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

A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schrödinger equation on the two-dimensional torus T². 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 Hˢ(T²) 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 τ s/2 in L²(T²) at this regularity level are proved. Numerical examples illustrate that these convergence results are sharp.