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Fourth-order time-stepping compact finite difference method for multi-dimensional space-fractional coupled nonlinear Schrödinger equations

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Abstract

In this work, an efficient fourth-order time-stepping compact finite difference scheme is devised for the numerical solution of multi-dimensional space-fractional coupled nonlinear Schrödinger equations. Some existing numerical schemes for these equations lead to full and dense matrices due to the non-locality of the fractional operator. To overcome this challenge, the spatial discretization in our method is carried out by using the compact finite difference scheme and matrix transfer technique in which FFT-based computations can be utilized. This avoids storing the large matrix from discretizing the fractional operator and also significantly reduces the computational costs. The amplification symbol of this scheme is investigated by plotting its stability regions, which indicates the stability of the scheme. Numerical experiments show that this scheme preserves the conservation laws of mass and energy, and achieves the fourth-order accuracy in both space and time.

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Acknowledgements

The authors would like to thank the editor and referees for their valuable comments and suggestions which improved the paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Mustafa Almushaira & Fei Liu

  2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, China

    Mustafa Almushaira & Fei Liu

  3. Department of Mathematics, Faculty of Science, Sana’a University, Sanaa, Yemen

    Mustafa Almushaira

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  1. Mustafa Almushaira
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  2. Fei Liu
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Correspondence to Fei Liu.

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This article is part of the section “Computational Approaches” edited by Siddhartha Mishra.

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Almushaira, M., Liu, F. Fourth-order time-stepping compact finite difference method for multi-dimensional space-fractional coupled nonlinear Schrödinger equations. SN Partial Differ. Equ. Appl. 1, 45 (2020). https://doi.org/10.1007/s42985-020-00048-6

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  • DOI: https://doi.org/10.1007/s42985-020-00048-6

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