Resonances and instabilities in symmetric multistep methods

TL;DR

This paper reveals that symmetric multistep methods, though useful for long-term integrations like planetary orbits, can suffer from resonances and instabilities at specific stepsizes, especially in high-order versions, which was previously overlooked.

Contribution

The study uncovers the resonance and instability issues in symmetric multistep methods that are not apparent from linear stability analysis, highlighting the need for cautious application.

Findings

Resonances occur at specific stepsizes in symmetric multistep methods.

High-order methods are more prone to instabilities than low-order ones.

Number of problematic stepsizes increases with solution frequencies.

Abstract

The symmetric multistep methods developed by Quinlan and Tremaine (1990) are shown to suffer from resonances and instabilities at special stepsizes when used to integrate nonlinear equations. This property of symmetric multistep methods was missed in previous studies that considered only the linear stability of the methods. The resonances and instabilities are worse for high-order methods than for low-order methods, and the number of bad stepsizes increases with the number frequencies present in the solution. Symmetric methods are still recommended for some problems, including long-term integrations of planetary orbits, but the high-order methods must be used with caution.