Unconditional and Optimal Pointwise Error Estimates of Finite Difference Methods for the Two-Dimensional Complex Ginzburg-Landau Equation

Abstract:

      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.