For the solution of the cubic nonlinear Schr\"odinger equation in one space dimension, we propose and analyse a fully discrete low-regularity integrator. The scheme is explicit and can easily be implemented using the fast Fourier transform with a complexity of $\mathcal{ O}(N\log N)$ operations per time step, where $N$ denotes the degrees of freedom in the spatial discretisation. We prove that the new scheme provides an $\mathcal{ O}(\tau^{ \frac32\gamma-\frac12-\varepsilon}+N^{ -\gamma})$ error bound in $L^2$ for any initial data belonging to $H^\gamma$, $\frac12<\gamma\leq 1$, where $\tau$ denotes the temporal step size. Numerical examples illustrate this convergence behavior.
Categories: math.NA cs.NA