Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical Orbits
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...
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MDPI AG
2021-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/12/1342 |
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| author | Yu-Cheng Shen Chia-Liang Lin Theodore E. Simos Charalampos Tsitouras |
| author_facet | Yu-Cheng Shen Chia-Liang Lin Theodore E. Simos Charalampos Tsitouras |
| author_sort | Yu-Cheng Shen |
| collection | DOAJ |
| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1.8</mn></mrow></semantics></math></inline-formula> digits of accuracy is gained on average over conventional pairs, which is truly remarkable for methods coming from the same family and order. |
| first_indexed | 2024-03-10T10:32:40Z |
| format | Article |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T10:32:40Z |
| publishDate | 2021-06-01 |
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| series | Mathematics |
| spelling | doaj.art-db1be540710945daafdb7bd5d64b23332023-11-21T23:32:31ZengMDPI AGMathematics2227-73902021-06-01912134210.3390/math9121342Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical OrbitsYu-Cheng Shen0Chia-Liang Lin1Theodore E. Simos2Charalampos Tsitouras3Department of Preschool Education, School of Educational Sciences, Huaiyin Campus, Huaiyin Normal University, Huaian City 223300, ChinaDepartment of Visual Communications, School of Arts, Huzhou University, Huzhou 313000, ChinaCollege of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, ChinaGeneral Department, GR34-400 Euripus Campus, National & Kapodistrian University of Athens, 15772 Athens, GreeceWe 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1.8</mn></mrow></semantics></math></inline-formula> digits of accuracy is gained on average over conventional pairs, which is truly remarkable for methods coming from the same family and order.https://www.mdpi.com/2227-7390/9/12/1342initial value problemKepler-type orbitsRunge–Kuttadifferential evolution |
| spellingShingle | Yu-Cheng Shen Chia-Liang Lin Theodore E. Simos Charalampos Tsitouras Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical Orbits Mathematics initial value problem Kepler-type orbits Runge–Kutta differential evolution |
| title | Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical Orbits |
| title_full | Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical Orbits |
| title_fullStr | Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical Orbits |
| title_full_unstemmed | Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical Orbits |
| title_short | Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical Orbits |
| title_sort | runge kutta pairs of orders 6 5 with coefficients trained to perform best on classical orbits |
| topic | initial value problem Kepler-type orbits Runge–Kutta differential evolution |
| url | https://www.mdpi.com/2227-7390/9/12/1342 |
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