Runge-Kutta-Nyström Pairs of Orders 8(6) with Coefficients Trained to Perform Best on Classical Orbits

In this study, we consider eight stages per step family of explicit Runge-Kutta-Nyström pairs of orders eight and six. The pairs from this family effectively use eight stages for each step. The coefficients provided by such a method are much less than the number of non linear order conditions requir...

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Bibliographic Details
Published in:Mathematics (Basel) 2022-02, Vol.10 (4), p.654
Main Authors: Jerbi, Houssem, Omri, Mohamed, Kchaou, Mourad, Simos, Theodore E., Tsitouras, Charalampos
Format: Article
Language:English
Subjects:
Coefficients
Control algorithms
differential evolution
Evolutionary computation
initial value problem
Kepler orbits
Mathematics
Orbits
Parameters
Ratios
Runge-Kutta method
Runge-Kutta-Nyström pairs
Subsystems
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Summary:In this study, we consider eight stages per step family of explicit Runge-Kutta-Nyström pairs of orders eight and six. The pairs from this family effectively use eight stages for each step. The coefficients provided by such a method are much less than the number of non linear order conditions required to be solved. Thus, we traditionally apply various simplified assumptions in order to address this drawback. The assumptions taken in the family we consider here deliver a subsystem where all the coefficients are evaluated successively and explicitly with respect to five free parameters. We train (adjust) these free parameters in order to derive a certain pair that outperforms other similar pairs of orders 8(6) in Keplerian type orbits, e.g., Kepler, perturbed Kepler, Arenstorf orbit, or Pleiades. Differential evolution technique is used for the training. The pair that we finally present offers about an additional digit of accuracy in a variety of orbits.
ISSN:2227-7390
2227-7390
DOI:10.3390/math10040654