# Error estimates of a Fourier integrator for the cubic Schr\"odinger equation at low regularity

## TL;DR

This paper introduces a novel filtered Fourier integrator for the cubic Schrödinger equation that achieves improved low-regularity error estimates, surpassing classical schemes by handling rougher solutions with better convergence rates.

## Contribution

The authors develop a new low-regularity Fourier integrator with rigorous error analysis, enabling better convergence rates at low regularity than existing classical schemes.

## Key findings

- Achieves a global L^2 error estimate of order τ^{1/2 + (5-d)/12} for H^1 solutions in dimensions d ≤ 3.
- Breaks the natural order barrier of τ^{1/2} for classical schemes at low regularity.
- Handles rougher solutions directly at the L^2 level using discrete Strichartz estimates.

## Abstract

We present a new filtered low-regularity Fourier integrator for the cubic nonlinear Schr\"odinger equation based on recent time discretization and filtering techniques. For this new scheme, we perform a rigorous error analysis and establish better convergence rates at low regularity than known for classical schemes in the literature so far. In our error estimates, we combine the better local error properties of the new scheme with a stability analysis based on general discrete Strichartz-type estimates. The latter allow us to handle a much rougher class of solutions as the error analysis can be carried out directly at the level of $L^2$ compared to classical results \black in dimension $d$, \black which are limited to higher-order (sufficiently smooth) Sobolev spaces $H^s$ with $s>d/2$. In particular, we are able to establish a global error estimate in $L^2$ for $H^1$ solutions which is roughly of order $\tau^{ {1\over 2} + { 5-d \over 12} }$ in dimension $d \leq 3$ ($\tau$ denoting the time discretization parameter). This breaks the "natural order barrier" of $\tau^{1/2}$ for $H^1$ solutions which holds for classical numerical schemes (even in combination with suitable filter functions).