Error Analysis of a First-Order IMEX Scheme for the Logarithmic Schrödinger Equation
Li-Lian Wang, Jingye Yan, Xiaolong Zhang
Abstract
The logarithmic Schrödinger equation (LogSE) has a logarithmic nonlinearity f(u) = uln |u|2 that is not differentiable at u = 0. Compared with its counterpart with a regular nonlinear term, it possesses richer and unusual dynamics, though the low regularity of the nonlinearity brings about significant challenges in both analysis and computation. Among very limited numerical studies, the semi-implicit regularized method via regularizing f(u) as u\varepsilon ln(\varepsilon+ |u\varepsilon|)2 to overcome the blowup of ln |u|2 at u = 0 has been investigated recently in the literature. With the understanding of f(0) = 0, we analyze the nonregularized first-order implicit-explicit scheme for the LogSE. We introduce some new tools for the error analysis that include the characterization of the Hölder continuity of the logarithmic term, and a nonlinear Grönwall's inequality. We provide ample numerical results to demonstrate the expected convergence. We position this work as the first to study the direct linearized scheme for the LogSE as far as we can tell.