In this paper, we conduct rigorous error analysis of the Lie-Totter time-splitting Fourier spectral scheme for the nonlinear Schr\"odinger equation with a logarithmic nonlinear term $f(u)=u\ln|u|^2$ (LogSE) and periodic boundary conditions on a $d$-dimensional torus $\mathbb T^d$. Different from existing works based on regularisation of the nonlinear term $ f(u)\approx f^\varepsilon(u)=u\ln (|u| + \varepsilon )^2,$ we directly discretize the LogSE with the understanding $f(0)=0.$ Remarkably, in the time-splitting scheme, the solution flow map of the nonlinear part: $g(u)= u {\rm e}^{-{\rm} i t \ln|u|^{2}}$ has a higher regularity than $f(u)$ (which is not differentiable at $u=0$ but H\"older continuous), where $g(u)$ is Lipschitz continuous and possesses a certain fractional Sobolev regularity with index $0<s<1$. Accordingly, we can derive the $L^2$-error estimate: $O\big((\tau^{s/2} + N^{-s})\ln\! N\big)$ of the proposed scheme for the LogSE with low regularity solution $u\in C((0,T]; H^s( \mathbb{T}^d)\cap L^\infty( \mathbb{T}^d)).$ Moreover, we can show that the estimate holds for $s=1$ with more delicate analysis of the nonlinear term and the associated solution flow maps. Furthermore, we provide ample numerical results to demonstrate such a fractional-order convergence for initial data with low regularity. This work is the first one devoted to the analysis of splitting scheme for the LogSE without regularisation in the low regularity setting, as far as we can tell.

翻译：本文严格分析了具有对数非线性项$f(u)=u\ln|u|^2$的LogSE在$d$维环面$\mathbb T^d$上周期边界条件下的Lie-Totter时间分裂傅里叶谱格式的误差。不同于现有基于非线性项正则化$ f(u)\approx f^\varepsilon(u)=u\ln (|u| + \varepsilon )^2$的工作，我们直接离散LogSE并理解$f(0)=0$。值得注意的是，在时间分裂格式中，非线性部分的解流映射：$g(u)= u {\rm e}^{-{\rm} i t \ln|u|^{2}}$具有比$f(u)$更高的正则性（$f(u)$在$u=0$处不可微但满足Hölder连续性），其中$g(u)$是Lipschitz连续的且具有指数$0<s<1$的某种分数阶Sobolev正则性。据此，对于低正则性解$u\in C((0,T]; H^s( \mathbb{T}^d)\cap L^\infty( \mathbb{T}^d))$的LogSE，我们可推导所提格式的$L^2$误差估计：$O\big((\tau^{s/2} + N^{-s})\ln\! N\big)$。进一步，通过对非线性项及相关解流映射的更精细分析，可证明该估计对$s=1$也成立。此外，我们提供了大量数值结果来展示低正则性初值下这种分数阶收敛性。据我们所知，这是首个在低正则性设定下不依赖正则化分析LogSE分裂格式的工作。