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 analyzed 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 $\ddot{I}{„}^{s/2}$ in $L^2({\mathbb{T}}^2)$ at this regularity level are proved. Numerical examples illustrate that these convergence results are sharp.