A high-order well-balanced discontinuous galerkin method based on the hydrostatic reconstruction for the ripa model

In this work, we present a high-order discontinuous Galerkin method for the shallow water equations incorporating horizontal temperature gradients (also known as the Ripa model), which exactly maintains the lake at rest steady state. Herein, we propose original numerical fluxes defined on the basis of the hydrostatic reconstruction idea and a simple source term approximation. This novel approach allows us to achieve the well-balancing of the discontinuous Galerkin method without complication. Moreover, the proposed method retains genuinely high-order accuracy for smooth solutions and it shows good resolution for discontinuous solutions at the same time. Rigorous numerical analysis as well as extensive numerical results all verify the good performances of the proposed method.