[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Two energy-conserving and compact finite difference schemes for two-dimensional Schrödinger-Boussinesq equations

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

Abstract

In this paper, we study two compact finite difference schemes for the Schrödinger-Boussinesq (SBq) equations in two dimensions. The proposed schemes are proved to preserve the total mass and energy in the discrete sense. In our numerical analysis, besides the standard energy method, a “cut-off” function technique and a “lifting” technique are introduced to establish the optimal H1 error estimates without any restriction on the grid ratios. The convergence rate is proved to be of O(τ2 + h4) with the time step τ and mesh size h. In addition, a fast finite difference solver is designed to speed up the numerical computation of the proposed schemes. The numerical results are reported to verify the error estimates and conservation laws.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Makhankov, V.G.: On stationary solutions of the Schrödinger equation with a self-consistent potential satisfying Boussinesq’s equation. Phys. Lett. 50, 42–44 (1974)

    Article  Google Scholar 

  2. Yajima, N., Satsuma, J.: Soliton solutions in a diatomic lattice system. Prog. Theor. Phys. 62, 370–378 (1979)

    Article  Google Scholar 

  3. Rao, N.N.: Coupled scalar field equations for nonlinear wave modulations in dispersive media. Pramana J. Pyhs. 46, 161–202 (1991)

    Article  Google Scholar 

  4. Zhang, L.M., Bai, D.M., Wang, S.S.: Numerical analysis for a conservative difference scheme to solve the Schrödinger-Boussinesq equation. J. Comput. Appl. Math. 235, 4899–4915 (2011)

    Article  MathSciNet  Google Scholar 

  5. Liao, F., Zhang, L.M.: Conservative compact finite difference scheme for the coupled Schrödinger-Boussinesq equation. Numer. Methods Partial Differ. Equ. 32, 1667–1688 (2016)

    Article  Google Scholar 

  6. Liao, F., Zhang, L.M.: Numerical analysis of a conservative linear compact difference scheme for the coupled Schrödinger-Boussinesq equations. Inter. J. Comput. Math. 95, 961–978 (2018)

    Article  Google Scholar 

  7. Liao, F., Zhang, L.M., Wang, T.C.: Unconditional \(l^{\infty }\) convergence of a conservative compact finite difference scheme for the N-coupled Schrödinger-Boussinesq equations. Appl. Numer. Math. 138, 54–77 (2019)

    Article  MathSciNet  Google Scholar 

  8. Zheng, J.D., Xiang, X.M.: The finite element analysis for the equation system coupling the complex Schrödinger and real Boussinesq fields. Math. Numer. Sin. 2, 344–355 (1984). (in Chinese)

    Google Scholar 

  9. Bai, D.M., Zhang, L.M.: The quadratic B-spline finite element method for the coupled Schrödinger-Boussinesq equations. Inter. J. Comput. Math. 88, 1714–1729 (2011)

    Article  Google Scholar 

  10. Huang, L.Y., Jiao, Y.D., Liang, D.M.: Multi-sympletic scheme for the coupled Schrödinger-Boussinesq equations. Chin. Phys. B. 22, 1–5 (2013)

    Google Scholar 

  11. Bai, D.M., Wang, J.L.: The time-splitting Fourier spectral method for the coupled Schrödinger -Boussinesq equations. Commun. Nonlinear Sci. Numer. Simulat. 17, 1201–1210 (2012)

    Article  Google Scholar 

  12. Liao, F., Zhang, L.M., Wang, S.S.: Time-splitting combined with exponential wave integrator fourier pseudospectral method for Schrödinger-Boussinesq system. Commun. Nonlinear Sci. Numer. Simulat. 55, 93–104 (2018)

    Article  Google Scholar 

  13. Liao, F., Zhang, L.M., Wang, S.S.: Numerical analysis of cubic orthogonal spline collocation methods for the coupled Schrödinger-Boussinesq equations. Appl. Numer. Math. 119, 194–212 (2017)

    Article  MathSciNet  Google Scholar 

  14. Cai, J., Yang, B., Zhang, C.: Efficient mass and energy preserving schemes for the coupled nonlinear Schrödinger-Boussinesq system. Appl. Math. Lett. 91, 76–82 (2019)

