Two-step hybrid methods adapted to the numerical integration of perturbed oscillators

Abstract

Two-step hybrid methods specially adapted to the numerical integration of perturbed oscillators are obtained. The formulation of the methods is based on a refinement of classical Taylor expansions due to Scheifele [{\em Z. Angew. Math. Phys.}, {\bf 22}, 186--210 (1971)]. The key property is that those algorithms are able to integrate exactly harmonic oscillators with frequency $ω$ and that, for perturbed oscillators, the local error contains the (small) perturbation parameter as a factor. The methods depend on a parameter $ν=ωh$, where $h$ is the stepsize. Based on the B2-series theory of Coleman [{\em IMA J. Numer. Anal.}, {\bf 23}, 197--220 (2003)] we derive the order conditions of this new type of methods. The linear stability and phase properties are examined. The theory is illustrated with some fourth- and fifth-order explicit schemes. Numerical results carried out on an assortment of test problems (such as the integration of the orbital motion of earth satellites) show the relevance of the theory.