Multistep numerical methods for the integration of oscillatory problems in several frequencies

The perturbed harmonic oscillator appear frequently in the mathematical modelling of many problems in physics and engineering. The harmonic oscillator has a special purpose in Astrodynamics, because the Kunstaanheimo–Stiefel (KS) and Burdet–Ferrándiz (BF) transformations reduce the Kepler problem to harmonic oscillators.A new multi-step methods of numerical integration are introduced that generalize SMF ones. They are defined for arbitrary order and have similar properties to the former methods.Modified methods allowing step variations, whose coefficients are computed from relations of recurrence, are derived, what considerably improve the implementation of the algorithms.These methods are based in a sequence of analytical φφ-functions dependent on two parameters αα and ββ that generalizes the Scheifele’s G-functions and that, under wide hypothesis, allow us obtain the solution of harmonic oscillator.In this paper a new methodology is generated to solve the problem which the φφ-functions series create regarding the calculus of recurrence relations transforming the method into a multistep scheme. The methodology is implemented in a computational algorithm, which lets us solve in a general way the problems of physics and engineering which are modalized by means of the study of the harmonic oscillators.Numerical examples already used by other authors are presented. They show how the new developed methods in this paper may compete in accuracy or efficiency with other well-reputed algorithms.