Optimal Convergence Rate And Applications Of The Corrected Explicit Euler Methods

In this paper,a corrected explicit Euler scheme with optimal convergence rate is con-structed for the convection-diffusion equation by combining the explicit Euler method for the temporal direction with the central difference discretization for the spatial direction.Theoretical analysis and numerical simulations are given.Finally this technique is extended to the nonlinear convective diffusion equation and the high-dimensional case.The paper consists of the following main parts.In Chapter 1,we introduce the background and progress of the explicit Euler method and convection-diffusion-reaction equations.In Chapter 2,we first consider the one-dimensional linear diffusion equation and construct a corrected explicit Euler scheme based on the further analysis of the local truncation error in the process of establishing its classical explicit Euler scheme.This scheme owns fourth-order convergence rate globally for a particular step ratior=1/6,i.e.it achieves an optimal rate of convergence.We then apply this technique to the one-dimensional linear convection-diffusion equation and construct corrected explicit Euler schemes for the convection-diffusion equations with the constant and variable convection coefficients,respectively,and give the priori estimates and convergence theorems.The corrected explicit Euler schemes are then constructed for the one-dimensional nonlinear convection-diffusion-reaction equations using the Fisher equation and the viscous Burgers’equation as examples.Finally we carry out numerical simulations of the above schemes and verify that the corrected explicit Euler schemes have an optimal rate of convergence(τ~2+h~4).In Chapter 3,we extend this technique to the generalized convection-diffusion equations in the high-dimensional cases.Firstly,we construct a corrected explicit Euler scheme for the anisotropic convection-diffusion equation with variable convection coefficients and give a priori estimate with optimal convergence.Secondly,the corrected explicit Euler schemes are applied to the two-dimensional nonlinear convection-diffusion-reaction equations under Dirichlet and Neumann boundaries and to the three-dimensional linear diffusion equation,respectively.Fi-nally,the optimal convergence rate of the schemes are verified by numerical simulations and the CFL conditions are explored in comparison with their corresponding classical explicit Euler schemes.In Chapter 4,we summarize our research content on this paper and point out subsequent research plan.