Well-balanced schemes to capture non-explicit steady states: Ripa model

The present paper concerns the derivation of numerical schemes to approximate the weak solutions of the Ripa model, which is an extension of the shallow-water model where a gradient of temperature is considered. Here, the main motivation lies in the exact capture of the steady states involved in the model. Because of the temperature gradient, the steady states at rest, of prime importance from the physical point of view, turn out to be very nonlinear and their exact capture by a numerical scheme is very challenging. We propose a relaxation technique to derive the required scheme. In fact, we exhibit an approximate Riemann solver that satisfies all the needed properties (robustness and well-balancing). We show three relaxation strategies to get a suitable interpretation of this adopted approximate Riemann solver. The resulting relaxation scheme is proved to be positive preserving, entropy satisfying and to exactly capture the nonlinear steady states at rest. Several numerical experiments illustrate the relevance of the method.