The Modified Camassa-Holm Equation Of Local Discontinuous Galerkin Method

Numerical methods for evolution equations have been important tools in scientific research areas, such as mechanics, control systems, engineering techniques and economic systems etc. The researches on the numerical methods of evolution equations have important theoretical and practical values. The local discontinuous Galerkin (LDG) method is an extension of the Runge-Kutta discontinuous Galerkin method. It has many advantages such as local conservation, easy handling of complicated geometries, allowance of discontinuous, format high order accuracy, easy realization of parallel and h-p adaptive calculation. These features make it suitable to solve evolution equations.In this paper, under the guidance of the AC=BD theory and by means of computation software Matlab, we study the LDG method for the modified Camassa-Holm (mCH) equation. The cell entropy inequality and the L2stability are proved. An error estimate for smooth solution is given. Numerical simulations illustrate the accuracy of LDG method. The main works are as follows:In Chapter1, we briefly introduce the background and significance of this paper. The history and development of discontinuous Galerkin method are described. The information of mCH equation is presented.In Chapter2, we illustrate essential ideas of DG method by one-dimensional hyperbolic conservation law equation, including mesh generation, semi-discrete scheme and process of finding a solution. Numerical flux, stability and other key factors involved are described.In Chapter3, we state some definitions and theorems of AC=BD theory. Furthermore, we give some applications on solving numerical solutions.In Chapter4, we apply the LDG method for the modified CH equation. Details related to the implementation of the method are described. We give a proof of the cell entropy inequality and L2stability, and present a prior error estimate. Finally, we provide numerical experiments for different types of solutions of the mCH equation in order to illustrate the accuracy and capability of LDG method.