Unconditional And Optimal Pointwise Error Estimates Of Finite Difference Methods For The Two-Dimensional Complex Ginzburg-Landau Equation

The complex Ginzburg-Landau equation is an important class of nonlinear evolution equa-tions,and has a very wide range of applications in many fields such as superconductivity and superfluidity.Therefore,solving the numerical solution of the complex Ginzburg-Landau equa-tion is a very meaningful thing.In this paper,the numerical solution of the cubic nonlinear complex Ginzburg-Landau equation in the two-dimensional case is studied.The two finite-difference schemes in the existing literature are reviewed and analyzed in depth,and a more in-depth analysis is obtained.Specifically,the convergence analysis of the finite difference scheme of the two-dimensional complex Ginzburg-Landau equation in the existing literature either has strict requirements on the grid ratio,or only has error estimates for discrete L~2norm.Using the analysis techniques in these literatures,it is difficult to obtain the optimal error esti-mate of the algorithm in the sense of the maximum norm without requiring the grid ratio.The main research work of this paper mainly includes the following two aspects:First,a linearized finite-difference scheme of the complex GL equation is analyzed in depth,and the optimal error estimate of the algorithm in the maximum norm sense is estab-lished without any requirement on the grid ratio by using energy analysis methods combined with mathematical induction and lifting techniques.It is proved that the scheme is convergent in both time and space directions with 2 order accuracy in the maximum norm sense.Second,a nonlinear finite difference scheme is analyzed in depth.The existence and uniqueness of the nonlinear scheme's solution is discussed?then the optimal error estimate of the scheme in the maximum sense is established by using the energy analysis method and the cut-off technique combined with the lifting technique,and the algorithm is proved to be accurate to the 2 order in both space-time directions.Finally,the error estimates are verified by numerical examples.