Skip to main content
SpringerLink
Log in

The locally extrapolated exponential splitting scheme for multi-dimensional nonlinear space-fractional Schrödinger equations

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

An efficient local extrapolation of the exponential operator splitting scheme is introduced to solve the multi-dimensional space-fractional nonlinear Schrödinger equations. Stability of the scheme is examined by investigating its amplification factor and by plotting the boundaries of the stability regions. Empirical convergence analysis and calculation of the local truncation error exhibit the second-order accuracy of the proposed scheme. The performance and reliability of the proposed scheme are tested by implementing it on two- and three-dimensional space-fractional nonlinear Schrödinger equations including the space-fractional Gross-Pitaevskii equation, which is used to model optical solitons in graded-index fibers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alkahtani, B.S.T.: Chua circuit model with Atangana-Baleanu derivative with fractional order. Chaos. Soliton. Fract. 89, 547–551 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Antoine, X., Tang, Q., Zhang, Y.: On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions. J. Comput. Phys. 325, 74–97 (2016)

  3. Atangana, A., Cloot, A.H.: Stability and convergence of the space fractional variable-order Schrödinger equation. Adv. Differ. Equ. 80, 1–10 (2013)

  4. Atangana, A., Baleanu, D.: New fractional derivatives with non-local and non-singular kernel, theory and application to heat transfer model. Therm. Sci. 20, 763–769 (2016)

    Article  Google Scholar 

  5. Atangana, A., Koca, I.: Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order. Chaos. Soliton. Fract. 89, 447–454 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bao, W., Jaksch, D., Markowich, P.A.: Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation. J. Comput. Phys. 187, 318–342 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Besse, C., Mauser, N.J., Stimming, H.P.: Numerical study of the Davey-Stewartson system. ESAIM: M2AN 38, 1035–1054 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Beylkin, G., Keiser, J.M., Vozovoi, L.: A new class of time discretization schemes for the solution of nonlinear PDEs. J. Comp. Phys. 147, 362–387 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bhatt, H.P., Khaliq, A.Q.M.: The locally extrapolated exponential time differencing LOD scheme for multidimensional reaction-diffusion systems. J. Comput. Appl. Math. 285, 256–278 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, M., Wang, Y., Cheng, X., Deng, W.: Second-order LOD multigrid method for multidimensional Riesz fractional diffusion equation. BIT Numer. Math. 54, 623–647 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cox, S.M., Matthews, P.C.: Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430–455 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Du, Q., Zhu, W.: Analysis and applications of the exponential time differencing schemes and their contour integration modifications. BIT Numer. Math. 45, 307–328 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jia, J., Wang, H.: Fast finite difference methods for space-fractional diffusion equations with fractional derivative boundary conditions. J. Comput. Phys. 293, 359–369 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Khaliq, A.Q.M., Liang, X., Furati, K.M.: A fourth-order implicit-explicit scheme for the space fractional nonlinear Schrödinger equations. Numer. Algor. doi:10.1007/s11075-016-0200-1 (2016)

  15. Lawson, J.D., Morris, J.L.I.: The extrapolation of first order methods for parabolic partial differential equations. SIAM J. Numer. Anal. 15, 1212–1224 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  16. Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E 66, 056108 (2002)

    Article  MathSciNet  Google Scholar 

  17. Norsett, S.P., Wolfbrandt, A.: Attainable order of rational approximations to the exponential function with only real poles. BIT 17, 200–208 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ortigueira, M.D.: Riesz potential operators and inverses via fractional centred derivatives. 1C12 (2006)

  19. Pitaevskii, L., Stringari, S.: Bose-Einstein Condensation. Oxford University Press, Oxford (2003)

    MATH  Google Scholar 

  20. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  21. Podlubny, I., Chechkin, A., Skovranek, T., Chen, Y., Jara, B.M.V.: Matrix approach to discrete fractional calculus II: partial fractional differential equations. J. Comput. Phys. 228, 3137C3153 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Renninger, W.H., Wise, F.W.: Optical solitons in graded-index multimode fiber. Nat. Commun. 4, Article number: 1719, doi:10.1038/ncomms2739

  23. Ran, M., Zhang, C.: A conservative difference scheme for solving the strongly coupled nonlinear fractional Schrödinger equations. Commun. Nonlinear. Sci. Numer. Simulat 41, 64C83 (2016)

    Article  Google Scholar 

  24. Wang, H., Du, N.: Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations. J. Comput. Phys. 258, 305–318 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wang, P., Huang, C.: Split-step alternating direction implicit difference scheme for the fractional Schrödinger equation in two dimensions. Comput. Math. Appl. 71, 1114–1128 (2016)

    Article  MathSciNet  Google Scholar 

  26. Wang, P., Huang, C., Zhao, L.: Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation. J. Comput. Appl. Math. 306, 231–247 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang, D., Xiao, A., Yang, W.: A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations. J. Comput. Phys. 272, 644–655 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Yu, B., Jiang, X.Y., Xu, H.Y.: A novel compact numerical method for solving the two-dimensional non-linear fractional reaction-subdiffusion equation. Numer. Algor. 68, 923–950 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zaslavsky, G.M.: Chaos, fractional kinetics, and anomalous transport. Phy. Rep. 371, 461–580 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhao, X., Sun, Z., Hao, Z.: A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrödinger equation. SIAM J. SCI. Comput. 36, A2865–A2886 (2014)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Computer Science, Hubei University of Arts and Science, Xiangyang, Hubei, 441053, People’s Republic of China

    X. Liang

  2. Department of Mathematical Sciences and Center for Computational Science, Middle Tennessee State University, Murfreesboro, TN, 37132-0001, USA

    A. Q. M. Khaliq

  3. Department of Mathematics and Statistics, Texas A&M University, Corpus Christi, TX, 78412, USA

    H. Bhatt

  4. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    K. M. Furati

Authors
  1. X. Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Q. M. Khaliq
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Bhatt
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. K. M. Furati
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to X. Liang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liang, X., Khaliq, A.Q.M., Bhatt, H. et al. The locally extrapolated exponential splitting scheme for multi-dimensional nonlinear space-fractional Schrödinger equations. Numer Algor 76, 939–958 (2017). https://doi.org/10.1007/s11075-017-0291-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-017-0291-3

Keywords

Search

Navigation