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Advances in Numerical Analysis
Volume 2012 (2012), Article ID 913429, 24 pages
http://dx.doi.org/10.1155/2012/913429
Research Article

Convergence of an Eighth-Order Compact Difference Scheme for the Nonlinear Schrödinger Equation

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

Received 24 May 2012; Accepted 7 August 2012

Academic Editor: Ting-Zhu Huang

Copyright © 2012 Tingchun Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A new compact difference scheme is proposed for solving the nonlinear Schrödinger equation. The scheme is proved to conserve the total mass and the total energy and the optimal convergent rate, without any restriction on the grid ratio, at the order of in the discrete -norm with time step τ and mesh size h. In numerical analysis, beside the standard techniques of the energy method, a new technique named “regression of compactness” and some lemmas are proposed to prove the high-order convergence. For computing the nonlinear algebraical systems generated by the nonlinear compact scheme, an efficient iterative algorithm is constructed. Numerical examples are given to support the theoretical analysis.