On the L_∞ convergence of a nonlinear difference scheme for Schr?dinger equations

摘要

In this article, a nonlinear difference scheme for Schr?dinger equations is studied. The existence of the difference solution is proved by Brouwer fixed point theorem. With the aid of the fact that the difference solution satisfies two conservation laws, the difference solution is proved to be bounded in the L∞ norm. Then, the difference solution is shown to be unique and second order convergent in the L∞ norm. Finally, a convergent iterative algorithm is presented.