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Fast Huygens Sweeping Methods for a Class of Nonlocal Schrödinger Equations

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Abstract

We present efficient numerical methods for solving a class of nonlinear Schrödinger equations involving a nonlocal potential. Such a nonlocal potential is governed by Gaussian convolution of the intensity modeling nonlocal mutual interactions among particles. The method extends the Fast Huygens Sweeping Method (FHSM) that we developed in Leung et al. (Methods Appl Anal 21(1):31–66, 2014) for the linear Schrödinger equation in the semi-classical regime to the nonlinear case with nonlocal potentials. To apply the methodology of FHSM effectively, we propose two schemes by using the Lie’s and the Strang’s operator splitting, respectively, so that one can handle the nonlinear nonlocal interaction term using the fast Fourier transform. The resulting algorithm can then enjoy the same computational complexity as in the linear case. Numerical examples demonstrate that the two operator splitting schemes achieve the expected first-order and second-order accuracy, respectively. We will also give one-, two- and three-dimensional examples to demonstrate the efficiency of the proposed algorithm.

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Acknowledgements

The work of Leung was supported in part by the Hong Kong RGC Grant 16302819. Qian is partially supported by NSF.

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Authors and Affiliations

  1. Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

    Kaho Ho & Shingyu Leung

  2. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Jianliang Qian

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  1. Kaho Ho
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  2. Shingyu Leung
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  3. Jianliang Qian
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Correspondence to Shingyu Leung.

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Ho, K., Leung, S. & Qian, J. Fast Huygens Sweeping Methods for a Class of Nonlocal Schrödinger Equations. J Sci Comput 88, 54 (2021). https://doi.org/10.1007/s10915-021-01549-1

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