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On time-splitting methods for nonlinear Schrödinger equation with highly oscillatory potential

Chunmei Su Zentrum Mathematik, Technische Universität München, 85748 Garching bei München, Germany Xiaofei Zhao School of Mathematics and Statistics & Hubei Key Laboratory of Computational Science, Wuhan University, 430072 Wuhan, P.R. China

Analysis of PDEs mathscidoc:2205.03009

ESAIM: Mathematical Modelling and Numerical Analysis, 54, (5), 1491-1508, 2020.6
In this work, we consider the numerical solution of the nonlinear Schrödinger equation with a highly oscillatory potential (NLSE-OP). The NLSE-OP is a model problem which frequently occurs in recent studies of some multiscale dynamical systems, where the potential introduces wide temporal oscillations to the solution and causes numerical difficulties. We aim to analyze rigorously the error bounds of the splitting schemes for solving the NLSE-OP to a fixed time. Our theoretical results show that the Lie–Trotter splitting scheme is uniformly and optimally accurate at the first order provided that the oscillatory potential is integrated exactly, while the Strang splitting scheme is not. Our results apply to general dispersive or wave equations with an oscillatory potential. The error estimates are confirmed by numerical results.
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@inproceedings{chunmei2020on,
  title={On time-splitting methods for nonlinear Schrödinger equation with highly oscillatory potential},
  author={Chunmei Su, and Xiaofei Zhao},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220519165155670911298},
  booktitle={ESAIM: Mathematical Modelling and Numerical Analysis},
  volume={54},
  number={5},
  pages={1491-1508},
  year={2020},
}
Chunmei Su, and Xiaofei Zhao. On time-splitting methods for nonlinear Schrödinger equation with highly oscillatory potential. 2020. Vol. 54. In ESAIM: Mathematical Modelling and Numerical Analysis. pp.1491-1508. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220519165155670911298.
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