Error estimates of local energy regularization for the logarithmic Schrödinger equation

Weizhu Bao Department of Mathematics, National University of Singapore, 119076 Singapore, Singapore Rémi Carles CNRS, IRMAR - UMR 6625, University of Rennes, F-35000 Rennes, France Chunmei Su Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, P. R. China Qinglin Tang School of Mathematics, State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610064, P. R. China

Numerical Analysis and Scientific Computing mathscidoc:2205.25009

Mathematical Models and Methods in Applied Sciences, 32, (01), 101-136, 2022.1
The logarithmic nonlinearity has been used in many partial differential equations (PDEs) for modeling problems in various applications. Due to the singularity of the logarithmic function, it introduces tremendous difficulties in establishing mathematical theories, as well as in designing and analyzing numerical methods for PDEs with such nonlinearity. Here, we take the logarithmic Schrödinger equation (LogSE) as a prototype model. Instead of regularizing f(ρ)=lnρ in the LogSE directly and globally as being done in the literature, we propose a local energy regularization (LER) for the LogSE by first regularizing F(ρ)=ρlnρ−ρ locally near ρ=0+ with a polynomial approximation in the energy functional of the LogSE and then obtaining an energy regularized logarithmic Schrödinger equation (ERLogSE) via energy variation. Linear convergence is established between the solutions of ERLogSE and LogSE in terms of a small regularization parameter 0<ε≪1. Moreover, the conserved energy of the ERLogSE converges to that of LogSE quadratically, which significantly improves the linear convergence rate of the regularization method in the literature. Error estimates are also presented for solving the ERLogSE by using Lie–Trotter splitting integrators. Numerical results are reported to confirm our error estimates of the LER and of the time-splitting integrators for the ERLogSE. Finally, our results suggest that the LER performs better than regularizing the logarithmic nonlinearity in the LogSE directly.
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@inproceedings{weizhu2022error,
  title={Error estimates of local energy regularization for the logarithmic Schrödinger equation},
  author={Weizhu Bao, Rémi Carles, Chunmei Su, and Qinglin Tang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220519154848687113280},
  booktitle={Mathematical Models and Methods in Applied Sciences},
  volume={32},
  number={01},
  pages={101-136},
  year={2022},
}
Weizhu Bao, Rémi Carles, Chunmei Su, and Qinglin Tang. Error estimates of local energy regularization for the logarithmic Schrödinger equation. 2022. Vol. 32. In Mathematical Models and Methods in Applied Sciences. pp.101-136. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220519154848687113280.
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