A weighted surface-depth gradient method for the solution of the 2D shallow water equations

The two dimensional shallow water equations (SWE) are currently accepted as a mathematical basis for the study of several rapidly varying flows, such as those induced by dam breaking or embankment failure. Among the many methodologies available for the numerical integration of SWE, the finite volume MUSCL-Hancock cell-centred schemes are nowadays frequently used. These schemes should also include a robust and efficient treatment of the bottom source term in order to track the wetting and drying fronts and to preserve the static condition of a quiescent fluid over an irregular topography (C–property). In particular the Surface Gradient Method (SGM – Zhou et al. 2001) computes water depth at the intercells from the extrapolation of the water surface level and introduces the bottom slope source term through a centred unsplit discretization; it can satisfy the C-property on a Cartesian grid even on irregular topographies, but it is not efficient in tracking wetting and drying fronts. On the other hand, many schemes that evaluate water depth at the cell interfaces through the extrapolation of the same conserved variable (Depth Gradient Method, DGM) and perform the splitting of the bed slope source term, do not satisfy the C-property, but are robust and stable near moving boundaries. In this paper a Weighted Surface-Depth Gradient Method, capable of preserving the good capabilities of the previously mentioned methods, is proposed. In the framework of the SGM scheme, the water depth at cell interfaces is estimated through a weighted average of the boundary extrapolated values deriving from MUSCL DGM and SGM reconstructions. The weight parameter is evaluated on the basis of the local Froude number through a formulation that allows a smooth transition between a fully SGM and a fully DGM treatment. The numerical scheme is validated through the application to some reference test cases whose exact solution is available in literature. The first set of tests concerns 1D steady flows on a steep parabolic bump in a rectangular frictionless channel (Liska & Wendroff 1998); the second deals with two exact solutions (Thacker 1981) related to the periodic motion of a volume of water in a frictionless basin whose shape is a parabola of revolution. In all the tests the results obtained by the application of the Weighted Surface-Depth Gradient Method are better or, at least, equal to those obtained by the SGM or DGM schemes.