Recent advances on the discontinuous Galerkin method for shallow water equations with topography source terms

We consider in this work the discontinuous Galerkin discretization of the nonlinear Shallow Water equations on unstructured triangulations. In the recent years, several improvements have been made in the quality of the discontinuous Galerkin approximations for the Shallow Water equations. In this paper, we first perform a review of the recent methods introduced to ensure the preservation of motionless steady states and robust computations. We then suggest an efficient combination of ingredients that leads to a simple high-order robust and well-balanced scheme, based on the alternative formulation of the equations known as the {\it pre-balanced} shallow water equations. We show that the preservation of the motionless steady states can be achieved, for an arbitrary order of polynomial expansion. Additionally, the preservation of the positivity of the water height is ensured using the recent method introduced in [J. Comput.Phys, 50, pp 29-62, 2012]. An extensive set of numerical validations is performed to highlight the efficiency of these approaches. Some accuracy, CPU-time and convergence studies are performed, based on comparisons with analytical solutions or validations against experimental data, for several test cases involving steady states and occurrence of dry areas. Some comparisons with a recent finite-volume MUSCL approach are also performed.