Numerical Algorithms

, Volume 66, Issue 1, pp 147–176

Order conditions for RKN methods solving general second-order oscillatory systems

  • Xiong You
  • Jinxi Zhao
  • Hongli Yang
  • Yonglei Fang
  • Xinyuan Wu
Original Paper

DOI: 10.1007/s11075-013-9728-5

Cite this article as:
You, X., Zhao, J., Yang, H. et al. Numer Algor (2014) 66: 147. doi:10.1007/s11075-013-9728-5

Abstract

This paper proposes and investigates the multidimensional extended Runge-Kutta-Nyström (ERKN) methods for the general second-order oscillatory system y″ + My = f(y, y′) where M is a positive semi-definite matrix containing implicitly the frequencies of the problem. The work forms a natural generalization of our previous work on ERKN methods for the special system y″ + My = f(y) (H. Yang et al. Extended RKN-type methods for numerical integration of perturbed oscillators, Comput. Phys. Comm. 180 (2009) 1777–1794 and X. Wu et al., ERKN integrators for systems of oscillatory second-order differential equations, Comput. Phys. Comm. 181 (2010) 1873–1887). The new ERKN methods, with coefficients depending on the frequency matrix M, incorporate the special structure of the equation brought by the term My into both internal stages and updates. In order to derive the order conditions for the ERKN methods, an extended Nyström tree (EN-tree) theory is established. The results of numerical experiments show that the new ERKN methods are more efficient than the general-purpose RK methods and the adapted RKN methods with the same algebraic order in the literature.

Keywords

Extended Runge-Kutta-Nyström type methods Extended Nyström trees Order conditions Second-orderoscillatory systems 

Mathematics Subject Classifications (2010)

65L05 65L06 

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Xiong You
    • 1
    • 2
  • Jinxi Zhao
    • 1
  • Hongli Yang
    • 3
  • Yonglei Fang
    • 4
  • Xinyuan Wu
    • 5
  1. 1.State Key Laboratory for Novel Software Technology at Nanjing UniversityNanjing UniversityNanjingPeople’s Republic of China
  2. 2.Department of Applied MathematicsNanjing Agricultural UniversityNanjingPeople’s Republic of China
  3. 3.Institute of MathematicsNanjing Institute of TechnologyNanjingPeople’s Republic of China
  4. 4.Department of Mathematics and Information ScienceZaozhuang UniversityZaozhuangPeople’s Republic of China
  5. 5.Department of MathematicsNanjing UniversityNanjingPeople’s Republic of China

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