    Article  MathSciNet  Google Scholar 

  15. Gao, Z., Xie, S.: Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations. Appl. Numer. Math. 61, 593–614 (2011)

    Article  MathSciNet  Google Scholar 

  16. Wang, T.C., Guo, B.L., Xu, Q.B.: Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions. J. Comput. Phys. 243, 383–399 (2013)

    MATH  Google Scholar 

  17. Liao, H.L., Sun, Z.Z., Shi, H.S.: Error estimate of fourth-order compact scheme for linear Schrödinger equations. SIAM J. Numer. Anal. 47, 4381–4401 (2010)

    Article  MathSciNet  Google Scholar 

  18. Bao, W.Z., Cai, Y.Y.: Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation. Math. Comput. 281, 99–128 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Wang, T.C., Zhao, X.F.: Optimal \(l^{\infty }\) error estimates of finite difference methods for the coupled Gross-Pitaevskii equations in high dimensions. Sci. China Math. 57, 2189–2214 (2014)

    Article  MathSciNet  Google Scholar 

  20. Bao, W.Z., Cai, Y.Y.: Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator. SIAM J. Numer. Anal. 50, 492–521 (2012)

    Article  MathSciNet  Google Scholar 

  21. Bao, W.Z., Cai, Y.Y.: Uniform and optimal error estimates of an exponential wave integrator Sine pseudospectral method for the nonlinear Schrödinger equation with wave operator. SIAM J. Numer. Anal. 52, 1103–1127 (2014)

    Article  MathSciNet  Google Scholar 

  22. Bao, W., Dong, X.: Analysis and comparison of numerical methods for Klein-Gordon equation in nonrelativistic limit regime. Numer. Math. 120, 189–229 (2012)

    Article  MathSciNet  Google Scholar 

  23. Akrivis, G., Dougalis, V., Karakashian, O.: On fully discrete Galerkin methods of second order temporal accuracy for the nonlinear Schrödinger equation. Numer. Math. 59, 31–53 (1991)

    Article  MathSciNet  Google Scholar 

  24. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (1997)

    Book  Google Scholar 

  25. Wang, T.C., Zhao, X.F., Jiang, J.P.: Unconditional and optimal H2 error estimate of two linear and conservative finite difference schemes for the Klein-Gordon-Schrödinger equation in high dimensions. Adv. Comput. Math. 44, 477–503 (2018)

    Article  MathSciNet  Google Scholar 

  26. Wang, T.C., Jiang, J.P., Xue, X.: Unconditional and optimal H1 error estimate of a Crank-Nicolson finite difference scheme for the nonlinear Schrödinger equation. J. Math. Anal. Appl. 459, 945–958 (2018)

    Article  MathSciNet  Google Scholar 

  27. Gong, Y.Z., Wang, Q., Wang, Y.S., Cai, J.X.: A conservative Fourier pseudo-spectral method for the nonlinear Schrödinger equation. J. Comput. Phys. 328, 354–370 (2017)

    Article  MathSciNet  Google Scholar 

  28. Gray, R.M.: Toeplitz and circulant matrix, ISL, Tech. Rep. Stanford Univ. , CA, Aug 2002 [Online]. Avaliable: http://ee-www.standford.edu/gray/toeplitz.html

  29. Zhou, Y.L.: Application of Discrete Functional Analysis to the Finite Difference Method. International Academic publishers (1990)

  30. Sun, Z.Z.: A note on finite difference method for generalized Zakharov equations. J. South. Univ. (English Edition) 16 (2) (2000)

  31. Wang, T.C., Guo, B.L.: Unconditional convergence of two conservative compact difference schemes for nonlinear Schrödinger equation in one dimension. Sci. Sin. Math. 41, 207–233 (2011). (in Chinese)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions that helped greatly to improved the quality of this article.

Funding

This work is financially supported by the National Natural Science Foundation of China under Grant No.11571181.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Changshu Institute of Technology, Changshu, 215500, China

    Feng Liao

  2. College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China

    Luming Zhang

  3. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Tingchun Wang

Authors
  1. Feng Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luming Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tingchun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tingchun Wang.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liao, F., Zhang, L. & Wang, T. Two energy-conserving and compact finite difference schemes for two-dimensional Schrödinger-Boussinesq equations. Numer Algor 85, 1335–1363 (2020). https://doi.org/10.1007/s11075-019-00867-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-019-00867-8

Keywords

Search

Navigation