A robust high order VFRoe scheme for shallow water equations

The VFRoe scheme has been recently introduced to approximate the solutions of the shallow water equations. It performs approximate solutions in a good agreement with the expected ones. However, the robustness of this numerical procedure has not been proposed. To achieve this, a relevant relaxation method is introduced. This new relaxation scheme is shown to coincide with the VFRoe scheme. Then, the robustness of the relaxation scheme is established and the non-negativity of the water height is ensured. A robust high order MUSCL extension is also proposed. Finally, we consider the recent hydrostatic reconstruction to approximate the topography source terms. Numerical experiments are performed to attest the interest of the procedure.