Skip to main content
SpringerLink
Log in

Error Analysis and Numerical Simulations of Strang Splitting Method for Space Fractional Nonlinear Schrödinger Equation

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

Abstract

In this paper, we study a fast explicit operator splitting method for space fractional nonlinear Schrödinger equation in one (1D), two (2D) and three dimensions (3D) with periodic boundary conditions. The equation is split into linear and nonlinear parts: the linear part is solved by the Fourier spectral method, which is based on the exact solution and thus has no stability restriction on the time-step size; the nonlinear subequation is then solved analytically due to the availability of a closed-form solution. The rigorous analysis of the discrete mass conservation principle and the convergence rate of the proposed algorithm are proved. The theoretical results show the proposed method is \(L^2\) unconditionally stable, second order accurate in time, whereas the spatial accuracy depends on the regularity of the solution. Numerical experiments for 1D, 2D and 3D cases demonstrate the efficiency of the proposed method.

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115(3), 485–491 (1959)

    Article  MathSciNet  Google Scholar 

  2. Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268, 298–305 (2000)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Guo, X., Xu, M.: Some physical applications of fractional Schrödinger equation. J. Math. Phys. 47, 082104 (2006)

    Article  MathSciNet  Google Scholar 

  5. Longhi, S.: Fractional Schrödinger equation in optics. Opt. Lett. 40, 1117 (2015)

    Article  Google Scholar 

  6. Pinsker, F., Bao, W., Zhang, Y., Ohadi, H., Dreismann, A., Baumberg, J.J.: Fractional quantum mechanics in polariton condensates with velocity dependent mass. Phys. Rev. B 92, 195310 (2015)

    Article  Google Scholar 

  7. Hu, Y., Kallianpur, G.: Schrödinger equations with fractional Laplacians. Appl. Math. Optim. 42, 281–290 (2000)

    Article  MathSciNet  Google Scholar 

  8. Chen, W., Pang, G.F.: A new definition of fractional Laplacian with application to modeling three-dimensional nonlocal heat conduction. J. Comput. Phys. 309(15), 350–367 (2016)

    Article  MathSciNet  Google Scholar 

  9. Yang, Q.Q., Turner, I., Liu, F.W., Ili’c, M.: Novel numerical methods for solving the time-space fractional diffusion equation in 2D. SIAM J. Sci. Comput. 33, 1159–1180 (2011)

    Article  MathSciNet  Google Scholar 

  10. Zhai, S.Y., Weng, Z.F., Feng, X.L.: Fast explicit operator splitting method and time-step adaptivity for fractional non-local Allen-Cahn model. Appl. Math. Model. 40(2), 1315–1324 (2016)

    Article  MathSciNet  Google Scholar 

  11. Song, F.Y., Xu, C.J., Karniadakis, G.E.: Computing fractional Laplacians on complex-geometry domains: algorithms and simulations. SIAM J. Sci. Comput. 39, A1320–A1344 (2017)

    Article  MathSciNet  Google Scholar 

  12. Wang, D.L., Xiao, A.G., 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  Google Scholar 

  13. Wang, D.L., Xiao, A.G., 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  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Ran, M.H., Zhang, C.J.: A conservative difference scheme for solving the strongly coupled nonlinear fractional Schrödinger equations. Commun. Nonlinear Sci. Numer. Simul. 41, 64–83 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  18. Aboelenen, T.: A high-order nodal discontinuous Galerkin method for nonlinear fractional Schrödinger type equations. Commun. Nonlinear Sci. Numer. Simul. 54, 428–452 (2018)

    Article  MathSciNet  Google Scholar 

  19. Bhrawy, A.H., Zaky, M.A.: An improved collocation method for multi-dimensional space time variable-order fractional Schrödinger equations. Appl. Numer. Math. 111, 197–218 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Cheng, Y.Z., Kurganov, A., Qu, Z.L., Tang, T.: Fast and stable explicit operator splitting methods for phase-field models. J. Comput. Phys. 303, 45–65 (2015)

    Article  MathSciNet  Google Scholar 

  22. Weng, Z.F., Zhuang, Q.Q.: Numerical approximation of the conservative Allen–Cahn equation by operator splitting method. Math. Method Appl. Sci. 40, 4462–4480 (2017)

    Article  MathSciNet  Google Scholar 

  23. Antoine, X.A., Bao, W.Z., Besse, C.: Computational methods for the dynamics of the nonlinear Schrodinger/Gross–Pitaevskii equations. Comput. Phys. Commun. 184, 2621–2633 (2013)

    Article  MathSciNet  Google Scholar 

  24. Bao, W.Z., Dong, X.C.: Numerical methods for computing ground state and dynamics of nonlinear relativistic Hartree equation for boson stars. J. Comput. Phys. 230, 5449–5469 (2011)

