Advertisement

A conservative numerical method for the fractional nonlinear Schröinger equation in two dimensions

  • Rongpei ZhangEmail author
  • Yong-Tao Zhang
  • Zhen Wang
  • Bo Chen
  • Yi Zhang
Articles
  • 6 Downloads

Abstract

This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schrödinger (NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grünwald-Letnikov difference (WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schrödinger (FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations (ODEs) in matrices formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson (CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor (cIIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O2). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up.

Keywords

fractional nonlinear Schrödinger equation weighted and shifted Grünwald-Letnikov difference compact integration factor method conservation 

MSC(2010)

65N06 35R11 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 61573008 and 61703290), the Foundation of LCP (Grant No. 6142A0502020717) and National Science Foundation of USA (Grant No. DMS-1620108).

References

  1. 1.
    Aboelenen T. A high-order nodal discontinuous Galerkin method for nonlinear fractional Schrödinger type equations. Commun Nonlinear Sci Numer Simul, 2018, 54: 428–452MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bhrawy A H, Abdelkawy M A. A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations. J Comput Phys, 2015, 294: 462–483MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boulenger T, Himmelsbach D, Lenzmann E. Blowup for fractional NLS. J Funct Anal, 2016, 271: 2569–2603MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Çelik C, Duman M. Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J Comput Phys, 2012, 231: 1743–1750MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chang Q, Jia E, Sun W. Difference schemes for solving the generalized nonlinear Schrödinger equation. J Comput Phys, 1999, 148: 397–415MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dehghan M, Taleei A. A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients. Comput Phys Comm, 2010, 181: 43–51MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Duan B, Zheng Z, Cao W. Finite element method for a kind of two-dimensional space-fractional diffusion equation with its implementation. Amer J Comput Math, 2015, 5: 135CrossRefGoogle Scholar
  8. 8.
    Duo S, Zhang Y. Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation. Comput Math Appl, 2016, 71: 2257–2271MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fröhlich J, Jonsson B L G, Lenzmann E. Boson stars as solitary waves. Comm Math Phys, 2007, 274: 1–30MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gong Y, Wang Q, Wang Y, et al. A conservative Fourier pseudo-spectral method for the nonlinear Schrödinger equation. J Comput Phys, 2017, 328: 354–370MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hou T, Tang T, Yang J. Numerical analysis of fully discretized Crank-Nicolson scheme for fractional-in-space Allen-Cahn equations. J Sci Comput, 2017, 72: 1214–1231MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ionescu A D, Pusateri F. Nonlinear fractional Schrödinger equations in one dimension. J Funct Anal, 2014, 266: 139–176MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ji C C, Sun Z Z. A high-order compact finite difference scheme for the fractional sub-diffusion equation. J Sci Comput, 2015, 64: 959–985MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Khaliq A Q M, Liang X, Furati K M. A fourth-order implicit-explicit scheme for the space fractional nonlinear Schröodinger equations. Numer Algorithms, 2017, 75: 147–172MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kirkpatrick K, Lenzmann E, Staffilani G. On the continuum limit for discrete NLS with long-range lattice interactions. Comm Math Phys, 2013, 317: 563–591MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Klein C, Sparber C, Markowich P. Numerical study of fractional nonlinear Schröodinger equations. Proc Math Phys Eng Sci, 2014, 470: 20140364CrossRefzbMATHGoogle Scholar
  17. 17.
    Laskin N. Fractional quantum mechanics. Phys Rev E, 2000, 62: 3135–3145CrossRefzbMATHGoogle Scholar
  18. 18.
    Laskin N. Fractional Schroödinger equation. Phys Rev E, 2002, 66: 056108MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lenzmann E. Well-posedness for semi-relativistic Hartree equations of critical type. Math Phys Anal Geom, 2007, 10: 43–64MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Li M, Huang C, Wang P. Galerkin finite element method for nonlinear fractional Schrödinger equations. Numer Algorithms, 2017, 74: 499–525MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Liang X, Khaliq A Q M, Bhatt H, et al. The locally extrapolated exponential splitting scheme for multi-dimensional nonlinear space-fractional Schrödinger equations. Numer Algorithms, 2017, 76: 939–958MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Magin R L, Abdullah O, Baleanu D, et al. Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation. J Magnet Reson, 2008, 190: 255–270CrossRefGoogle Scholar
  23. 23.
    Meerschaert M M, Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations. J Comput Appl Math, 2004, 172: 65–77MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Merle F, Tsutsumi Y. L 2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity. J Differential Equations, 1990, 84: 205–214MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Moler C, Loan C V. Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev, 2003 45: 3–49MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Nie Q, Wan F Y M, Zhang Y, et al. Compact integration factor methods in high spatial dimensions. J Comput Phys, 2008, 227: 5238–5255MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nie Q, Zhang Y, Zhao R. Efficient semi-implicit schemes for stiff systems. J Comput Phys, 2006, 214: 521–537MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Obrecht C. Remarks on the full dispersion Davey-Stewartson systems. Commun Pure Appl Anal, 2015, 14: 1547–1561MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ortigueira M D. Riesz potential operators and inverses via fractional centred derivatives. Int J Math Math Sci, 2006, 2006: 1–12MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tian W, Zhou H, Deng W. A class of second order difference approximations for solving space fractional diffusion equations. Math Comput, 2015, 84: 1703–1727MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Wang D, Zhang L, Nie Q. Array-representation integration factor method for high-dimensional systems. J Comput Phys, 2014, 258: 585–600MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Wang P, Huang C. An energy conservative difference scheme for the nonlinear fractional Schrödinger equations. J Comput Phys, 2015, 293: 238–251MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Wang P, Huang C. Split-step alternating direction implicit difference scheme for the fractional Schrödinger equation in two dimensions. Comput Math Appl, 2016, 71: 1114–1128MathSciNetCrossRefGoogle Scholar
  34. 34.
    Xu Y, Shu C. Local discontinuous Galerkin methods for nonlinear Schröodinger equations. J Comput Phys, 2005, 205: 72–97MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Yang Q, Liu F, Turner I. Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl Math Model, 2010, 34: 200–218MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Yang Q, Turner I, Liu F, et al. Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions. SIAM J Sci Comput, 2011, 33: 1159–1180MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Yang Z. A class of linearized energy-conserved finite difference schemes for nonlinear space-fractional Schrödinger equations. Int J Comput Math, 2016, 93: 609–626MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Zhang R, Yu X, Li M, et al. A conservative local discontinuous Galerkin method for the solution of nonlinear Schröodinger equation in two dimensions. Sci China Math, 2017, 60: 2515–2530MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Zhang R, Zhu J, Yu X, et al. A conservative spectral collocation method for the nonlinear Schröodinger equation in two dimensions. Appl Math Comput, 2017, 310: 194–203MathSciNetGoogle Scholar
  40. 40.
    Zhao X, Sun Z, Hao Z. A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schröodinger equation. SIAM J Sci Comput, 2014, 36: 2865–2886MathSciNetCrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Rongpei Zhang
    • 1
    Email author
  • Yong-Tao Zhang
    • 2
  • Zhen Wang
    • 3
  • Bo Chen
    • 4
  • Yi Zhang
    • 1
  1. 1.School of Mathematics and Systematic SciencesShenyang Normal UniversityShenyangChina
  2. 2.Department of Applied and Computational Mathematics and StatisticsUniversity of Notre DameNotre DameUSA
  3. 3.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoChina
  4. 4.College of Mathematics and StatisticsShenzhen UniversityShenzhenChina

Personalised recommendations