Multistep Numerical Methods Based on the Scheifele G-Functions with Application to Satellite Dynamics

Authors:
Pablo Martín;José M. Ferrándiz
Affiliations:
-;-
Venue:
SIAM Journal on Numerical Analysis
Year:
1997

Citing 0
Cited 8

Increasing the accuracy in the numerical integration of perturbed oscillators with new methods

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New methods for oscillatory problems based on classical codes

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A variable-stepsize variable-order multistep method for the integration of perturbed linear problems

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Runge-Kutta methods adapted to the numerical integration of oscillatory problems

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Scheifele two-step methods for perturbed oscillators

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

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Accurate Numerical Integration of Perturbed Oscillatory Systems in Two Frequencies

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An algorithm for exact integration of some forced and damped oscillatory problems, based in the τ-functions

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Abstract

In 1971 Scheifele designed a method for the integration of perturbed oscillators by refining the classical method of power series. This method consists of defining a certain sequence of entire functions and finding the sought solution in the form of a linear combination of them using coefficients obtained through recurrence formulas. In spite of the good behavior of the Scheifele method, it is only practical for use in a few particular cases, in which the force function is very simple, given the complexity of the preliminary calculations needed to obtain the recurrence formulas. This problem is resolved in this article by converting the G-functions method into a multistep scheme. The key property is that those algorithms are able to integrate, without truncation error, harmonic oscillators involving various frequencies, and that, for perturbed problems, the local error contains the (small) perturbation parameter as a factor. Therefore, the methods are useful for finding highly accurate solutions to problems whose solutions are close to quasi-periodic ones. An application is performed in investigating the behavior of the proposed schemes for the integration of the orbital motion of Earth satellites.