The non-differentiability of the singular nonlinearity (such as $f=\ln|u|^2$) at $u=0$ presents significant challenges in devising accurate and efficient numerical schemes for the logarithmic Schr\"{o}dinger equation (LogSE). To address this singularity, we propose an energy regularization technique for the LogSE. For the regularized model, we utilize Implicit-Explicit Relaxation Runge-Kutta methods, which are linearly implicit, high-order, and mass-conserving for temporal discretization, in conjunction with the Fourier pseudo-spectral method in space. Ultimately, numerical results are presented to validate the efficiency of the proposed methods.

翻译：奇异非线性项（如$f=\ln|u|^2$）在$u=0$处的不可微性，为对数薛定谔方程（LogSE）高精度高效数值格式的构建带来了显著挑战。为处理该奇异性，本文针对LogSE提出了一种能量正则化技术。对于正则化模型，我们采用具有线性隐式、高阶及质量守恒特性的隐式-显式松弛Runge-Kutta方法进行时间离散，并结合空间上的傅里叶伪谱方法。最终，数值结果验证了所提方法的有效性。