A fully discrete low-regularity integrator for the 1D periodic cubic nonlinear Schrödinger equation

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