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Regularized numerical methods for the logarithmic Schrödinger equation

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Abstract

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 \(0<\varepsilon \ll 1\) is adopted to approximate the LogSE with linear convergence rate \(O(\varepsilon )\). Then we use the Lie–Trotter splitting integrator to solve the RLogSE and establish its error bound \(O(\tau ^{1/2}\ln (\varepsilon ^{-1}))\) with \(\tau >0\) the time step, which implies an error bound at \(O(\varepsilon +\tau ^{1/2}\ln (\varepsilon ^{-1}))\) 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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Acknowledgements

The authors acknowledge the support from Ministry of Education of Singapore Grant R-146-000-223-112 (MOE2015-T2-2-146) (W. Bao) and by the Fundamental Research Funds for the Central Universities (YJ201807) (Q. Tang).

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Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, Singapore, 119076, Singapore

    Weizhu Bao

  2. CNRS, IRMAR - UMR 6625, Univ Rennes, 35000, Rennes, France

    Rémi Carles

  3. Zentrum Mathematik, Technische Universität München, 85748, Garching bei München, Germany

    Chunmei Su

  4. School of Mathematics, State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu, 610064, People’s Republic of China

    Qinglin Tang

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  1. Weizhu Bao
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Correspondence to Qinglin Tang.

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Bao, W., Carles, R., Su, C. et al. Regularized numerical methods for the logarithmic Schrödinger equation. Numer. Math. 143, 461–487 (2019). https://doi.org/10.1007/s00211-019-01058-2

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