Some approximate Godunov schemes to compute shallow-water equations with topography

We study here the computation of shallow-water equations with topography by Finite Volume methods, in a one-dimensional framework (though all methods introduced may be naturally extended in two dimensions). All methods are based on a discretisation of the topography by a piecewise function constant on each cell of the mesh, from an original idea of Le Roux et al. Whereas the Well-Balanced scheme of Le Roux is based on the exact resolution of each Riemann problem, we consider here approximate Riemann solvers. Several single step methods are derived from this formalism, and numerical results are compared to a fractional step method. Some test cases are presented: convergence towards steady states in subcritical and supercritical configurations, occurrence of dry area by a drain over a bump and occurrence of vacuum by a double rarefaction wave over a step. Numerical schemes, combined with an appropriate high-order extension, provide accurate and convergent approximations.