Low regularity estimates of the Lie-Totter time-splitting Fourier spectral method for the logarithmic Schrödinger equation
Xiaolong Zhang, Li-Lian Wang
TL;DR
The paper analyzes a nonregularised Lie-Trotter time-splitting Fourier spectral method for the logarithmic Schrödinger equation on the torus, proving L^2-error bounds that scale like $O((\tau^{s/2}+N^{-s})\ln N)$ for solutions with low regularity $u_0\in L^2(\mathbb{T}^d)$ and $u\in C((0,T]; H^s(\mathbb{T}^d)\cap L^\infty(\mathbb{T}^d))$ with $0<s<1$, and extending to $s=1$ with refined analysis giving $\|u^{m+1}-u_N^{m+1}\|\le C e^{2|\lambda|T}(\mathcal{C}+\ln N)(\sqrt{\tau}+N^{-1})|u|_{C(H^1)}$. The method splits evolution into a linear dispersive part and a nonlinear logarithmic part, analyzes stability and regularity of the flow maps, and establishes a full-discretization error bound driven by fractional Sobolev regularity. Numerical experiments corroborate fractional-order convergence for low-regularity data and mass conservation, demonstrating the practical viability of nonregularised splitting for LogSE. The work advances numerical analysis for nonlinear Schrödinger equations with singular nonlinearities and opens avenues for further refinement of low-regularity integrators and extension to broader settings.
Abstract
In this paper, we conduct rigorous error analysis of the Lie-Totter time-splitting Fourier spectral scheme for the nonlinear Schrödinger 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 Hölder 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: $O\big((τ^{s/2} + N^{-s})\ln\! N\big)$ 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.