High order well-balanced asymptotic preserving IMEX RKDG schemes for the two-dimensional nonlinear shallow water equations.

In this paper, a high order well-balanced asymptotic preserving scheme is presented for the two-dimensional nonlinear shallow water equations over variable bottom topography in all Froude number regimes. To obtain the well-balanced property, the system is first reformulated as a new form by introducing an auxiliary parameter. The flux is then split into a linear stiff part to be treated implicitly and a nonlinear non-stiff part to be treated explicitly, and the source term is treated explicitly. An implicit-explicit Runge-Kutta discontinuous Galerkin scheme is designed for solving the equations. The proposed scheme can be proved to be well-balanced, asymptotic preserving and asymptotically accurate. Finally, several numerical tests are carried out to validate the performance of our proposed scheme. • A new splitting scheme is presented for the 2D nonlinear shallow water equations over variable bottom. • A high order IXEM RKDG method is designed to solve the equations. • The method is proved to be well-balanced, asymptotic preserving and asymptotically accurate. [ABSTRACT FROM AUTHOR]