Regularized numerical methods for the logarithmic Schrödinger equation

Weizhu Bao Department of Mathematics, National University of Singapore, Singapore, 119076, Singapore Rémi Carles CNRS, IRMAR - UMR 6625, Univ Rennes, 35000, Rennes, France Chunmei Su Zentrum Mathematik, Technische Universität München, 85748, Garching bei München, Germany Qinglin Tang School of Mathematics, State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu, 610064, People’s Republic of China

Analysis of PDEs mathscidoc:2205.03007

Numerische Mathematik, 143, 461–487q, 2019.7
We present and analyze two numerical methods for the logarithmic Schrödinger equation (LogSE) consisting of a regularized splitting method and a regularized conservative Crank–Nicolson finite difference method (CNFD). In order to avoid numerical blow-up and/or to suppress round-off error due to the logarithmic nonlinearity in the LogSE, a regularized logarithmic Schrödinger equation (RLogSE) with a small regularized parameter 0<ε≪1 is adopted to approximate the LogSE with linear convergence rate O(ε). Then we use the Lie–Trotter splitting integrator to solve the RLogSE and establish its error bound O(τ^{1/2}ln(ε^{−1})) with τ>0 the time step, which implies an error bound at O(ε+τ^{1/2}ln(ε^{−1})) for the LogSE by the Lie–Trotter splitting method. In addition, the CNFD is also applied to discretize the RLogSE, which conserves the mass and energy in the discretized level. Numerical results are reported to confirm our error bounds and to demonstrate rich and complicated dynamics of the LogSE.
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@inproceedings{weizhu2019regularized,
  title={Regularized numerical methods for the logarithmic Schrödinger equation},
  author={Weizhu Bao, Rémi Carles, Chunmei Su, and Qinglin Tang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220519164402420451296},
  booktitle={Numerische Mathematik},
  volume={143},
  pages={461–487q},
  year={2019},
}
Weizhu Bao, Rémi Carles, Chunmei Su, and Qinglin Tang. Regularized numerical methods for the logarithmic Schrödinger equation. 2019. Vol. 143. In Numerische Mathematik. pp.461–487q. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220519164402420451296.
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