Runge–Kutta Pairs of Orders 5(4) Trained to Best Address Keplerian Type Orbits

Vladislav N. Kovalnogov, Ruslan V. Fedorov, Tamara V. Karpukhina, Theodore E. Simos and Charalampos Tsitouras
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Vladislav N. Kovalnogov: Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
Ruslan V. Fedorov: Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
Tamara V. Karpukhina: Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
Theodore E. Simos: Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
Charalampos Tsitouras: General Department, Euripus Campus, National & Kapodistrian University of Athens, 34400 Athens, Greece

Mathematics, 2021, vol. 9, issue 19, 1-12

Abstract: The derivation of Runge–Kutta pairs of orders five and four that effectively uses six stages per step is considered. The coefficients provided by such a method are 27 and have to satisfy a system of 25 nonlinear equations. Traditionally, various solutions have been tried. Each of these solutions makes use of some simplified assumptions and offers different families of methods. Here, we make use of the most celebrated family to appear in the literature, where we may use as the last stage the first function evaluation from the next step (FSAL property). The family under consideration has the advantage of being solved explicitly. Actually, we arrive at a subsystem where all the coefficients are found with respect to five free parameters. These free parameters are adjusted (trained) in order to deliver a pair that outperforms other similar pairs of orders 5 ( 4 ) in Keplerian type orbits, e.g., Kepler, perturbed Kepler, Arenstorf orbit or Pleiades. The training uses differential evolution technique. The finally proposed pair has a remarkable performance and offers on average more than a digit of accuracy in a variety of orbits.

Keywords: initial value problem; Runge–Kutta pairs; differential evolution; Kepler orbits (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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