Skip to main content
SpringerLink
Log in

Split-step cubic B-spline collocation methods for nonlinear Schrödinger equations in one, two, and three dimensions with Neumann boundary conditions

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

Abstract

In this paper, split-step cubic B-spline collocation (SS3BC) schemes are constructed by combining the split-step approach with the cubic B-spline collocation (3BC) method for the nonlinear Schrödinger (NLS) equation in one, two, and three dimensions with Neumann boundary conditions. Unfortunately, neither of the advantages of the two methods can be maintained for the multi-dimensional problems, if one combines them in the usual manner. For overcoming the difficulty, new medium quantities are introduced in this paper to successfully reduce the multi-dimensional problems into one-dimensional ones, which are essential for the SS3BC methods. Numerical tests are carried out, and the schemes are verified to be convergent with second-order both in time and space. The proposed method is also compared with the split-step finite difference (SSFD) scheme. Finally, the present method is applied to two problems of the Bose-Einstein condensate. The proposed SS3BC method is numerically verified to be effective and feasible.

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
Fig. 3

Similar content being viewed by others

References

  1. Fairweather, G., Meade, D.: A survey of spline collocation methods for the numerical solution of differential equations. In: Diaz, J.C. (ed.) Mathematics for Large Scale Computing, Lecture Notes in Pure and Applid Mathematics, vol. 120. Marcel Dekker, New York (1989)

  2. Daǧ, İ., Saka, B., Irk, D.: Application of cubic B-splines for numerical solution of the RLW equation. Appl. Math. Comput. 159, 373–389 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Khalifa, A.K., Raslan, K.R., Alzubaidi, H.M.: A collocation method with cubic B-splines for solving the MRLW equation. J. Comput. Appl. Math. 212, 406–418 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Mittal, R.C., Bhatia, R.: Numerical solution of nonlinear system of Klein-Gordon equations by cubic B-spline collocation method. Inter. J. Comput. Math. 92(10), 2139–2159 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Pourgholi, R., Saeedi, A.: Applications of cubic B-splines collocation method for solving nonlinear inverse parabolic partial differential equations. Numer. Methods Partial Diff. Eq. 33, 88–104 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Wang, H.: Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations. Appl. Math. Comput. 170, 17–35 (2005)

    MathSciNet  MATH  Google Scholar 

  7. Wang, S., Zhang, L.: An efficient split-step compact finite difference method for cubic-quintic complex Ginzburg-Landau equations. Comput. Phys. Commun. 184, 1511–1521 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Wang, H., Ma, X., Lu, J., et al.: An efficient time-splitting compact finite difference method for Gross-Pitaevskii equation. Appl. Math. Comput. 297, 131–144 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Gao, Y., Mei, L., Li, R.: A time-splitting Galerkin finite element method for Davey-Stewartson equations. Comput. Phys. Commun. 197, 35–42 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bao, W., Shen, J.: A fourth-order time-splitting Laguerre-Hermite pseudospectral method for Bose-Einstein condensates. SIAM J. Sci. Comput. 26(6), 2010–2028 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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 

  12. Hofstätter, H., Koch, O., Thalhammer, M.: Convergence analysis of high-order time-splitting pseudo-spectral methods for rotational Gross-Pitaevskii equations. Numer. Math. 127, 315–364 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Wang, S., Zhang, L.: Split-step orthogonal spline collocation methods for nonlinear Schrödinger equations in one, two, and three dimensions. Appl. Math. Comput. 218, 1903–1916 (2011)

    MathSciNet  MATH  Google Scholar 

  14. Yağmurlu, N.M., Uçar, Y., Çelikkaya, İ. : Operator splitting for numerical solutions of the RLW equation. J. Appl. Anal. Comput. 8(5), 1494–1510 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Saka, B., Daǧ, İ.: Quartic B-spline collocation method to the numerical solutions of the Burgers’ equation. Chaos Soliton Fract. 32, 1125–1137 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Saka, B.: Algorithms for numerical solution of the modified equal width wave equation using collocation method. Math. Comput. Model. 45, 1096–1117 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Botella, O.: On a collocation B-spline method for the solution of the Navier-Stokes equations. Comput. Fluids 31, 397–420 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Johnson, R.W.: A B-spline collocation method for solving the incompressible Navier-Stokes equations using an hoc method: the boundary residual method. Comput. Fluids 34, 121–149 (2005)

    Article  MATH  Google Scholar 

  19. Hidayat, M.I.P., Ariwahjoedi, B., Parman, S.: B-spline collocation with domain decomposition method. J. Phys.: Conf. Ser. 423, 012012 (2013)

    Google Scholar 

  20. Taha, T.R., Ablowitz, M.J.: Analytical and numerical aspects of certain nonlinear Schrödinger equations. II: Numerical. J. Comput. Phys. 55, 203–230 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fairweather, G., Khebchareon, M.: Numerical Methods for Schrödinger-Type Problems, Trends in Industrial and Applied Mathematics. Kluwer Academic, Netherlands (2002)

    Google Scholar 

  22. Prenter, P.M.: Splines and Variational Methods. Wiley, New York (1975)

    MATH  Google Scholar 

  23. Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5(3), 506–517 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yu, S., Zhao, S., Wei, G.W.: Local spectral time splitting method for first- and second-order partial differential equations. J. Comput. Phys. 206, 727–780 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Fernandes, R.I.: Efficient orthogonal spline collocation methods for solving linear second order hyperbolic problems on rectangles. Numer. Math. 77, 223–241 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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 

Download references

Funding

This work is supported by the National Natural Science Foundation of China under Grant No. 11701280.

Author information

Authors and Affiliations

  1. College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Shanshan Wang & Luming Zhang

Authors
  1. Shanshan Wang
    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

Corresponding author

Correspondence to Shanshan 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

Wang, S., Zhang, L. Split-step cubic B-spline collocation methods for nonlinear Schrödinger equations in one, two, and three dimensions with Neumann boundary conditions. Numer Algor 81, 1531–1546 (2019). https://doi.org/10.1007/s11075-019-00762-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-019-00762-2

Keywords

Search

Navigation