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

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 [Z. Angew. Math. Phys., 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 = ω h, where h is the stepsize. Based on the B2-series theory of Coleman [IMA J. Numer. Anal., 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 fourthand 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. AMS Classification : 65L05