论文|Weizhu Bao, Remi Carles, Chunmei Su and Qinglin Tang. Reguarized Numerical Methods for the Logarithmic Schrödinger Equation

本文原载 Numerische Mathematik，由四川大学数学学院唐庆粦教授等科研人员创作，系四川大学“智慧法治”超前部署学科系列学术成果。后续会持续分享四川大学“智慧法治”超前部署学科系列学术成果，欢迎大家阅读。

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＜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.

Mathematics Subject Classification: 35Q40·35Q55·65M15·81Q05

Weizhu Bao, Remi Carles, Chunmei Su and Qinglin Tang. Reguarized Numerical Methods for the Logarithmic Schrödinger Equation, Numerische Mathematik, Vol. 143, pp. 461--487, 2019.（论文下载）