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Numerical Algorithms

, Volume 79, Issue 1, pp 337–356 | Cite as

Galerkin-Legendre spectral schemes for nonlinear space fractional Schrödinger equation

  • Hui Zhang
  • Xiaoyun JiangEmail author
  • Chu Wang
  • Wenping Fan
Original Paper

Abstract

In the paper, we first propose a Crank-Nicolson Galerkin-Legendre (CN-GL) spectral scheme for the one-dimensional nonlinear space fractional Schrödinger equation. Convergence with spectral accuracy is proved for the spectral approximation. Further, a Crank-Nicolson ADI Galerkin-Legendre spectral method for the two-dimensional nonlinear space fractional Schrödinger equation is developed. The proposed schemes are shown to be efficient with second-order accuracy in time and spectral accuracy in space which are higher than some recently studied methods. Moreover, some numerical results are demonstrated to justify the theoretical analysis.

Keywords

Galerkin-Legendre spectral method Crank-Nicolson difference method Nonlinear space fractional Schrödinger equation Convergence analysis 

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Notes

Acknowledgements

We would like to express our gratitude to the editor for taking time to handle the manuscript and to anonymous referees whose constructive comments are very helpful for improving the quality of our paper. This work has been supported by the National Natural Science Foundation of China (Grants Nos. 11472161, 11771254, 11672163), and the Natural Science Foundation of Shandong Province (Grant ZR2015AM011, ZR2017MA030).

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  • Hui Zhang
    • 1
  • Xiaoyun Jiang
    • 1
    Email author
  • Chu Wang
    • 2
  • Wenping Fan
    • 3
  1. 1.School of MathematicsShandong UniversityJinanPeople’s Republic of China
  2. 2.Nokia Bell LabsMurray HillUSA
  3. 3.School of ScienceJiangnan UniversityWuxiPeople’s Republic of China

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