Error analysis of a first-order IMEX scheme for the logarithmic Schrödinger equation
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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 regularising f(u) as u^εln (ε+ |u^ε|)^2 to overcome the blowup of ln |u|^2 at u=0 has been investigated recently in literature. With the understanding of f(0)=0, we analyze the non-regularized first-order Implicit-Explicit (IMEX) 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 one to study the direct linearized scheme for the LogSE as far as we can tell.
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