A flux-vector splitting scheme for the shallow water equations extended to high-order on unstructured meshes

We present a flux vector splitting method for the one and two-dimensional shallow water equations following the approach first proposed by Toro and Vázquez for the compressible Euler equations. The resulting first-order schemes turn out to be exceedingly simple, with accuracy and robustness comparable to that of the sophisticated Godunov upwind method used in conjunction with complete non-linear Riemann solvers. The technique splits the full system into two subsystems, namely an advection system and a pressure system. The sought numerical flux results from fluxes for each of the subsystems. As to the source terms, there is potential for treating general source terms by incorporating them into either subsystem. In this article we show preliminary results for the case of a discontinuous bottom, incorporated into the pressure system. Results show that the resulting method is well balanced. The basic methodology, extended on 2D unstructured meshes, constitutes the building block for the construction of numerical schemes of very high order of accuracy following the ADER approach. The presented numerical schemes are systematically assessed on a carefully selected suite of test problems with reference solutions, in one and two space dimensions. The applicability of the schemes is illustrated through simulations of tsunami wave propagation in the Pacific Ocean.