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

A fully discrete and fully 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, and is proved to have an $L^2$-norm error bound of $O(τ\sqrt{\ln(1/τ)}+N^{-1})$ for $H^1$ initial data, without requiring any CFL condition, where $τ$ and $N$ denote the temporal stepsize and the degree of freedoms in the spatial discretisation, respectively.

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