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Article

Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical Orbits

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Department of Preschool Education, School of Educational Sciences, Huaiyin Campus, Huaian City 223300, China
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Department of Visual Communications, School of Arts, Huzhou 313000, China
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College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, China
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Scientific and Educational Center “Digital Industry”, South Ural State University, 76 Lenin Ave., 454 080 Chelyabinsk, Russia
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Department of Medical Research, China Medical University Hospital, China Medical University, Taichung City 40402, Taiwan
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Data Recovery Key Laboratory of Sichuan Province, Neijiang Normal University, Neijiang 641100, China
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Section of Mathematics, Department of Civil Engineering, Democritus University of Thrace, 67100 Xanthi, Greece
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General Department, GR34-400 Euripus Campus, National & Kapodistrian University of Athens, 15772 Athens, Greece
*
Author to whom correspondence should be addressed.
Academic Editor: Arsen Palestini
Mathematics 2021, 9(12), 1342; https://doi.org/10.3390/math9121342
Received: 26 May 2021 / Revised: 6 June 2021 / Accepted: 8 June 2021 / Published: 10 June 2021
We consider a family of explicit Runge–Kutta pairs of orders six and five without any additional property (reduced truncation errors, Hamiltonian preservation, symplecticness, etc.). This family offers five parameters that someone chooses freely. Then, we train them in order for the presented method to furnish the best results on a couple of Kepler orbits, a certain interval and tolerance. Consequently, we observe an efficient performance on a wide range of orbital problems (i.e., Kepler for a variety of eccentricities, perturbed Kepler with various disturbances, Arenstorf and Pleiades). About 1.8 digits of accuracy is gained on average over conventional pairs, which is truly remarkable for methods coming from the same family and order. View Full-Text
Keywords: initial value problem; Kepler-type orbits; Runge–Kutta; differential evolution initial value problem; Kepler-type orbits; Runge–Kutta; differential evolution
MDPI and ACS Style

Shen, Y.-C.; Lin, C.-L.; Simos, T.E.; Tsitouras, C. Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical Orbits. Mathematics 2021, 9, 1342. https://doi.org/10.3390/math9121342

AMA Style

Shen Y-C, Lin C-L, Simos TE, Tsitouras C. Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical Orbits. Mathematics. 2021; 9(12):1342. https://doi.org/10.3390/math9121342

Chicago/Turabian Style

Shen, Yu-Cheng; Lin, Chia-Liang; Simos, Theodore E.; Tsitouras, Charalampos. 2021. "Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical Orbits" Mathematics 9, no. 12: 1342. https://doi.org/10.3390/math9121342

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