Unconditional and Optimal Pointwise Error Estimates of Finite Difference Methods for the Two-Dimensional Complex Ginzburg-Landau Equation
Received:March 16, 2023  Revised:July 08, 2023
Key Words: complex Ginzburg-Landau equation   finite difference method   unconditional convergence   optimal estimates   pointwise error estimates  
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11571181) and the Research Start-Up Foundation of Nantong University (Grant No.135423602051).
Author NameAffiliation
Yue CHENG School of Mathematics and Statistics, Nantong University, Jiangsu 226019, P. R. China 
Dongsheng TANG Jiangsu Xinhai Senior High School, Jiangsu 222005, P. R. China 
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      In this paper, we give improved error estimates for linearized and nonlinear Crank-Nicolson type finite difference schemes of Ginzburg-Landau equation in two dimensions. For linearized Crank-Nicolson scheme, we use mathematical induction to get unconditional error estimates in discrete $L^2$ and $H^1$ norm. However, it is not applicable for the nonlinear scheme. Thus, based on a `cut-off' function and energy analysis method, we get unconditional $L^2$ and $H^1$ error estimates for the nonlinear scheme, as well as boundedness of numerical solutions. In addition, if the assumption for exact solutions is improved compared to before, unconditional and optimal pointwise error estimates can be obtained by energy analysis method and several Sobolev inequalities. Finally, some numerical examples are given to verify our theoretical analysis.
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