Study On The Equivalence Of Discontinuous Galerkin Method And Direct Flux Reconstruction Method For High Order Partial Differential Equations

Direct flux reconstruction method(DFR method)is a method that firstly uses interpolation method to rewrite partial differential equations into ordinary differential equations,and then uses Runge-Kutta method to program and solve ordinary differential equations so as to obtain numerical solutions.Discontinuous Galerkin method(DG method)is a method that firstly rewrites partial differential equations into ordinary differential equations by integral method,and then uses Runge-Kutta method to program and solve ordinary differential equations to obtain numerical solutions.This paper mainly studies the equivalence of DFR method and DG method in solving higher order partial differential equations.Firstly,this paper introduces the basic knowledge of DFR method and DG method and the main tools used in the research process,and then gives the specific formats of DFR method and direct discontinuous Galerkin method(DDG method)for solving parabolic equations.Two methods are used to prove the equivalence of DFR method and DDG method in solving parabolic equation: the first method mainly uses Gauss quadrature at K point with K12-order algebraic accuracy;The second method mainly uses the special properties of Legendre polynomials,Radau polynomial and Lobatto polynomial.Then,taking the parabolic equation as an example,the ordinary differential equations used in the two numerical algorithms are given to prove the equivalence of the two numerical algorithms.Secondly,the specific formats of DFR method and local discontinuous Galerkin method(LDG method)for solving linear convection-diffusion equation are given.The equivalence of DFR method and LDG method in solving linear convection-diffusion equation is proved by two methods.The main idea used in this section is to express the auxiliary variables used in LDG method directly by interpolation method by using K-1 degree polynomial with at most K-1different zero points.Then,taking the convection diffusion equation as an example,the ordinary differential equations used in the two numerical algorithms are given to prove the equivalence of the two numerical algorithms.Finally,the specific formats of local direct flux reconstruction(LDFR method)and ultra-weak discontinuous Galerkin method(UWDG method)for solving the fourth-order partial differential equation are given.By expanding the system of ordinary differential equations obtained by discretizing the ldfr method four times using the partial integration method and Gauss quadrature formula,the equivalence of the two numerical methods for solving the fourth-order partial differential equations is proved.Then,taking the Fourth order partial differential equation as an example,the ordinary differential equations used in the two numerical algorithms are given to prove the equivalence of the two numerical algorithms.All the above proofs are a further improvement of the equivalence theory of interpolation method and projection method for solving partial differential equations.