Local Discontinuous Galerkin Methods For Partial Differential Equations With Higher Order Spatial Derivatives

The discontinuous finite element method is a numerical method using com-pletely discontinuous basis functions which are usually chosen as piecewise polyno-mials, and the time discretization is achieved by the explicit Runge-Kutta method. Since the end of 1980's, it attracted the attention of mathematicians and was de-veloped well. In this paper,we develop local discontinuous Galerkin methods for solving two partial differential equations with higher order spatial derivatives. The stability and error estimates are obtained and numerical simulation is presented. This paper consists of four chapters.Chapter 1 is an introduction. In this part we briefly introduce the origin and development of the local discontinuous Galerkin finite element method and make a summary of the advantages of this method.In Chapter 2, we we apply the Local discontinuous Galerkin (LDG) method to solve linear Kdv-Burgers equation: The stability and error estimates are obtained, numerical simulation is presented. In Chapter 3, we apply the Local discontinuous Galerkin (LDG) method tosolve one-dimensional nonlinear Cahn-Hilliard equation:The energy stability is obtained and numerical simulation is presented. Chapter 4 is a conclusion of the paper.