A remark on finite volume methods for 2D shallow water equations over irregular bottom topography

The 2D shallow water equations are often solved through finite volume (FV) methods in the presence of irregular topography or in non-rectangular channels. The ability of FV schemes to preserve uniform flow conditions under these circumstances is herein analysed as a preliminary condition for more involved applications. The widely used standard Harten-Lax-Van Leer (HLL) and a well-balanced Roe method are considered, along with a scheme from the class of wave propagation methods. The results indicate that, differently from the other solvers, the standard HLL does not preserve the uniform condition. It is therefore suggested to benchmark numerical schemes under this condition to avoid inaccurate hydrodynamic simulations. The analysis of the discretized equations reveals an unbalance of the HLL transversal fluxes in the streamwise momentum equation. Finally, a modification of the HLL scheme, based on the auxiliary variable balance method, is proposed, which strongly improves the scheme performance.