A symmetric low-regularity integrator for the nonlinear Schr\"odinger equation

TL;DR

This paper presents a symmetric low-regularity numerical scheme for the nonlinear Schrödinger equation that achieves fractional convergence beyond classical methods, effective on various domains with lower regularity assumptions.

Contribution

The paper introduces a novel symmetric low-regularity integrator for NLS that surpasses classical techniques in regularity requirements and domain generality, with proven fractional convergence.

Findings

Achieves fractional convergence from first to second order in $L^2$-norm.

Works on both torus and bounded domains with Dirichlet boundary conditions.

Numerical experiments show improved structure preservation and error constants.

Abstract

We introduce and analyze a symmetric low-regularity scheme for the nonlinear Schr\"odinger (NLS) equation beyond classical Fourier-based techniques. We show fractional convergence of the scheme in $L^2$-norm, from first up to second order, both on the torus $\mathbb{T}^d$ and on a smooth bounded domain $\Omega \subset \mathbb{R}^d$, $d\le 3$, equipped with homogeneous Dirichlet boundary condition. The new scheme allows for a symmetric approximation to the NLS equation in a more general setting than classical splitting, exponential integrators, and low-regularity schemes (i.e. under lower regularity assumptions, on more general domains, and with fractional rates). We motivate and illustrate our findings through numerical experiments, where we witness better structure preserving properties and an improved error-constant in low-regularity regimes.