Discontinuous Galerkin method for two-dimensional bilayer shallow water equations

We present a numerical solution of the two-layer shallow water equations in two space dimensions, based on a discontinuous Galerkin finite element method. The continuous equations are discretized and solved locally using the finite element method of an unstructured computational domain using nodal polynomial basis functions of arbitrary order in space. To complete the discretization in space, we choose the numerical flux based on the local Lax–Friedrichs flux which establishes connection between elements. The numerical discretization results in a set of coupled nonlinear equations which can be solved efficiently and locally by integrating them using a Runge–Kutta scheme. The considered discontinuous Galerkin method is fully explicit, stable, highly accurate, and locally conservative finite element method whose approximate solutions are discontinuous across inter-element boundaries; this property renders the method ideally suitable for the hp-adaptivity. Several numerical results are presented to demonstrate the high resolution of the proposed method and to confirm its capability to provide accurate and efficient simulations to solve the two-dimensional bilayer shallow water equations.