A staggered semi-implicit hybrid finite volume / finite element scheme for the shallow water equations at all Froude numbers

Abstract

We present a novel staggered semi-implicit hybrid FV/FE method for the numerical solution of the shallow water equations at all Froude numbers on unstructured meshes. A semi-discretization in time of the conservative Saint-Venant equations with bottom friction terms leads to its decomposition into a first order hyperbolic subsystem containing the nonlinear convective term and a second order wave equation for the pressure. For the spatial discretization of the free surface elevation an unstructured mesh of triangular simplex elements is considered, whereas a dual grid of the edge-type is employed for the computation of the depth-averaged momentum vector. The first stage of the proposed algorithm consists in the solution of the nonlinear convective subsystem using an explicit Godunov-type FV method on the staggered grid. Next, a classical continuous FE scheme provides the free surface elevation at the vertex of the primal mesh. The semi-implicit strategy followed circumvents the contribution of the surface wave celerity to the CFL-type time step restriction making the proposed algorithm well-suited for low Froude number flows. The conservative formulation of the governing equations also allows the discretization of high Froude number flows with shock waves. As such, the new hybrid FV/FE scheme is able to deal simultaneously with both, subcritical as well as supercritical flows. Besides, the algorithm is well balanced by construction. The accuracy of the overall methodology is studied numerically and the C-property is proven theoretically and validated via numerical experiments. The solution of several Riemann problems attests the robustness of the new method to deal also with flows containing bores and discontinuities. Finally, a 3D dam break problem over a dry bottom is studied and our numerical results are successfully compared with numerical reference solutions and experimental data