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

Houssem Jerbi, Mohamed Omri, Mourad Kchaou, Theodore E. Simos and Charalampos Tsitouras
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Houssem Jerbi: Department of Industrial Engineering, College of Engineering, University of Ha’il, Hail 1234, Saudi Arabia
Mohamed Omri: Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah 21589, Saudi Arabia
Mourad Kchaou: Department of Electrical Engineering, College of Engineering, University of Ha’il, Hail 1234, Saudi Arabia
Theodore E. Simos: College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, China
Charalampos Tsitouras: General Department, Euripus Campus, National & Kapodistrian University of Athens, GR34-400 Psachna, Greece

Mathematics, 2022, vol. 10, issue 4, 1-12

Abstract: 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.

Keywords: initial value problem; Runge-Kutta-Nyström pairs; differential evolution; Kepler orbits (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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