Forward Symplectic Integrators and the Long Time Phase Error in Periodic Motions

TL;DR

This paper demonstrates that using correctable forward symplectic integrators significantly reduces phase and energy errors in periodic motion simulations, achieving higher order accuracy.

Contribution

It introduces the use of correctable algorithms with forward symplectic integrators to enhance accuracy in long-term periodic motion simulations.

Findings

Fourth order forward integrators achieve sixth order phase accuracy.

Energy errors are reduced to eighth order with correctable algorithms.

The methods are validated on harmonic oscillator and Kepler problem.

Abstract

We show that when time-reversible symplectic algorithms are used to solve periodic motions, the energy error after one period is generally two orders higher than that of the algorithm. By use of correctable algorithms, we show that the phase error can also be eliminated two orders higher than that of the integrator. The use of fourth order forward time step integrators can result in sixth order accuracy for the phase error and eighth accuracy in the periodic energy. We study the 1-D harmonic oscillator and the 2-D Kepler problem in great details, and compare the effectiveness of some recent fourth order algorithms.