High order well-balanced discontinuous Galerkin methods based on hydrostatic reconstruction for shallow water equations

In this paper, we introduce high order well-balanced discontinuous Galerkin methods for shallow water equations over non-flat bottom topography, which preserve the lake at rest steady state. To achieve the well-balanced property, we propose to construct the numerical fluxes based on the hydrostatic reconstruction idea and to be in combination with a novel source term approximation as well as a decomposition algorithm. Rigorous theoretical analysis and extensive numerical results all verify that the current methods maintain the well-balanced property. In addition, numerical results also indicate that the resulting methods enjoy the ability to accurately capture small perturbations of the lake at rest steady state and keep the genuine high order accuracy for smooth solutions.