A two-dimensional method for a family of dispersive shallow water models

Source abstract

We propose a numerical method for a family of two-dimensional dispersive shallow water systems with topography. The considered models consist in shallow water approximations – without the hydrostatic assumption – of the incompressible Euler system with free surface. Hence, the studied models appear as extensions of the classical shallow water system enriched with dispersive terms. The model formulation motivates us to use a prediction-correction scheme for its numerical approximation. The prediction part leads to solving a classical shallow water system with topography while the correction part leads to solving an elliptic-type problem. The numerical approximation of the considered dispersive models in the two-dimensional case over unstructured meshes is described, it requires to combine finite volume and finite element techniques. A special emphasis is given to the formulation and the numerical resolution of the correction step (variational formulation, inf-sup condition, boundary conditions,...). The numerical procedure is confronted with analytical and experimental test cases. Finally, an application to a real tsunami case is given.