A discontinuous Galerkin method for two-layer shallow water equations

Abstract

In this paper, we study a discontinuous Galerkin method to approximate solutions of the two-layer shallow water equations on non-flat topography. The layers can be formed in the shallow water model based on the vertical variation of water density which in general depends on the water temperature and salinity. For a water body with equal density the model reduces to the canonical single-layer shallow water equations. Thus, for a model with equal density on flat bottom, the method is equivalent to the discontinuous Galerkin method for conservation laws. The considered method is a stable, highly accurate and locally conservative finite element method whose approximate solutions are discontinuous across inter-element boundaries; this property renders the method ideally suited for the hp-adaptivity. Several numerical results illustrate the performance of the method and confirm its capability to solve two-layer shallow water flows including tidal conditions on the water free-surface and bed frictions on the bottom topography.