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.

翻译：非线性薛定谔方程低正则性解的数值逼近是众所周知的难题，若同时要求格式保持结构则更为困难。近期研究已成功为该方程建立了对称低正则性积分器。然而，迄今为止，所有先前的对称低正则性算法均为完全隐式，因此需要在每个时间步求解非线性方程，导致迭代过程中产生显著的计算开销。本工作首次为非线性薛定谔方程引入了完全显式（多步）对称低正则性积分器。我们展示了此类完整格式族的构造方法，其显著优势在于能够以显式形式对称化大量现有的低正则性积分器。我们提供了所提格式的严格收敛性分析，并通过数值算例证明了新格式在结构保持特性上的优越性，以及相较于隐式方法在计算成本上的显著降低。