A Weighted Average Flux (WAF) scheme applied to shallow water equations for real-life applications

This work focuses on the application of a Weighted Average Flux (WAF) approximation to the shallow water equations with friction and real bathymetry on unstructured two dimensional grids. The scheme obtained must satisfy severe numerical criteria so it can be implemented in the TELEMAC code, widely used in industry. Despite its interesting numerical properties (at least in theory, which includes second order in time and space), WAF approximation is rarely applied to unstructured meshes. Here, we provide a general method for applying the WAF scheme to a homogenous equation. The general and efficient discretization of the topographic source term (through hydrostatic reconstruction) and friction (with a semi-implicit way) is also presented. We applied the approximation using the HLLC Riemann solver within a vertex-centered finite volume framework. The discretization obtained is validated with several theoretical and real benchmarks. The WAF-HLLC scheme proposed is also compared to several other well-known schemes such as HLLC, Roe and Kinetic type schemes. The results obtained show that the behavior of the scheme is encouraging as they demonstrate well-balanceness, strict mass conservation, positivity of water depth, efficient treatment of wetting and drying phenomena, good shock-capturing and low numerical diffusion. Nevertheless, the second order of accuracy in space is not attained with sufficient rigor and remains an open problem requiring more detailed study.