ADER-preserving Discontinuous Galerkin Finite Element Method For Shallow Water Wave Equation

In this paper,we construct positivity-preserving high order Arbitrary DERivatives in time and space(ADER)and arbitrary Galerkin(DG)finite element method.This method can accurately maintain static water steady-state for shallow water equations with non-flat bottom topography.The ADER-DG method constructed in this paper adopts the DG method in space,which can accurately capture the small disturbance of water flow under steady state.In addition,in order to achieve the equilibrium attribute,we decompose the numerical solution into two parts based on the decomposition algorithm for approximation,and accordingly decompose the approximate source term into two parts,so that the dispersion of the source term and the flux gradient can reach a mutually balanced state,and design the numerical flux with Well-balanced properties.In this method,the ADER method is used in time,and the differential transformation program is used to replace the Cauchy-Kovalevskaya(C-K)program,which can deal with some special source items,save the running time of the computer,and avoid occupying too much memory.Compared with the traditional Runge-Kutta method,the ADER method can achieve arbitrarily high order precision.In addition,we have introduced a simple positive holding limiter into the constructed method,which maintains water depth non-negative and provides effective and robust simulations near wetting and drying fronts.The ADER-DG method constructed in this paper is single-step,single-stage and fully discrete,which can accurately capture small disturbances in the static state of the lake,maintain the non-negative water depth,and maintain the true high-order accuracy of the smooth solution.Extensive 1D and 2D numerical example results also show that the constructed method has the above advantages.