Error Estimates of a Regularized Finite Difference Method for the Logarithmic Schrödinger Equation

Weizhu Bao Department of Mathematics, National University of Singapore, Singapore 119076 Rémi Carles CNRS, IRMAR, Universit ́e de Rennes 1 & ENS Rennes, France Chunmei Su epartment of Mathematics, University of Innsbruck, Innsbruck 6020,Austria Qinglin Tang hool of Mathematics, State Key Laboratory of Hydraulics andMountain River Engineering, Sichuan University, Chengdu 610064, People’s Republic of China

Numerical Analysis and Scientific Computing mathscidoc:2205.25016

SIAM Journal on Numerical Analysis, 57, (2), 657-680, 2019.3
We present a regularized finite difference method for the logarithmic Schr ̈odingerequation (LogSE) and establish its error bound. Due to the blowup of the logarithmic nonlinearity, i.e., ln ρ→ −\infty when ρ→0+v with ρ = |u|^2 being the density and u being the complex-valuedwave function or order parameter, there are significant difficulties in designing numerical methodsand establishing their error bounds for the LogSE. In order to suppress the roundoff error and toavoid blowup, a regularized LogSE (RLogSE) is proposed with a small regularization parameter 0 < ε << 1 and linear convergence is established between the solutions of RLogSE and LogSE interm of ε. Then a semi-implicit finite difference method is presented for discretizing the RLogSEand error estimates are established in terms of the mesh sizehand time stepτas well as the smallregularization parameterε. Finally numerical results are reported to illustrate our error bounds.
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@inproceedings{weizhu2019error,
  title={Error Estimates of a Regularized Finite Difference Method for the Logarithmic Schrödinger Equation},
  author={Weizhu Bao, Rémi Carles, Chunmei Su, and Qinglin Tang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220519164112519093295},
  booktitle={SIAM Journal on Numerical Analysis},
  volume={57},
  number={2},
  pages={657-680},
  year={2019},
}
Weizhu Bao, Rémi Carles, Chunmei Su, and Qinglin Tang. Error Estimates of a Regularized Finite Difference Method for the Logarithmic Schrödinger Equation. 2019. Vol. 57. In SIAM Journal on Numerical Analysis. pp.657-680. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220519164112519093295.
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