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Derivation of Three-Derivative Two-Step Runge–Kutta Methods

Author

Listed:
  • Xueyu Qin

    (National Key Laboratory of Computational Fluid Dynamics, Beihang University, Beijing 100191, China
    School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China)

  • Jian Yu

    (National Key Laboratory of Computational Fluid Dynamics, Beihang University, Beijing 100191, China
    School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China)

  • Chao Yan

    (National Key Laboratory of Computational Fluid Dynamics, Beihang University, Beijing 100191, China
    School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China)

Abstract

In this paper, we develop explicit three-derivative two-step Runge–Kutta (ThDTSRK) schemes, and propose a simpler general form of the order accuracy conditions ( p ≤ 7 ) by Albrecht’s approach, compared to the order conditions in terms of rooted trees. The parameters of the general high-order ThDTSRK methods are determined by utilizing the order conditions. We establish a theory for the A -stability property of ThDTSRK methods and identify optimal stability coefficients. Moreover, ThDTSRK methods can achieve the intended order of convergence using fewer stages than other schemes, making them cost-effective for solving the ordinary differential equations.

Suggested Citation

  • Xueyu Qin & Jian Yu & Chao Yan, 2024. "Derivation of Three-Derivative Two-Step Runge–Kutta Methods," Mathematics, MDPI, vol. 12(5), pages 1-16, February.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:5:p:711-:d:1347908
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