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

Abstract

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.