    Article  MathSciNet  Google Scholar 

  25. Bao, W., Cai, Y.: Mathematical theory and numerical methods for Bose–Einstein condensation. Kinet. Relat. Models 6, 1–135 (2013)

    Article  MathSciNet  Google Scholar 

  26. Shen, J., Tang, T., Wang, L.L.: Spectral Methods Algorithms: Analyses and Applications, 1st edn. Springer, Berlin (2010)

    Google Scholar 

  27. Antoine, X., Tang, Q.L., Zhang, J.W.: On the numerical solution and dynamical laws of nonlinear fractional Schrödinger/Gross–Pitaevskii equations. Int. J. Comput. Math. 95, 1423–1443 (2018)

    Article  MathSciNet  Google Scholar 

  28. Lubich, C.: On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations. Math. Comput. 77, 2141–2153 (2008)

    Article  Google Scholar 

  29. Thalhammer, M.: Convergence analysis of high-order time-splitting pseudospectral methods for nonlinear Schrödinger equations. SIAM J. Numer. Anal. 50, 3231–3258 (2012)

    Article  MathSciNet  Google Scholar 

  30. Einkemmer, L., Ostermann, A.: Convergence analysis of Strang splitting for Vlasov-type equations. SIAM J. Numer. Anal. 52, 140–155 (2014)

    Article  MathSciNet  Google Scholar 

  31. Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2011)

    Book  Google Scholar 

  32. Ainsworth, M., Mao, Z.P.: Analysis and approximation of a fractional Cahn–Hilliard equation. SIAM J Numer Anal. 55, 1689–1718 (2017)

    Article  MathSciNet  Google Scholar 

  33. Hu, J.Q., Xin, J., Lu, H.: The global solution for a class of systems of fractional nonlinear Schrödinger equations with periodic boundary condition. Comput. Math. Appl. 62, 1510–1521 (2011)

    Article  MathSciNet  Google Scholar 

  34. Hong, Y., Yang, C.: Strong convergence for discrete nonlinear Schrödinger equations in the Continuum Limit. SIAM J. Math. Anal. 51, 1297–1320 (2019)

    Article  MathSciNet  Google Scholar 

  35. Felmer, P., Quaas, A., Tan, J.: Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinb. Sect. A 142, 1237–1262 (2012)

    Article  Google Scholar 

  36. Jahnke, T., Lubich, C.: Error bounds for exponential operator splitting. BIT Numer. Math. 40, 735–744 (2000)

    Article  MathSciNet  Google Scholar 

  37. Willoughby, M.R.: High-order time-adaptive numerical methods for the Allen–Cahn and Cahn–Hilliard equations. Master’s thesis, Mathematics, University of British Columbia, Vancouver (2011)

  38. Zhang, Y.Q., Liu, X., Belić, M.R., Zhong, W.P., Zhang, Y.P., Xiao, M.: Propagation dynamics of a light beam in a fractional Schrödinger equation. Phys. Rev. Lett. 115, 180403 (2015)

    Article  Google Scholar 

  39. Tian, Z.F., Yu, P.X.: High-order compact ADI (HOC-ADI) method for solving unsteady 2D Schrödinger equation. Comput. Phys. Commun. 181, 861–868 (2010)

    Article  Google Scholar 

  40. 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, 382–399 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to gratefully thank Professor Wei Wei from Northwest University for the helpful discussions on the regularity of SFNLS equation in this work. The authors also thank the anonymous referees for their valuable comments and suggestions, which were very helpful in improving the quality of the paper.

Author information

Authors and Affiliations

  1. Fujian Province University Key Laboratory of Computation Science, School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, People’s Republic of China

    Shuying Zhai & Zhifeng Weng

  2. Center for Nonlinear Studies, School of Mathematics, Northwest University, Xi’an, 710127, Shaanxi, People’s Republic of China

    Dongling Wang

  3. School of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Xuan Zhao

Authors
  1. Shuying Zhai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dongling Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhifeng Weng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xuan Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xuan Zhao.

Additional information

Publisher's Note

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

This work is in part supported by the National Natural Science Foundation of China (Nos. 11701196, 11871057, 11501447, 11701197, 11701081, 1186010285), the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (No. ZQN-YX502), the Fundamental Research Funds for the Central Universities (Nos. ZQN-702, 2242019K40111), Natural Science Youth Foundation of Jiangsu Province (No. BK20160660), the Jiangsu Provincial Key Laboratory of Networked Collective Intelligence (No. BM2017002), Key Project of Natural Science Foundation of China (No. 61833005).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhai, S., Wang, D., Weng, Z. et al. Error Analysis and Numerical Simulations of Strang Splitting Method for Space Fractional Nonlinear Schrödinger Equation. J Sci Comput 81, 965–989 (2019). https://doi.org/10.1007/s10915-019-01050-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-019-01050-w

Keywords

Mathematics Subject Classification

Search

Navigation