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Advances in Condensed Matter Physics
Volume 2015, Article ID 127580, 7 pages
Research Article

An Efficient Compact Finite Difference Method for the Solution of the Gross-Pitaevskii Equation

1School of Sciences, Liaoning Shihua University, Fushun 113001, China
2School of Foreign Language, Liaoning Shihua University, Fushun 113001, China
3Faculty of Mathematics, Baotou Teachers College, Baotou 014030, China

Received 25 March 2015; Accepted 20 May 2015

Academic Editor: Sergei Sergeenkov

Copyright © 2015 Rongpei Zhang et al. 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.


We present an efficient, unconditionally stable, and accurate numerical method for the solution of the Gross-Pitaevskii equation. We begin with an introduction on the gradient flow with discrete normalization (GFDN) for computing stationary states of a nonconvex minimization problem. Then we present a new numerical method, CFDM-AIF method, which combines compact finite difference method (CFDM) in space and array-representation integration factor (AIF) method in time. The key features of our methods are as follows: (i) the fourth-order accuracy in space and rth () accuracy in time which can be achieved and (ii) the significant reduction of storage and CPU cost because of array-representation technique for efficient handling of exponential matrices. The CFDM-AIF method is implemented to investigate the ground and first excited state solutions of the Gross-Pitaevskii equation in two-dimensional (2D) and three-dimensional (3D) Bose-Einstein condensates (BECs). Numerical results are presented to demonstrate the validity, accuracy, and efficiency of the CFDM-AIF method.