A well-balanced discontinuous Galerkin method for the shallow water flows on erodible bottom

Abstract

In this paper, we investigate the shallow water (SW) flow over the erodible layer using a fully coupled mathematical model in two-dimensional (2D) space. A well-balanced discontinuous Galerkin (DG) scheme is proposed for solving the SW equations with sediment transport and bed evolution. To achieve the well-balanced property of the numerical scheme easily, the nonlinear SW equations are first reformulated into a new form by introducing an auxiliary variable. Then the DG method is used to discretize the model, in which the FORCE flux is used. By choosing an appropriate value of the auxiliary variable, we can prove that the numerical method can accurately maintain the steady solution in still water, so it is indeed an equilibrium preserving scheme. The well-balanced property can be extended to any numerical flux which satisfies the consistency. Moreover, we investigate the impact on the numerical results if the appropriate value of auxiliary variable is slightly modified. The effectiveness of the numerical method is finally verified by numerical experiments.