The Discontinuous Galerkin Method 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),The model is widely used in the hydrodynamic.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.First,we construct the well-balanced numerical flux based on the idea of hydrostatic reconstruction.With this unique way of numerical flow volume construction,we should simply adopt a simple Gaussian quadrature formula for the source term,without the complex source term discretization,to accurately balance the discretization of the hyperbolic term and for the source term,thus realizing the well-balanced property.After implementing the discretization for both the hyperbolic and source terms,we obtain semi-discrete methods for the Ripa model.Time discrete with the third order Runge-Kutta method and finally obtain well-balanced discontinuous Galerkin method.Both rigorous numerical analysis and extensive numerical results show that the proposed method maintains a constant solution to machine accuracy.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.