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Two second-order and linear numerical schemes for the multi-dimensional nonlinear time-fractional Schrödinger equation

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Abstract

This paper presents two second-order and linear finite element schemes for the multi-dimensional nonlinear time-fractional Schrödinger equation. In the first numerical scheme, we adopt the L2-1σ formula to approximate the Caputo derivative. However, this scheme requires storing the numerical solution at all previous time steps. In order to overcome this drawback, we develop the \(\mathcal {F}L2\)-1σ formula to construct the second numerical scheme, which reduces the computational storage and cost. We prove that both the L2-1σ and \(\mathcal {F}L2\)-1σ formulas satisfy the three assumptions of the generalized discrete fractional Grönwall inequality. Furthermore, combining with the temporal-spatial error splitting argument, we rigorously prove the unconditional stability and optimal error estimates of these two numerical schemes, which do not require any time-step restrictions dependent on the spatial mesh size. Numerical examples in two and three dimensions are given to illustrate our theoretical results and show that the second scheme based on \(\mathcal {F}L2\)-1σ formula can reduce CPU time significantly compared with the first scheme based on L2-1σ formula.

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References

  1. Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)

    Article  MathSciNet  Google Scholar 

  2. Antoine, X., Bao, W., Besse, C.: Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations. Comput. Phys. Commun. 184(12), 2621–2633 (2013)

    Article  MathSciNet  Google Scholar 

  3. Antoine, X., Tang, Q., Zhang, J.: On the numerical solution and dynamical laws of nonlinear fractional Schrödinger/Gross-Pitaevskii equations. Int. J. Comput. Math. 95(6-7), 1423–1443 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Bhrawy, A.H., Abdelkawy, M.A.: A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations. J. Comput. Phys. 294, 462–483 (2015)

    Article  MathSciNet  Google Scholar 

  7. Chen, X., Di, Y., Duan, J., Li, D.: Linearized compact ADI schemes for nonlinear time-fractional Schrödinger equations. Appl. Math. Lett. 84, 160–167 (2018)

    Article  MathSciNet  Google Scholar 

  8. Cheng, M.: Bound state for the fractional Schrodinger̈ equation with unbounded potential. J. Math. Phys. 53(4), 043,507 (2012)

    Article  MathSciNet  Google Scholar 

  9. Gao, G., Alikhanov, A.A., Sun, Z.: The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations. J. Sci. Comput. 73 (1), 93–121 (2017)

    Article  MathSciNet  Google Scholar 

  10. Gao, Y., Mei, L.: Implicit-explicit multistep methods for general two-dimensional nonlinear Schrödinger equations. Appl. Numer. Math. 109, 41–60 (2016)

    Article  MathSciNet  Google Scholar 

  11. Guo, B., Han, Y., Xin, J.: Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation. Appl. Math. Comput. 204(1), 468–477 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Guo, B., Huo, Z.: Well-posedness for the nonlinear fractional Schrodinger̈ equation and inviscid limit behavior of solution for the fractional Ginzburg-Landau equation. Fract. Calc. Appl. Anal. 16(1), 226–242 (2013)

    Article  MathSciNet  Google Scholar 

  13. Guo, X., Xu, M.: Some physical applications of fractional Schrodinger̈ equation. J. Math. Phys. 47(8), 082,104 (2006)

    Article  MathSciNet  Google Scholar 

  14. Hu, J., Xin, J., Lu, H.: The global solution for a class of systems of fractional nonlinear Schrodinger̈ equations with periodic boundary condition. Comput. Math. Appl. 62(3), 1510–1521 (2011)

    Article  MathSciNet  Google Scholar 

  15. Iomin, A.: Fractional-time Schrödinger equation: fractional dynamics on a comb. Chaos Solitons Fractals 44(4-5), 348–352 (2011)

    Article  MathSciNet  Google Scholar 

  16. Laskin, N.: Fractional quantum mechanics. Phys. Rev. E. 62(3), 3135 (2000)

    Article  MathSciNet  Google Scholar 

  17. Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A. 268(4), 298–305 (2000)

    Article  MathSciNet  Google Scholar 

  18. Li, D., Wang, J., Zhang, J.: Unconditionally convergent L1-Galerkin FEMs for nonlinear time-fractional Schrödinger equations. SIAM J. Sci. Comput. 39(6), A3067–A3088 (2017)

    Article  Google Scholar 

  19. Liao, H., McLean, W., Zhang, J.: A discrete Grönwall inequality with applications to numerical schemes for subdiffusion problems. SIAM J. Numer. Anal. 57(1), 218–237 (2019)

