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

TL;DR

This paper introduces a filtered Lie splitting scheme for the 2D cubic nonlinear Schrödinger equation that achieves low regularity error estimates, allowing for initial data with minimal smoothness and demonstrating optimal convergence rates.

Contribution

The paper develops a novel filtered Lie splitting method analyzed in discrete Bourgain spaces, enabling low regularity error estimates for the 2D NLS, surpassing previous smoothness restrictions.

Findings

Convergence rate of order τ^{s/2} in L^2 for initial data in H^s, s>0.

Numerical results confirm the sharpness of the theoretical convergence rates.

Overcomes stability restrictions to smooth Sobolev spaces with s>1.

Abstract

A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schr\"odinger 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 $\tau^{s/2}$ in $L^2(\mathbb{T}^2)$ at this regularity level are proved. Numerical examples illustrate that these convergence results are sharp.