About this Journal Submit a Manuscript Table of Contents
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.

Linked References

  1. H.-Q. Zhang, X.-H. Meng, T. Xu, L.-L. Li, and B. Tian, “Interactions of bright solitons for the (2+1)-dimensional coupled nonlinear Schrödinger equations from optical fibres with symbolic computation,” Physica Scripta, vol. 75, no. 4, pp. 537–542, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  2. M. S. Ismail and T. R. Taha, “Numerical simulation of coupled nonlinear Schrödinger equation,” Mathematics and Computers in Simulation B, vol. 56, no. 6, pp. 547–562, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. S. Ismail, “A fourth-order explicit schemes for the coupled nonlinear Schrödinger equation,” Applied Mathematics and Computation, vol. 196, no. 1, pp. 273–284, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. S. Ismail, “Numerical solution of coupled nonlinear Schrödinger equation by Galerkin method,” Mathematics and Computers in Simulation, vol. 78, no. 4, pp. 532–547, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. M. S. Ismail and S. Z. Alamri, “Highly accurate finite difference method for coupled nonlinear Schrödinger equation,” International Journal of Computer Mathematics, vol. 81, no. 3, pp. 333–351, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. T. R. Taha and X. Xu, “Parallel split-step fourier methods for the coupled nonlinear Schrödinger type equations,” The Journal of Supercomputing, vol. 32, no. 1, pp. 5–23, 2005. View at Publisher · View at Google Scholar · View at Scopus
  7. Z. Gao and S. Xie, “Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations,” Applied Numerical Mathematics, vol. 61, no. 4, pp. 593–614, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. L. Kong, Y. Duan, L. Wang, X. Yin, and Y. Ma, “Spectral-like resolution compact ADI finite difference method for the multi-dimensional Schrödinger equations,” Mathematical and Computer Modelling, vol. 55, no. 5-6, pp. 1798–1812, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Z. F. Tian and P. X. Yu, “High-order compact ADI (HOC-ADI) method for solving unsteady 2D Schrödinger equation,” Computer Physics Communications, vol. 181, no. 5, pp. 861–868, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. T. Wang, B. Guo, and Q. Xu, “Fourth order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions,” Journal of Computational Physics, vol. 243, pp. 382–399, 2013.
  11. Y. Xu and L. Zhang, “Alternating direction implicit method for solving two-dimensional cubic nonlinear Schrödinger equation,” Computer Physics Communications, vol. 183, no. 5, pp. 1082–1093, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  12. S. Karaa and J. Zhang, “High order ADI method for solving unsteady convection-diffusion problems,” Journal of Computational Physics, vol. 198, no. 1, pp. 1–9, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. Sapagovas and K. Jakubeliene, “Alternating direction method for two-dimensional parabolic equation with nonlocal integral condition,” Nonlinear Analysis, vol. 17, no. 1, pp. 91–98, 2012. View at MathSciNet
  14. M. Subasi, “On the finite difference schemes for the numerical solution of two dimensional Schrödinger equation,” Numerical Methods for Partial Differential Equations, vol. 18, pp. 124–134, 2002.
  15. D. You, “A high-order Padé ADI method for unsteady convection-diffusion equations,” Journal of Computational Physics, vol. 214, no. 1, pp. 1–11, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A. Biswas, “1-soliton solution of 1+2 dimensional nonlinear Schrödinger equations in power law media,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, pp. 1830–1833, 2009.
  17. A. R. Mitchell and D. F. Griffiths, The Finite Difference Method in Partial Differential Equations, John Wiley & Sons, Chichester, UK, 1980. View at MathSciNet
  18. C. Besse, N. J. Mauser, and H. P. Stimming, “Numerical studies for nonlinear Schrodinger equations: the Schrodinger-Poisson-Xα model and Davey Stewartson systems,” In press.
  19. A. V. Terekhov, “A fast parallel algorithm for solving block-tridiagonal systems of liner equations including the domain decomposition method,” Parallel Computing, vol. 39, pp. 245–258, 2013.
  20. A. V. Terekhov, “Parallel dichotomy algorithm for solving tridiagonal system of linear equations with multiple right-hand sides,” Parallel Computing, vol. 36, no. 8, pp. 423–438, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet