On the Parametric Instability Caused by Step Size Variation in Runge-Kutta-Nystr\"om Methods

TL;DR

This paper investigates how varying step sizes in Runge-Kutta-Nyström methods can cause parametric instability in numerical solutions of ODEs, identifying critical step sizes through perturbation analysis.

Contribution

It provides a theoretical analysis of parametric instability in RKN methods with variable step sizes, including quantification of critical step sizes using perturbation techniques.

Findings

Solutions remain nonincreasing in some norm for all positive step sizes with A-stable RKN methods.

Perturbation methods effectively quantify critical step sizes for instability.

The study enhances understanding of stability issues in variable step size ODE integration.

Abstract

The parametric instability arising when ordinary differential equations (ODEs) are numerically integrated with Runge-Kutta-Nystr\"om (RKN) methods with varying step sizes is investigated. It is shown that when linear constant coefficient ODEs are integrated with RKN methods that are based on A-stable Runge-Kutta methods, the solution is nonincreasing in some norm for all positive step sizes, constant or varying. Perturbation methods are used to quantify the critical step sizes associated with parametric instability.