The Local Discontinuous Galerkin Method For A Class Of KdV Equations

The local discontinuous Galerkin (LDG) method is an extension of the dis-continuous Galerkin (DG) method which is studied by the scholars represented by Cockburn and Shu to convection-diffusion equations. It was generalized to solve convection-diffusion equations by Cockburn and Shu, and from then on, it was developed well. Just as the RKDG method, the LDG method uses a completely discontinuous piecewise polynomial space as its finite element space.In this paper, we apply the LDG method to solve a fifth-order linear KdV equation:(where a,(3,7are arbitrary constants). We develop the LDG scheme. The stability and error estimate are obtained, at the same time, numerical simula-tions are presented.This paper has four chapters.In Chapter1, we briefly introduce several numerical methods for solving the generalized fifth-order KdV equation We also introduce the DG and LDG methods’history and their advantages. In Chapter2, we develop the semi-discrete LDG scheme for the fifth-order linear KdV equation shown before by introducing auxiliary variables. We give a detailed description of the scheme. Then the stability analysis and error estimate are presented. The most important point in this chapter is to estimate the error. The main techniques we use are the standard L2-projection and the special projection.In Chapter3, we carry out the fully discrete LDG scheme with explicit Runge-Kutta time discretization. And we apply this scheme to solve several numerical examples, we show the numerical results and the comparing figures of the numerical solutions and the exact solutions. At last, by combining the numerical results and the figures, we gave a further analysis of the effectiveness and precision of the fully discrete LDG method in solving KdV equations.The fourth chapter is the conclusion of this paper.