Efficient well-balanced hydrostatic upwind schemes for shallow-water equations

The proposed work concerns the numerical approximations of the shallow-water equations with varying topography. The main objective is to introduce an easy and systematic technique to enforce the well-balance property and to make the scheme able to deal with dry areas. To access such an issue, the derived numerical method is obtained by involving the free surface instead of the water height and this produces the scheme well-balanced independently from the numerical flux function associated with the homogeneous problem. As a consequence, we obtain an easy well-balanced scheme which preserves non-negative water height. When compared with the well-known hydrostatic reconstruction, the presented topography discretization does not involve any max function known to introduce some numerical errors as soon as the topography admits very strong variations or discontinuities. A second-order MUSCL accurate reconstruction is adopted. The proposed hydrostatic upwind scheme is next extended for considering 2D simulations performed over unstructured meshes. Several 1D and 2D numerical experiments are performed to exhibit the relevance of the scheme.