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Symplecticity-preserving continuous-stage Runge–Kutta–Nyström methods

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  • Tang, Wensheng
  • Zhang, Jingjing

Abstract

In this paper, we develop continuous-stage Runge–Kutta–Nyström (csRKN) methods for numerical integration of second-order ordinary differential equations (ODEs) written in the form q¨=f(t,q). Numerous ODEs in such form can be reduced to first-order ODEs with the separable form of Hamiltonian systems and symplecticity-preserving discretizations of these systems are of interest. For the sake of designing symplectic csRKN methods, we explore the sufficient conditions for symplecticity, and we show a simple way to derive symplectic RKN-type integrators by using Legendre polynomial expansion. Numerical results show the efficiency of the presented methods.

Suggested Citation

  • Tang, Wensheng & Zhang, Jingjing, 2018. "Symplecticity-preserving continuous-stage Runge–Kutta–Nyström methods," Applied Mathematics and Computation, Elsevier, vol. 323(C), pages 204-219.
    • Handle: RePEc:eee:apmaco:v:323:y:2018:i:c:p:204-219
    DOI: 10.1016/j.amc.2017.11.054
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    References listed on IDEAS

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    1. Tang, Wensheng & Lang, Guangming & Luo, Xuqiong, 2016. "Construction of symplectic (partitioned) Runge-Kutta methods with continuous stage," Applied Mathematics and Computation, Elsevier, vol. 286(C), pages 279-287.
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    Cited by:

    1. Tang, Wensheng & Zhang, Jingjing, 2019. "Symmetric integrators based on continuous-stage Runge–Kutta–Nyström methods for reversible systems," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 1-12.
    2. Amodio, Pierluigi & Brugnano, Luigi & Iavernaro, Felice, 2019. "A note on the continuous-stage Runge–Kutta(–Nyström) formulation of Hamiltonian Boundary Value Methods (HBVMs)," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
    3. Tang, Wensheng & Sun, Yajuan & Zhang, Jingjing, 2019. "High order symplectic integrators based on continuous-stage Runge-Kutta-Nyström methods," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 670-679.
    4. Tang, Wensheng, 2018. "A note on continuous-stage Runge–Kutta methods," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 231-241.
    5. Jun Zhang & Jingjing Zhang & Shangyou Zhang, 2023. "Explicit Symplectic Runge–Kutta–Nyström Methods Based on Roots of Shifted Legendre Polynomial," Mathematics, MDPI, vol. 11(20), pages 1-13, October.
    6. Zhang, Jingjing, 2020. "An improved Störmer-Verlet method based on exact discretization for nonlinear oscillators," Applied Mathematics and Computation, Elsevier, vol. 386(C).

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    1. Tang, Wensheng & Sun, Yajuan & Zhang, Jingjing, 2019. "High order symplectic integrators based on continuous-stage Runge-Kutta-Nyström methods," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 670-679.
    2. Tang, Wensheng, 2018. "A note on continuous-stage Runge–Kutta methods," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 231-241.
    3. Tang, Wensheng & Zhang, Jingjing, 2019. "Symmetric integrators based on continuous-stage Runge–Kutta–Nyström methods for reversible systems," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 1-12.

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