Some Efficient Numerical Schemes For Two-dimensional Ginzburg-landau Equation

In this thesis,we mainly propose some efficient numerical schemes for 2-dimensional Ginzburg-Landau(GL)equation.Their properties like stability and convergence are analyzed,the detailed numerical results confirm with our theoretical conclusion.In Chapter 1,we firstly introduce some basic knowledge about GL equation and some notations used in this paper.Then we prove the solutions for the GL equation is bounded.Lastly,we study the preliminaries about high-order compact(HOC)scheme.In Chapter 2,we discuss the basic idea and principle of split step method and propose an efficient numerical scheme for 2D GL equation.We firstly utilize splitting method to alter the 2D GL equation into a linear problem and a nonlinear problem in order to avoid solving a coupled nonlinear algebraic system.Then,in order to reduce storage and computational cost,we decompose the linear problem into two one dimensional problems by local one-dimensional(LOD)method.The scheme has the advantages such as high efficiency and high accuracy.Lastly,on the basis of previous work,we propose two another efficient numerical schemes for the 2D GL equation.The space is approximated by HOC methods to improve the computational efficiency.Based on Crank-Nicholson method in time,several temporal approximations are used which are starting from different viewpoints.In Chapter 3,The numerical characters of the schemes proposed in Chapter 2 such as the existence and uniqueness,stability,convergence are investigated.It shows that the differential equations obtained by discretization of partial differential equations can be used to approximate the exact solution.In Chapter 4,Some numerical illustrations are reported to confirm the advantages of the new schemes by comparing with other existing works.In the numerical experiments,the role of some parameters in the model is considered and tested.The main result of this paper is some new schemes for the 2D GL equation based on the existing numerical schemes.These new schemes can improve the accuracy,the computation and storage capacity is decreased at the same time.Through the work of this paper,we can get more efficient and accurate numerical solution of the 2D GL equation.