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 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^{\varepsilon}\ln ({\varepsilon}+ |u^{\varepsilon}|)^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 H\"older continuity of the logarithmic term, and a nonlinear Gr\"{o}nwall'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.

翻译：对数薛定谔方程（LogSE）具有对数非线性项 $f(u)=u\ln |u|^2$，该非线性项在 $u=0$ 处不可微。与具有规则非线性项的对应方程相比，尽管非线性项的低正则性给分析和计算带来了重大挑战，但LogSE展现出更丰富且不寻常的动力学行为。在极为有限的数值研究中，近期文献探讨了半隐式正则化方法：通过将 $f(u)$ 正则化为 $ u^{\varepsilon}\ln ({\varepsilon}+ |u^{\varepsilon}|)^2$ 来克服 $\ln |u|^2$ 在 $u=0$ 处的爆炸。基于对 $f(0)=0$ 的理解，我们针对LogSE分析了非正则化的一阶隐式-显式（IMEX）格式。我们引入了若干误差分析新工具，包括对数项Hölder连续性的刻画以及非线性Grönwall不等式。我们提供了丰富的数值结果以展示预期收敛性。据我们所知，本研究是首次针对LogSE的线性化直接格式展开探讨。