    Article  MathSciNet  Google Scholar 

  20. Liao, H., Mclean, W., Zhang, J.: A second-order scheme with nonuniform time steps for a linear reaction-sudiffusion problem. arXiv:1803.09873v4(2019)

  21. Longhi, S.: Fractional Schrodinger̈ equation in optics. Opt. Lett. 40(6), 1117–1120 (2015)

    Article  Google Scholar 

  22. Mohebbi, A., Abbaszadeh, M., Dehghan, M.: The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics. Eng. Anal. Bound. Elem. 37(2), 475–485 (2013)

    Article  MathSciNet  Google Scholar 

  23. Naber, M.: Time fractional Schrödinger equation. J. Math. Phys. 45(8), 3339–3352 (2004)

    Article  MathSciNet  Google Scholar 

  24. Petroni, N.C., Pusterla, M.: Levy processes and Schrödinger equatioń. Phys. A. 388(6), 824–836 (2009)

    Article  MathSciNet  Google Scholar 

  25. Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations to methods of their solution and some of their applications, vol. 198. Academic press, San Diego (1998)

    Google Scholar 

  26. Secchi, S.: Ground state solutions for nonlinear fractional Schrodinger̈ equations in RN. J. Math. Phys. 54(3), 031,501 (2013)

    Article  Google Scholar 

  27. Secchi, S., Squassina, M.: Soliton dynamics for fractional Schrodinger̈ equations. Appl. Anal. 93(8), 1702–1729 (2014)

    Article  MathSciNet  Google Scholar 

  28. Stickler, B.A.: Potential condensed-matter realization of space-fractional quantum mechanics: The one-dimensional Levý crystal. Phys. Rev. E. 88(1), 012,120 (2013)

    Article  Google Scholar 

  29. Stynes, M.: Too much regularity may force too much uniqueness. Fract. Calc. Appl. Anal. 19(6), 1554–1562 (2016)

    Article  MathSciNet  Google Scholar 

  30. Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55(2), 1057–1079 (2017)

    Article  MathSciNet  Google Scholar 

  31. Wang, Y., Li, Q., Mei, L.: A linear, symmetric and energy-conservative scheme for the space-fractional Klein-Gordon-Schrödinger equations. Appl. Math. Lett. 95, 104–113 (2019)

    Article  MathSciNet  Google Scholar 

  32. Wang, Y., Mei, L., Li, Q., Bu, L.: Split-step spectral Galerkin method for the two-dimensional nonlinear space-fractional Schrodinger̈ equation. Appl. Numer. Math. 136, 257–278 (2019)

    Article  MathSciNet  Google Scholar 

  33. Wei, L., He, Y., Zhang, X., Wang, S.: Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrödinger equation. Finite Elem. Anal. Des. 59, 28–34 (2012)

    Article  MathSciNet  Google Scholar 

  34. Yan, Y., Sun, Z.Z., Zhang, J.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations: a second-order scheme. Commun. Comput. Phys. 22(4), 1028–1048 (2017)

    Article  MathSciNet  Google Scholar 

  35. Zhang, J., Wang, J., Zhou, Y.: Numerical analysis for time-fractional Schrödinger equation on two space dimensions. Adv. Difference Equ. 53, 16 (2020)

    Google Scholar 

  36. Zheng, M., Liu, F., Jin, Z.: The global analysis on the spectral collocation method for time fractional Schrödinger equation. Appl. Math. Comput. 365(124), 689 (2020)

    Google Scholar 

  37. Zhu, X., Yuan, Z., Wang, J., Nie, Y., Yang, Z.: Finite element method for time-space-fractional Schrödinger equation. Electron. J. Differential Equations 166, 18 (2017)

    MATH  Google Scholar 

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Funding

This work was supported by the Science Challenge Project (No.TZ2016002) and the Fundamental Research Funds for the Central Universities (No. G2019KY05104).

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Authors and Affiliations

  1. School of Science, Xi’an University of Architecture and Technology, Xi’an, Shaanxi, 710055, People’s Republic of China

    Ying Wang

  2. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, Shaanxi, 710129, People’s Republic of China

    Gang Wang

  3. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, People’s Republic of China

    Linlin Bu & Liquan Mei

Authors
  1. Ying Wang
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  2. Gang Wang
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Correspondence to Liquan Mei.

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Wang, Y., Wang, G., Bu, L. et al. Two second-order and linear numerical schemes for the multi-dimensional nonlinear time-fractional Schrödinger equation. Numer Algor 88, 419–451 (2021). https://doi.org/10.1007/s11075-020-01044-y

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