Energy-preserving trigonometrically fitted continuous stage Runge-Kutta-Nyström methods for oscillatory Hamiltonian systems

  • Jiyong LiEmail author
  • Yachao Gao
Original Paper


Recently, continuous-stage Runge-Kutta-Nyström (CSRKN) methods for solving numerically second-order initial value problem \(q^{\prime \prime }= f(q)\) have been proposed and developed by Tang and Zhang (Appl. Math. Comput. 323, 204–219, 2018). This problem is equivalent to a separable Hamiltonian system when f(q) = −∇U(q) with smooth function U(q). Symplecticity-preserving discretizations of this system were studied in that paper. However, as an important representation of the Hamiltonian system, energy preservation has not been studied. In addition, many Hamiltonian systems in practical applications often have oscillatory characteristics so we should design special algorithms adapted to this feature. In this paper, we propose and study energy-preserving trigonometrically fitted CSRKN methods for oscillatory Hamiltonian systems. We extend the theory of trigonometrical fitting to CSRKN methods and derive sufficient conditions for energy preservation. We also study the symmetry and stability of the methods. Two symmetric and energy-preserving trigonometrically fitted schemes of order two and four, respectively, are constructed. Some numerical experiments are provided to confirm the theoretical expectations.


Trigonometrically fitted methods Continuous-stage Runge-Kutta–Nyström methods Order conditions Energy-preserving Oscillatory Hamiltonian systems 


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The authors are sincerely thankful to the anonymous referees for their constructive comments and valuable suggestions.

Funding Information

The research was supported in part by the Natural Science Foundation of China under Grant No: 11401164, by Hebei Natural Science Foundation of China under Grant No: A2014205136 and by Science Foundation of Hebei Normal University No:L2018J01.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceHebei Normal UniversityShijiazhuangPeople’s Republic of China
  2. 2.Hebei Key Laboratory of Computational Mathematics and ApplicationsShijiazhuangPeople’s Republic of China

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