Extended phase properties and stability analysis of RKN-type integrators for solving general oscillatory second-order initial value problems

Abstract

In this paper, we study in detail the phase properties and stability of numerical methods for general oscillatory second-order initial value problems whose right-hand side functions depend on both the position y and velocity y '. In order to analyze comprehensively the numerical stability of integrators for oscillatory systems, we introduce a novel linear test model y ?(t) + ? 2 y(t) + µ y '(t)=0 with µ<2?. Based on the new model, further discussions and analysis on the phase properties and stability of numerical methods are presented for general oscillatory problems. We give the new definitions of dispersion and dissipation which can be viewed as an essential extension of the traditional ones based on the linear test model y ?(t) + ? 2 y(t)=0. The numerical experiments are carried out, and the numerical results showthatthe analysisofphase properties and stability presentedinthispaper ismoresuitableforthenumericalmethodswhentheyareappliedtothe generaloscillatory second-order initial value problem involving both the position and velocity.