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

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.