The Error Estimates Of Discontinuous Galerkin Methods Based On Upwind-Biased Fluxes

Discontinuous Galerkin methods have many advantages such as high precision and high resolution in solving partial differential equations problems.In the process of calculation,researchers developed methods such as local discontinuous Galerkin method and direct discontinuous Galerkin method for different equations and different problems.It is important for discontinuous Galerkin method to select the appropriate numerical fluxes.This paper mainly studied the error estimates of discontinuous Galerkin methods based on upwind-baised fluxes.In this paper we first introduce the basic knowledge of discontinuous Galerkin methods,then a global Gauss-Radau projection based on generalized alternating fluxes is defined for the fifth order partial differential equations and prove the existence and uniqueness of the projection,then we prove that the ~2Lerror estimates of fifth-order partial differential equations in semi-discrete scheme can reach to order k(10)1.Secondly for linear convection diffusion equations,we obtain the semi-discrete scheme of direct discontinuous Galerkin method by using some interface corrections.The convection term uses upwind-baised fluxes and the convection term uses the first order diversion fluxes.Then,we define the discrete energy norm and prove the stability of the numerical solutions.Finally,we prove that the ~2Lerror estimates of direct discontinuous Galerkin method can reach to order k(10)1.At last,we use the SSP Runge-Kutta method for the fifth-order partial differential equations and the third-order explicit Runge-Kutta method for the second-order linear convection-diffusion equations in time discretization.Numerical experiments showed that the error estimates of both the fifth-order partial differential equations and the second-order linear convection-diffusion equation can reach to order k(10)1,that verify the validity of the theory.