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The logarithmic Schr\"odinger equation (LogSE) has a logarithmic nonlinearity$f(u)=u\ln |u|^2$ that is not differentiable at $u=0.$ Compared with itscounterpart with a regular nonlinear term, it possesses richer and unusualdynamics, though the low regularity of the nonlinearity brings aboutsignificant challenges in both analysis and computation. Among very limitednumerical studies, the semi-implicit regularized method via regularising $f(u)$as $ u^{\varepsilon}\ln ({\varepsilon}+ |u^{\varepsilon}|)^2$ to overcome theblowup 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-orderImplicit-Explicit (IMEX) scheme for the LogSE. We introduce some new tools forthe error analysis that include the characterization of the H\"older continuityof the logarithmic term, and a nonlinear Gr\"{o}nwall's inequality. We provideample numerical results to demonstrate the expected convergence. We positionthis work as the first one to study the direct linearized scheme for the LogSEas far as we can tell.

Error analysis of a first-order IMEX scheme for the logarithmic Schr\"odinger equation