Some Research About The Compact Finite Difference Scheme For One Dimensional Complex Ginzburg-Landau Equation

In the paper, based on the finite difference scheme, we not only establish three compact finite difference schemes to solve the one-dimensional complex Ginzburg-Landau equation but also discuss the associated theoretical analysis and the numerical experiments.A periodic initial value problem of the the one-dimensional complex Ginzburg-Landau equation is presented. For the convenience of constructing the difference schemes in the next part, the second chapter explain the definition about the inner product and some related notations. Some lemmas are also proved.In the third chapter, we propose the first difference method which is a two level nonlinear scheme for the problem.In theory, the unique solvability, the prior estimations and convergence in maximum norm are proved by using the knowledge of matrix analysis. An iterative algorithm should be used to get the numerical results. The convergence and the stability of scheme are confirmed by the numerical experiment.Then we construct the second method in this paper is a three level nonlinear difference scheme. Similarily, the convergence and the stability in maximum norm of the scheme are verified by the numerical experiments. Taking account for the previous two difference schemes solved by the iterative algorithm which are time-consuming, we carry out the third scheme which is a three level linear difference method in this paper. We also use the knowledge of matrix analysis to prove the unique solvability, the prior estimations and convergence in maximum norm. An iterative algorithm and numerical results are reported to support our theoretical results.Finally, we compare the proposed three difference schemes in two aspects of computing time and error results in maximum norm.