Convergence Analysis Of Finite Difference Scheme For Ginzburg-landau Equation (or Equations)

In this paper,based on the finite difference method,some finite difference schemes are proposed for solving the general Ginzburg-Landau equation and the Ginzburg-Landau equations with quintic term.And also we give the convergence analysis for all the finite difference schemes,then the numerical examples are given to support the theoretical analysis.This paper includes four chapters,and it is organized as follows.The first chapter is an introduction,it includes research background and status,basic notations and auxiliary lemmas,mainly works of this paper.In the second chapter,we propose a difference scheme for the general Ginzburg-Landau equation,and prove that the difference solution converges to the exact solution with second order in the discrete L? norm.Numerical examples support the theoretical analysis.In the third chapter,we construct a finite difference scheme(CN)for Ginzburg-Landau equations with quintic term,then prove that the difference solution converges to the exact solution with second order in the discrete L? norm on the basis of the prior estimate.In the fourth chapter,we still study the Ginzburg-Landau equations.In order to improve the calculation speed and the order of the nonlinear difference scheme in the third chapter,we optimize the scheme by a new technique.We can propose a high order difference scheme and then obtain the prior estimate on the basis of knowledge about matrix.At last,the difference solution can be proved to converge to the exact solution with four order in the discrete L? norm without any restrictions.