A new approach for multistep numerical methods in several frequencies for perturbed oscillators

Abstract

This article presents a new multi-step numerical method based on φ-function series and designed to integrate forced oscillators with precision. The new algorithm retains the good properties of the MDFpPC methods while presenting the advantages of greater precision and that of integrating the non-perturbed problem without any discretization error. In addition, this new method permits a single formulation to be obtained from the MDFpPC schemes independently of the parity of the number of steps, which facilitates the design of a computational algorithm thus permitting improved implementation in a computer. The construction of a new method for accurately integrating the homogenous problem is necessary if a method is sought which would be comparable to the methods based on Scheifele G-function series, very often used when problems of satellite orbital dynamics need to be resolved without discretization error. Greater precision compared to the MDFpPC methods and other known integrators is demonstrated by overcoming stiff and highly oscillatory problems with the new method and comparing approximations obtained with those calculated by means of other integrators.