Scheifele two-step methods for perturbed oscillators

Two-step 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 [G. Scheifele, On the numerical integration of perturbed linear oscillating systems, Z. Angew. Math. Phys. 22 (1971) 186–210]. The key property is that those algorithms are able to integrate exactly harmonic oscillators with frequency ωω. The methods depend on a parameter ν=ωh, where hh is the stepsize. Based on the B2-series theory of Coleman [J.P. Coleman, Order conditions for a class of two-step methods for y″=f(x,y)y″=f(x,y), IMA J. Numer. Anal. 23 (2003) 197–220] we derive the order conditions of this new type of method. 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.