Exactly well-balanced discontinuous Galerkin methods for the shallow water equations with moving water equilibrium

Hyperbolic conservation laws with source terms often admit steady state solutions where the fluxes and source terms balance each other. To capture this balance and near-equilibrium solutions, well-balanced methods have been introduced and performed well in many numerical tests. Shallow water equations have been extensively investigated as a prototype example. In this paper, we develop well-balanced discontinuous Galerkin methods for the shallow water system, which preserve not only the still water at rest steady state, but also the more general moving water equilibrium. The key idea is the recovery of well-balanced states, a special source term approximation, and the approximation of the numerical fluxes based on a generalized hydrostatic reconstruction. We also study the extension of the positivity-preserving limiter presented in [40] in this framework. Numerical examples are provided at the end to verify the well-balanced property and good resolution for smooth and discontinuous solutions.