneous system does. Numerical assessments, involving dry areas and complex geometry are performed.