Exact evolution of time-reversible symplectic integrators and their phase error for the harmonic oscillator

TL;DR

This paper derives exact solutions for the evolution of time-reversible symplectic integrators applied to harmonic oscillators, revealing their phase error characteristics and advantages over non-reversible methods.

Contribution

It provides a closed-form analytical solution for the evolution and phase error of time-reversible symplectic integrators, enhancing understanding of their accuracy.

Findings

Modified Hamiltonians converge via Lie series

Time-reversible integrators have less phase distortion

Analytical phase error can be precisely assessed

Abstract

The evolution of any factorized time-reversible symplectic integrators, when applied to the harmonic oscillator, can be exactly solved in a closed form. The resulting modified Hamiltonians demonstrate the convergence of the Lie series expansions. They are also less distorted than modified Hamiltonian of non-reversible algorithms. The analytical form for the modified angular frequency can be used to assess the phase error of any time-reversible algorithm.