On the Parametric Instability Caused by Step Size Variation in Runge-Kutta-Nyström Methods

The parametric instability arising when ordinary differential equations (ODEs) are numerically integrated with Runge-Kutta-Nyström (RKN) methods with varying step sizes is investigated. Perturbation methods are used to quantify the critical step sizes associated with parametric instability. It is shown that there is no parametric instability for linear constant coefficient ODEs integrated with RKN methods that are based on A-stable Runge-Kutta methods, because the solution is nonincreasing in some norm for all positive step sizes, constant or varying.