Numerical treatment of the loss of hyperbolicity of the two-layer shallow-water system

Abstract

In this work, a characterization of the hyperbolicity region for the two layer shallow-water system is proposed and checked. Next, some path-conservative finite volume schemes (see [11]) that can be used even if the system is not hyperbolic are presented, but they are not in general L2 linearly stable in that case. Then, we introduce a simple but efficient strategy to enforce the hyperbolicity of the two-layer shallow-water system consisting in adding to the system an extra amount of friction at every cell in which complex eigenvalues are detected at a given time step. The implementation is performed by a predictor/corrector strategy: first a numerical scheme is applied to the unmodified two-layer system, regardless of the hyperbolic character of the system. Next, we check if the predicted cell averages are in the hyperbolic region or not. If not, the mass-fluxes are corrected by adding a quadratic friction law between layers whose coefficient is computed so that the corrected cell average is as near as possible of the boundary of the hyperbolicity region. Finally, some numerical test have been performed to assess the efficiency of the proposed strategy