Models and methods for two-layer shallow water flows

Two-layer shallow water models present at least two fundamental difficulties that are addressed in the present contribution. The first one is related to the lack of hyperbolicity of most existing models. By considering weak compressibility of the phases, a strictly hyperbolic formulation with pressure relaxation is obtained. It is shown to tend to the conventional two-layer model in the stiff pressure relaxation limit. The second issue is related to the non-conservative terms in the momentum equations. Analyzing the Riemann problem structure, local constants appear precisely at locations where the non-conservative products need definition. Thanks to these local constants, a locally conservative formulation of the equations is obtained, simplifying the Riemann problem resolution through a HLL-type Riemann solver. The method is compared to literature data, showing accurate and oscillation free solutions. Additional numerical experiments show robustness and accuracy of the method.