Equivalence Between DFR Method and DG Method for Solving Parabolic Equation and Convection-diffusion Equation

Abstract

The equivalence between direct flux reconstruction method and discontinuous Galerkin method for solving parabolic equation and convection-diffusion equation is studied. The research process is divided into two parts: the first part gives two proofs for the equivalence of DFR method and direct discontinuous Galerkin method for solving parabolic equations. The first proof mainly uses K-point Gauss quadrature with 2K－1 order algebraic accuracy. The second proof mainly uses the special properties of Legendre polynomials, Radau polynomial and Lobatto polynomial. In the second part, the equivalence of DFR method and local discontinuous Galerkin method in solving convection-diffusion equation is proved. The main idea is that the polynomial of degree K－1 has at most K－1 different zeros, so that the auxiliary variables used in local discontinuous Galerkin method can be directly expressed by interpolation. The proof of equivalence between the two methods improves the equivalence theory of interpolation method and projection method for solving partial differential equations.

Keywords

• direct discontinuous galerkin methods
• local discontinuous galerkin methods
• direct flux reconstruction
• parabolic equation
• convection diffusion equation