A fully discrete low-regularity integrator for the 1D periodic cubic nonlinear Schr\"odinger equation

Abstract

A fully discrete and fully explicit low-regularity integrator is constructed for the one-dimensional periodic cubic nonlinear Schr\"odinger equation. The method can be implemented by using fast Fourier transform with $O(N\ln N)$ operations at every time level, and is proved to have an $L^2$-norm error bound of $O(\tau\sqrt{\ln(1/\tau)}+N^{-1})$ for $H^1$ initial data, without requiring any CFL condition, where $\tau$ and $N$ denote the temporal stepsize and the degree of freedoms in the spatial discretisation, respectively.