A Positive Preserving High Order VFRoe Scheme for Shallow Water Equations: A Class of Relaxation Schemes

The VFRoe scheme has been recently introduced to approximate the solutions of the shallow water equations. One of the main interest of this method is to be easily implemented. As a consequence, such a scheme appears as an interesting alternative to other more sophisticated schemes. The VFRoe methods perform approximate solutions in a good agreement with the expected ones. However, the robustness of this numerical procedure has not been proposed. Following the ideas introduced by Jin-Xin [Comm. Pure Appl. Math., 45, 235–276 (1995)], a relevant relaxation method is derived. The interest of this relaxation scheme is twofold. In the first hand, the relaxation scheme is shown to coincide with the considered VFRoe scheme. In the second hand, the robustness of the relaxation scheme is established and thus the non-negativity of the water height, obtained involving the VFRoe approach, is ensured. Following the same idea, a family of relaxation schemes is exhibited. Next, robust high order MUSCL extensions are proposed. The final scheme is obtained when considering the hydrostatic reconstruction to approximate the topography source terms. Numerical experiments are performed to attest the interest of the procedure.