A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives

Abstract

In this paper, we develop a new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Stability is ensured by a careful choice of interface numerical fluxes. The method can be designed for quite general nonlinear PDEs and we prove stability and give error estimates for a few representative classes of PDEs up to fifth order. Numerical examples show that our scheme attains the optimal $(k+1)$-th order of accuracy when using piecewise $k$-th degree polynomials, under the condition that $k+1$ is greater than or equal to the order of the equation.