Regularized numerical methods for the logarithmic Schrodinger equation
This work provides rigorous error analysis for numerical solutions of a challenging nonlinear PDE, benefiting computational scientists studying quantum mechanics and nonlinear waves.
The authors propose two regularized numerical methods for the logarithmic Schrödinger equation, achieving linear convergence in the regularization parameter and error bounds of O(ε + τ^{1/2} ln(ε^{-1})) for the splitting method. Numerical experiments confirm the error bounds and reveal complex dynamics.
We present and analyze two numerical methods for the logarithmic Schr{ö}dinger 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 Schr{ö}dinger equation (RLogSE) with a small regularized parameter 0 < $ε$ $\ll$ 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.