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

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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 O(τ2). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up.

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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).

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Authors and Affiliations

  1. School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang, 110034, China

    Rongpei Zhang & Yi Zhang

  2. Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Yong-Tao Zhang

  3. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China

    Zhen Wang

  4. College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China

    Bo Chen

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  1. Rongpei Zhang
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  2. Yong-Tao Zhang
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  3. Zhen Wang
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  4. Bo Chen
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  5. Yi Zhang
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Correspondence to Rongpei Zhang.

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Zhang, R., Zhang, YT., Wang, Z. et al. A conservative numerical method for the fractional nonlinear Schrödinger equation in two dimensions. Sci. China Math. 62, 1997–2014 (2019). https://doi.org/10.1007/s11425-018-9388-9

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