Explicit symmetric low-regularity integrators for the nonlinear Schr\"odinger equation

TL;DR

This paper introduces the first fully explicit symmetric low-regularity integrators for the nonlinear Schrödinger equation, offering reduced computational cost while preserving structure and ensuring convergence.

Contribution

It develops a new class of explicit symmetric low-regularity integrators, enabling structure preservation and efficiency for the nonlinear Schrödinger equation.

Findings

Proposed explicit schemes demonstrate favorable structure preservation.

Numerical results show significant cost reduction compared to implicit methods.

Rigorous convergence analysis confirms reliability of the schemes.

Abstract

The numerical approximation of low-regularity solutions to the nonlinear Schr\"odinger equation is notoriously difficult and even more so if structure-preserving schemes are sought. Recent works have been successful in establishing symmetric low-regularity integrators for this equation. However, so far, all prior symmetric low-regularity algorithms are fully implicit, and therefore require the solution of a nonlinear equation at each time step, leading to significant numerical cost in the iteration. In this work, we introduce the first fully explicit (multi-step) symmetric low-regularity integrators for the nonlinear Schr\"odinger equation. We demonstrate the construction of an entire class of such schemes which notably can be used to symmetrise (in explicit form) a large amount of existing low-regularity integrators. We provide rigorous convergence analysis of our schemes and numerical examples demonstrating both the favourable structure preservation properties obtained with our novel schemes, and the significant reduction in computational cost over implicit methods.