Optimal convergence and long-time conservation of exponential integration for Schr\"{o}dinger equations in a normal or highly oscillatory regime

TL;DR

This paper develops exponential integrators for nonlinear Schr"odinger equations that preserve energy, achieve optimal convergence, and maintain near conservation of physical quantities over long times, applicable in highly oscillatory regimes.

Contribution

It introduces continuous-stage exponential integrators with energy preservation and demonstrates their optimal convergence and long-term conservation properties.

Findings

Exact energy preservation for Hamiltonian systems.

Optimal convergence rates of the proposed integrators.

Near conservation of actions, momentum, and density over long times.

Abstract

In this paper, we formulate and analyse exponential integrations when applied to nonlinear Schr\"{o}dinger equations in a normal or highly oscillatory regime. A kind of exponential integrators with energy preservation, optimal convergence and long time near conservations of actions, momentum and density will be formulated and analysed. To this end, we derive continuous-stage exponential integrators and show that the integrators can exactly preserve the energy of Hamiltonian systems. Three practical energy-preserving integrators are presented. It is shown that these integrators exhibit optimal convergence and have near conservations of actions, momentum and density over long times. A numerical experiment is carried out to support all the theoretical results presented in this paper. Some applications of the integrators to other kinds of ordinary/partial differential equations are also presented.