Skip to main content
SpringerLink
Log in

Optimal Error Estimates of SAV Crank–Nicolson Finite Element Method for the Coupled Nonlinear Schrödinger Equation

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, we reformulate the coupled nonlinear Schrödinger (CNLS) equation by using the scalar auxiliary variable (SAV) approach and solve the resulting system by using Crank-Nicolson finite element method. The fully-discrete method is proved to be mass- and energy-conserved. However, if the convergence results are investigated by using the classical way, the presence of \(u_t\) and \(v_t\) in the equation of \(r'(t)\) may lead to not only a consistency error of sub-optimal order in time but also some difficulties in analysing the numerical stability. The mentioned difficulties are overcome technically by estimating the difference quotient of the error in the \(H^{-1}\)-norm and carefully analysising the connections of errors between the couple systems. Consequently, the numerical solution is shown to be convergent at the order of \({\mathcal {O}}( \tau ^2 + h^p) \) in the \(H^1\)-norm with time step \(\tau \), mesh size h and the degree of finite elements p. Several numerical examples are presented to confirm our theoretical results.

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

Fig. 1
Fig. 2

Similar content being viewed by others

Data availability

Enquiries about data availability should be directed to the authors.

References

  1. Aydın, A.: Multisymplectic integration of \(N\)-coupled nonlinear Schrödinger equation with destabilized periodic wave solutions. Chaos Solit. Fract. 41(2), 735–751 (2009)

    Article  MATH  Google Scholar 

  2. Akrivis, G.D.: Finite difference discretization of the cubic Schrödinger equation. IMA J. Numer. Anal. 13(1), 115–124 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Akrivis, G.D., Dougalis, V.A., Karakashian, O.A.: On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation. Numer. Math. 59(1), 31–53 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Akrivis, G., Li, B., Li, D.: Energy-decaying extrapolated RK-SAV methods for the Allen-Cahn and Cahn-Hilliard equations. SIAM J. Sci. Comput. 41(6), A3703–A3727 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Akrivis, G., Li, D.: Structure-preserving Gauss methods for the nonlinear Schrödinger equation. Calcolo 58(2), 17 (2021)

    Article  MATH  Google Scholar 

  6. Besse, C.: A relaxation scheme for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 42(3), 934–952 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Barletti, L., Brugnano, L., Frasca, Caccia G., Iavernaro, F.: Energy-conserving methods for the nonlinear Schrödinger equation. Appl. Math. Comput. 318, 3–18 (2018)

    MathSciNet  MATH  Google Scholar 

  8. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (2008)

    Book  MATH  Google Scholar 

  9. Cai, W., Jiang, C., Wang, Y., Song, Y.: Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions. J. Comput. Phys. 395, 166–185 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, H., Mao, J., Shen, J.: Optimal error estimates for the scalar auxiliary variable finite-element schemes for gradient flows. Numer. Math. 145(1), 167–196 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cai, J., Shen, J.: Two classes of linearly implicit local energy-preserving approach for general multi-symplectic Hamiltonian PDEs. J. Comput. Phys. 401, 108975 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. Delfour, M., Fortin, M., Payre, G.: Finite-difference solutions of a nonlinear Schrödinger equation. J. Comput. Phys. 44(2), 277–288 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  13. Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

  14. Feng, X., Li, B., Ma, S.: High-order mass- and energy-conserving SAV-Gauss collocation finite element methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 59(3), 1566–1591 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  15. Feng, X., Liu, H., Ma, S.: Mass- and energy-conserved numerical schemes for nonlinear Schrödinger equations. Commun. Comput. Phys. 26(5), 1365–1396 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Hong, J., Liu, Y., Munthe-Kaas, H., Zanna, A.: Globally conservative properties and error estimation of a multi-symplectic scheme for Schrödinger equations with variable coefficients. Appl. Numer. Math. 56(6), 814–843 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kanna, T., Lakshmanan, M.: Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations. Phys. Rev. Lett. 86, 5043–5046 (2001)

    Article  Google Scholar 

  19. Kemmochi, T., Sato, S.: Scalar auxiliary variable approach for conservative/dissipative partial differential equations with unbounded energy functionals. BIT 62(3), 903–930 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, H., Guo, Y.: Numerical solution of coupled nonlinear Schrödinger equations on unbounded domains. Appl. Math. Lett. 104, 106286 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, D., Li, X., Zhang, Z.: Implicit-explicit relaxation Runge-Kutta methods: construction, analysis and applications to PDEs. Math. Comput. 92(339), 117–146 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, D., Sun, W.: Linearly implicit and high-order energy-conserving schemes for nonlinear wave equations. J. Sci. Comput. 83(3), 65 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, X., Shen, J., Rui, H.: Energy stability and convergence of SAV block-centered finite difference method for gradient flows. Math. Comput. 88(319), 2047–2068 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, X., Wen, J., Li, D.: Mass- and energy-conserving difference schemes for nonlinear fractional Schrödinger equations. Appl. Math. Lett. 111, 106686 (2021)

    Article  MATH  Google Scholar 

  25. Sanz-Serna, J.M.: Methods for the numerical solution of the nonlinear Schroedinger equation. Math. Comput. 43(167), 21–27 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407–416 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61(3), 474–506 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Berlin (2006)

    MATH  Google Scholar 

  29. Wang, J.: A new error analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation. J. Sci. Comput. 60(2), 390–407 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang, T., Guo, B., Zhang, L.: New conservative difference schemes for a coupled nonlinear Schrödinger system. Appl. Math. Comput. 217(4), 1604–1619 (2010)

    MathSciNet  MATH  Google Scholar 

  31. Wang, P., Huang, C.: An energy conservative difference scheme for the nonlinear fractional Schrödinger equation. J. Comput. Phys. 293, 238–251 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, S., Wang, T., Zhang, L.: Numerical computations for \(N\)-coupled nonlinear Schrödinger equations by split step spectral methods. Appl. Math. Comput. 222, 438–452 (2013)

    MathSciNet  MATH  Google Scholar 

  33. Wang, D., Xiao, A., Yang, W.: Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative. J. Comput. Phys. 242, 670–681 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Dongfang Li & Xiaoxi Li

  2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, China

    Dongfang Li

  3. Department of Mathematics, University of Macau, Macao, China

    Hai-wei Sun

Authors
  1. Dongfang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoxi Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hai-wei Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaoxi Li.

Ethics declarations

Conflict of interest

The authors have not disclosed any competing interests.

Additional information

Publisher's Note

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

This work is supported in part by NSFC (No. 11771162, 12231003) and research grants of the Science and Technology Development Fund, Macau SAR (file no. 0122/2020/A3), and MYRG2020-00224-FST from University of Macau.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, D., Li, X. & Sun, Hw. Optimal Error Estimates of SAV Crank–Nicolson Finite Element Method for the Coupled Nonlinear Schrödinger Equation. J Sci Comput 97, 71 (2023). https://doi.org/10.1007/s10915-023-02384-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-023-02384-2

Keywords

Mathematics Subject Classification

Search

Navigation