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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 497439, 9 pages
http://dx.doi.org/10.1155/2013/497439
Research Article

Parallel Methods and Higher Dimensional NLS Equations

1Department of Mathematics, College of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, Saudi Arabia
2Department of Computer Science, University of Georgia, Athens, GA 30602-7404, USA

Received 3 June 2013; Accepted 28 July 2013

Academic Editor: Juan Carlos Cortés López

Copyright © 2013 M. S. Ismail and T. R. Taha. 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

Alternating direction implicit (ADI) schemes are proposed for the solution of the two-dimensional coupled nonlinear Schrödinger equation. These schemes are of second- and fourth-order accuracy in space and second order in time. The resulting schemes in each ADI computation step correspond to a block tridiagonal system which can be solved by using one-dimensional block tridiagonal algorithm with a considerable saving in computational time. These schemes are very well suited for parallel implementation on a high performance system with many processors due to the nature of the computation that involves solving the same block tridiagonal systems with many right hand sides. Numerical experiments on one processor system are conducted to demonstrate the efficiency and accuracy of these schemes by comparing them with the analytic solutions. The results show that the proposed schemes give highly accurate results.