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Eighth Order Two-Step Methods Trained to Perform Better on Keplerian-Type Orbits

Author

Listed:
  • Vladislav N. Kovalnogov

    (Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 432027 Ulyanovsk, Russia)

  • Ruslan V. Fedorov

    (Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 432027 Ulyanovsk, Russia)

  • Andrey V. Chukalin

    (Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 432027 Ulyanovsk, Russia)

  • Theodore E. Simos

    (Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 432027 Ulyanovsk, Russia
    College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, China
    Department of Medical Research, China Medical University Hospital, China Medical University, Taichung City 40402, Taiwan
    Data Recovery Key Laboratory of Sichuan Province, Neijiang Normal University, Neijiang 641100, China)

  • Charalampos Tsitouras

    (General Department, National & Kapodistrian University of Athens, 10679 Athens, Greece
    Department of Business Administration, Hellenic Open University, 26335 Patras, Greece)

Abstract

The family of Numerov-type methods that effectively uses seven stages per step is considered. All the coefficients of the methods belonging to this family can be expressed analytically with respect to four free parameters. These coefficients are trained through a differential evolution technique in order to perform best in a wide range of Keplerian-type orbits. Then it is observed with extended numerical tests that a certain method behaves extremely well in a variety of orbits (e.g., Kepler, perturbed Kepler, Arenstorf, Pleiades) for various steplengths used by the methods and for various intervals of integration.

Suggested Citation

  • Vladislav N. Kovalnogov & Ruslan V. Fedorov & Andrey V. Chukalin & Theodore E. Simos & Charalampos Tsitouras, 2021. "Eighth Order Two-Step Methods Trained to Perform Better on Keplerian-Type Orbits," Mathematics, MDPI, vol. 9(23), pages 1-19, November.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:23:p:3071-:d:690816
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    References listed on IDEAS

    as
    1. Ch. TSITOURAS, 2006. "Explicit Eighth Order Two-Step Methods With Nine Stages For Integrating Oscillatory Problems," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 17(06), pages 861-876.
    2. J. M. Franco & L. Rández, 2018. "Eighth-order explicit two-step hybrid methods with symmetric nodes and weights for solving orbital and oscillatory IVPs," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 29(01), pages 1-18, January.
    3. Franco, J.M. & Rández, L., 2016. "Explicit exponentially fitted two-step hybrid methods of high order for second-order oscillatory IVPs," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 493-505.
    Full references (including those not matched with items on IDEAS)

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