Error estimates at low regularity of splitting schemes for NLS

TL;DR

This paper develops error estimates for a filtered Lie splitting scheme applied to the cubic nonlinear Schrödinger equation, enabling convergence analysis at low regularity levels where traditional methods fail.

Contribution

It introduces a novel approach using discrete Bourgain spaces to achieve error estimates for data in $H^s$ with $0<s<1$, surpassing previous stability restrictions.

Findings

Achieves convergence rates of order τ^{s/2} in L^2 for low regularity data.

Extends the applicability of splitting schemes to rougher initial data.

Overcomes the standard stability restriction of s>1/2.

Abstract

We study a filtered Lie splitting scheme for the cubic nonlinear Schr\"{o}dinger equation. We establish error estimates at low regularity by using discrete Bourgain spaces. This allows us to handle data in $H^s$ with $0<s<1$ overcoming the standard stability restriction to smooth Sobolev spaces with index $s>1/2$ . More precisely, we prove convergence rates of order $\tau^{s/2}$ in $L^2$ at this level of regularity.