Low regularity full error estimates for the cubic nonlinear Schrödinger equation

Abstract

For the numerical solution of the cubic nonlinear Schrödinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to converge even for initial data with very low regularity. In particular, for data in $H^s({\mathbb{T}}^2)$, where $s>0$, convergence of order $\mathcal{O}(\ddot{I}{„}^{s/2}+N^{-s})$ is proved in $L^2$. Here $\ddot{I}„$ denotes the time step size and $N$ the number of Fourier modes considered. The proof of this result is carried out in an abstract framework of discrete Bourgain spaces, the final convergence result, however, is given in $L^2$. The stated convergence behavior is illustrated by several numerical examples.