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In this paper, we conduct rigorous error analysis of the Lie-Tottertime-splitting Fourier spectral scheme for the nonlinear Schr\"odinger equationwith a logarithmic nonlinear term $f(u)=u\ln|u|^2$ (LogSE) and periodicboundary conditions on a $d$-dimensional torus $\mathbb T^d$. Different fromexisting works based on regularisation of the nonlinear term $ f(u)\approxf^\varepsilon(u)=u\ln (|u| + \varepsilon )^2,$ we directly discretize the LogSEwith the understanding $f(0)=0.$ Remarkably, in the time-splitting scheme, thesolution 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 differentiableat $u=0$ but H\"older continuous), where $g(u)$ is Lipschitz continuous andpossesses 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 regularitysolution $u\in C((0,T]; H^s( \mathbb{T}^d)\cap L^\infty( \mathbb{T}^d)).$Moreover, we can show that the estimate holds for $s=1$ with more delicateanalysis of the nonlinear term and the associated solution flow maps.Furthermore, we provide ample numerical results to demonstrate such afractional-order convergence for initial data with low regularity. This work isthe first one devoted to the analysis of splitting scheme for the LogSE withoutregularisation in the low regularity setting, as far as we can tell.

Low regularity estimates of the Lie-Totter time-splitting Fourier spectral method for the logarithmic Schr\"odinger equation