Jump to ContentJump to Main Navigation
Show Summary Details
More options …

International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: 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) 2016: 0.251
Source Normalized Impact per Paper (SNIP) 2016: 0.624

Mathematical Citation Quotient (MCQ) 2016: 0.07

See all formats and pricing
More options …
Volume 18, Issue 1


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
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Chengjian Zhang
  • Corresponding author
  • School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Jingjing Zhang
Published Online: 2017-01-19 | DOI: https://doi.org/10.1515/ijnsns-2014-0064


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


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

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

  • [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.Google Scholar

  • [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.Google Scholar

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

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

  • [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.Google Scholar

  • [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.Google Scholar

  • [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.Google Scholar

  • [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.Google Scholar

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

  • [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.Google Scholar

  • [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.Google Scholar

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, Volume 18, Issue 1, Pages 1–7, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2014-0064.

Export Citation

©2017 by De Gruyter. Copyright Clearance Center

Comments (0)

Please log in or register to comment.
Log in