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Eighth-order explicit two-step hybrid methods with symmetric nodes and weights for solving orbital and oscillatory IVPs

Author

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  • J. M. Franco

    (IUMA, Departamento de Matemática Aplicada, Pza. San Francisco s/n., Universidad de Zaragoza, 50009 Zaragoza, Spain)

  • L. Rández

    (IUMA, Departamento de Matemática Aplicada, Pza. San Francisco s/n., Universidad de Zaragoza, 50009 Zaragoza, Spain)

Abstract

The construction of new two-step hybrid (TSH) methods of explicit type with symmetric nodes and weights for the numerical integration of orbital and oscillatory second-order initial value problems (IVPs) is analyzed. These methods attain algebraic order eight with a computational cost of six or eight function evaluations per step (it is one of the lowest costs that we know in the literature) and they are optimal among the TSH methods in the sense that they reach a certain order of accuracy with minimal cost per step. The new TSH schemes also have high dispersion and dissipation orders (greater than 8) in order to be adapted to the solution of IVPs with oscillatory solutions. The numerical experiments carried out with several orbital and oscillatory problems show that the new eighth-order explicit TSH methods are more efficient than other standard TSH or Numerov-type methods proposed in the scientific literature.

Suggested Citation

  • 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.
    • Handle: RePEc:wsi:ijmpcx:v:29:y:2018:i:01:n:s012918311850002x
    DOI: 10.1142/S012918311850002X
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    Citations

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    Cited by:

    1. Obaid Alshammari & Sondess Ben Aoun & Mourad Kchaou & Theodore E. Simos & Charalampos Tsitouras & Houssem Jerbi, 2024. "Eighth-Order Numerov-Type Methods Using Varying Step Length," Mathematics, MDPI, vol. 12(14), pages 1-14, July.
    2. Vladislav N. Kovalnogov & Ruslan V. Fedorov & Tamara V. Karpukhina & Theodore E. Simos & Charalampos Tsitouras, 2021. "Sixth Order Numerov-Type Methods with Coefficients Trained to Perform Best on Problems with Oscillating Solutions," Mathematics, MDPI, vol. 9(21), pages 1-12, October.
    3. 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.
    4. Lee, K.C. & Nazar, R. & Senu, N. & Ahmadian, A., 2024. "A promising exponentially-fitted two-derivative Runge–Kutta–Nyström method for solving y′′=f(x,y): Application to Verhulst logistic growth model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 219(C), pages 28-49.

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