Jump to ContentJump to Main Navigation
Show Summary Details
In This Section

International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board Member: Armbruster, Dieter / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

8 Issues per year


IMPACT FACTOR 2016: 0.890

CiteScore 2016: 0.84

SCImago Journal Rank (SJR) 2015: 0.298
Source Normalized Impact per Paper (SNIP) 2015: 0.476

Mathematical Citation Quotient (MCQ) 2015: 0.04

Online
ISSN
2191-0294
See all formats and pricing
In This Section
Volume 18, Issue 1 (Feb 2017)

Issues

Compact Discrete Gradient Schemes for Nonlinear Schrödinger Equations

Xiuling Yin
  • Corresponding author
  • School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
  • School of mathematical sciences, Dezhou University, Dezhou 253023, China
  • Email:
/ Chengjian Zhang
  • Corresponding author
  • School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
  • Email:
/ Jingjing Zhang
  • School of Science, East China Jiaotong University, Nanchang 330013, Jiangxi, China
Published Online: 2017-01-19 | DOI: https://doi.org/10.1515/ijnsns-2014-0064

Abstract

This paper proposes two schemes for a nonlinear Schrödinger equation with four-order spacial derivative by using compact scheme and discrete gradient methods. They are of fourth-order accuracy in space. We analyze two discrete invariants of the schemes. The numerical experiments are implemented to investigate the efficiency of the schemes.

Keywords: Schrödinger equation; compact scheme; discrete gradient method; discrete invariant

MSC 2010: 65M06; 65M12; 65Z05; 70H15

References

  • [1] He J., Homotopy perturbation method for bifurcation of nonlinear problems. Int. J. Nonlinear Sci. Numer. Simul. 6 (2005), 207–208.

  • [2] He J. and Wu H., Exp-function method for nonlinear wave equations. Chaos, Solitons Fractals 3 (2006), 700–708.

  • [3] He J. and Wu H., Construction of solitary solution and compacton-like solution by variational iteration method. Chaos, Solitons Fractals 1 (2006), 108–113.

  • [4] Abdou M. and Soliman A., Variational iteration method for solving Burger’s and coupled Burger’s equations. J. Comput. Appl. Math. 2 (2005), 245–251.

  • [5] He J., Variational approach for nonlinear oscillators. Chaos, Solitons Fractals 5 (2007), 1430–1439.

  • [6] He J., Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos, Solitons Fractals 4 (2004), 847–851.

  • [7] Phillips A., Introduction to Quantum Mechanics. Wiley, Chichester, 2003.

  • [8] Kong L., Hong J., Wang L. and Fang F., Symplectic integrator for nonlinear high order Schrödinger equation with a trapped term. J. Comput. Appl. Math. 231 (2009), 664–679.

  • [9] Hong J., Liu Y., Munthe-Kaas H. and Zanna A., Globally conservative properties and error estimation of a multi-symplectic scheme for Schrödinger equations with variable coefficients. Appl. Numer. Math. 56 (2006), 814–843.

  • [10] Hong J., Liu X. and Li C., Multi-symplectic Runge-Kutta- Nyström methods for nonlinear Schrödinger equations with variable coefficients. J. Comput. Phys. 226 (2007), 1968–1984.

  • [11] Xu Z., He J. and Han H., Semi-implicit operator splitting Padé method for higher-order nonlinear Schrödinger equations. Appl. Math. Comput. 179 (2006), 596–605.

  • [12] Chang Q., Jia E. and Sun W., Difference schemes for solving the generalized nonlinear Schrödinger equation. J. Comput. Phys. 148 (1999), 397–415.

  • [13] Lele S., Compact finite difference schemes with spectral-like solution. J. Comput. Phys. 103 (1992), 16–42.

  • [14] Fu Y., Compact fourth-order finite difference schemes for Helmholtz equation with high wave numbers. J. Comput. Math. 26 (2008), 98–111.

  • [15] Ma Y., Kong L. and Hong J., High-order compact splitting multisymplectic method for the coupled nonlinear Schrödinger equations. Comput. Math. Appl. 61 (2011), 319–333.

  • [16] Miyatake Y. and Matsuo T., Energy conservative/dissipative H1-Galerkin semi-discretizations for partial differential equations. AIP Conf. Proc. 1479 (2012), 1268–1271.

  • [17] Zhang J., Energy-preserving Sinc collocation method for Klein-Gordon-Schröinger equations. AIP Conf. Proc. 1479 (2012), 1272–1275.

  • [18] Kanazawa H., Matsuo T. and Yaguchi T., A conservative compact finite difference scheme for the KdV equation. JSIAM Lett. 4 (2012), 5–8.

About the article

Received: 2014-05-29

Accepted: 2016-12-14

Published Online: 2017-01-19

Published in Print: 2017-02-01


The authors are supported by the Director Innovation Foundation of ICMSEC and AMSS, the Foundation of CAS, the NNSFC (Nos. 11571128, 11501082, 11201125) and the NSF of Shandong Province (Nos. ZR2015AL016, ZR2016AQ07). The authors would like to thank the referees for valuable suggestions.



Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2014-0064. Export Citation

Comments (0)

Please log in or register to comment.
Log in