Fully-discrete Local Discontinuous Galerkin Methods For Solving Convection-diffusion Problems

Local discontinuous Galerkin (LDG) method is a new method for solving convection-diffusion equation. Since it was introduced, the LDG method has been widely applied in numerical solution of (higher order) partial differential equations.Compared with the rapid development of the numerical study, the theoretical research is relative backward. At present, there is not much theoretical analysis about the fully discrete LDG method. In practice, one usually adopts certain time-marching method to solve time-dependent problems. Thus, the study of the fully discretization estimates is significant. The purpose of this dissertation is to study the stability and error estimates for the fully discrete LDG method for solving convection-diffusion equations. It is mainly composed of three chapters:In the second chapter, we consider explicit Runge-Kutta (RK) fully discrete LDG (EXRK-LDG) method for solving convection-diffusion problems with Dirichlet boundaries. There are mainly two difficulties for Dirichlet boundary problems:one is the choice of nu-merical flux at the boundary, improper choice will affect the stability and accuracy of the scheme; the other is the boundary setting at each intermediate stage of RK method, uncorrect setting method will affect the total accuracy of the scheme. In this paper, we will present the error estimate of the third order EXRK-LDG method, by the aid of energy analysis. Also we will give an efficient choice of numerical flux, and a boundary setting method which will not destroy the accuracy. Meanwhile, we prove that, under these settings, the third order EXRK-LDG scheme can achieve optimal L2 accuracy in both space and time, if the time step τ satisfies the CFL conditionsc τ/h≤λc and dτ/h2≤λd. Here c and d are coefficients of convection and diffusion, respectively, and h is the size of the spatial mesh, λc and Ad are given CFL numbers.For convection dominant problem, the temporal-spatial condition for explicit time march-ing method is τ=O(h), hence it is a good choice for convection dominant problems. How- ever, for diffusion dominant problems,the time step restriction is τ= O(h2) for explicit time discretization. To overcome the small time step restriction, we will study a kind of implicit-explicit (IMEX) time discretization method. When using IMEX method for solv-ing convection-diffusion equations, we treat the convection term explicitly and the diffusion term implicitly. This kind of method can avoid the small time step restriction, hence it can efficiently solve convection-diffusion problems which are diffusion dominant, especially for convection-diffusion problems with linear diffusion parts and nonlinear convection parts.In the third chapter, we will consider RK type IMEX fully discrete LDG scheme (IMEX-RK-LDG) for solving one-dimensional linear convection-diffusion problems. By establishing the important relationship between the gradient and the jump of the numerical solution with the approximation solution of the gradient, and by the aid of energy analysis, we show that, the several specific carefully chosen IMEX-RK-LDG schemes are unconditionally stable, in the sense that the time step is only required to be upper bounded by a constant τ0 which is independent of the mesh size, but only depending on the coefficients of convection and diffusion. Rigorous analysis implies that t0 is proportional to d/c2. Under this condition, we also show the optimal L2 accuracy of the schemes under consideration.In the forth chapter, we will extend the stability and error analysis to convection-diffusion problems with nonlinear convection part and multi-dimensional convection-diffusion problems. Also we will study the multi-step IMEX fully discrete LDG schemes (IMEX-MS-LDG). This chapter is composed of three sections. In section one, by building up the similar properties of the LDG spatial discretization as linear case, and by the aid of the a priori assumption, we get the similar results as chapter three. In section two, by the aid of energy method, we prove that, under the similar condition as chapter three, the IMEX-MS-LDG schemes are stable in the energy norm, and they can achieve optimal L2 accuracy. In section three, by establishing the important relationship between the gradient and the jump of the nu-merical solution with the approximation solution of the gradient, in multi-dimensional space, we derive the similar stability result as chapter three, and by the aid of elliptic projection, we show the optimal L2 accuracy for IMEX-RK-LDG schemes.