Regularized numerical methods for the logarithmic Schrödinger equation

• Published in: Numerische Mathematik 2019-10, Vol.143 (2), p.461-487
• Main Authors: Bao, Weizhu, Carles, Rémi, Su, Chunmei, Tang, Qinglin
• Format: Article
• Language: English
• Subjects: Analysis of PDEs; Energy conservation; Finite difference method; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical and Computational Physics; Numerical methods; Roundoff error; Schrodinger equation; Simulation; Splitting; Theoretical
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-019-01058-2

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.