Low regularity error estimates for the time integration of 2D NLS

Abstract

A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schrödinger equation on the two-dimensional torus $\mathbb{T}^{2}$. The scheme is analysed in a framework of discrete Bourgain spaces, which allows us to consider initial data with low regularity; more precisely initial data in $H^{s}(\mathbb{T}^{2})$ with $s>0$. In this way, the usual stability restriction to smooth Sobolev spaces with index $s>1$ is overcome. Rates of convergence of order $\tau ^{s/2}$ in $L^{2}(\mathbb{T}^{2})$ at this regularity level are proved. Numerical examples illustrate that these convergence results are sharp.