Skip to main content
SpringerLink
Log in

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

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aubry, A., Chartier, P.: Pseudo-symplectic Runge-Kutta methods. BIT. 38, 439–461 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  2. Avdyushev, V. A.: Special perturbation theory methods in celestial mechanics, I. Principles for the construction and substantiation of the application. Russ. Phys. J. 49, 1344–1353 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  3. Franco, J. M.: Runge-Kutta-Nyström methods adapted to the numerical integration of perturbed oscillators. Comput. Phys. Comm. 147, 770–787 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  4. García-Alonso, F., Reyes, J., Ferrádiz, J., Vigo-Aguiar, J.: Accurate numerical integration of perturbed oscillatory systems in two frequencies. ACM Trans. Math. Softw. 36, 21–34 (2009).

    Article  Google Scholar 

  5. López, D. J., Martín, P.: A numerical method for the integration of perturbed linear problems. Appl. Math. Comput. 96, 65–73 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  6. Ramos, H., Vigo-Aguiar, J.: Variable-stepsize Chebyshev-type methods for the integration of second-order I.V.P.s. J. Comput. Appl. Math. 204, 102–113 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  7. Jordan, D. W., Smith, P.: Nonlinear ordinary differential equations, 4th edn. An introduction for scientists and engineers. Oxford University Press, Oxford, (2007)

  8. Wu, X., You, X., Li, J.: Note on derivation of order conditions for ARKN methods for perturbed oscillator. Comput. Phys. Comm. 180, 1545–1549 (2009).

    Article  MathSciNet  Google Scholar 

  9. Wu, X., You, X., Xia, J.: Order conditions for ARKN methods solving oscillatory systems. Comput. Phys. Comm. 180, 2250–2257 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  10. Wu, X., Wang, B.: Multidimensional adapted Runge-Kutta-Nystrom methods for oscillatory systems. Comput. Phys. Comm. 181, 1955–1962 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  11. Yang, H., Wu, X., You, X., Fang, Y.: Extended RKN-type methods for numerical integration of perturbed oscillators. Comput. Phys. Comm. 180, 1777–1794 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  12. Wu, X., You, X., Shi, W., Wang, B.: ERKN integrators for systems of oscillatory secondorder differential equations. Comput. Phys. Comm. 181, 1873–1887 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  13. Wu, X., You, X., Wang, B.: Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer-Verlag, Berlin, Heidelberg (2013).

    Book  MATH  Google Scholar 

  14. Hairer, E., Nørsett, S. P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problem. Springer-Verlag, Berlin, Heidelberg (1993).

    Google Scholar 

  15. Butcher, J. C.: The Numerical Analysis of Ordinary Differential Equations, 2nd edn. Wiley, New York, (2008).

  16. Johnson, W. P.: The curious history of Faà di Bruno’s Formula. Math. Assoc. Am. 109, 217–234 (2002).

    Article  MATH  Google Scholar 

  17. González, A. B., Martín, P., Farto, J. M.: A new family of Runge-Kutta type methods for the numerical integration of perturbed oscillators. Numer. Math. 82, 635–646 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  18. Weinberger, H. F.: A First Course in Partial Differential Equations with Complex Variables and Transform Methods. Dover Publications Inc., New York (1965).

    MATH  Google Scholar 

  19. Franco, J. M.: New methods for oscillatory systems based on ARKN methods. Appl. Numer. Math. 56, 1040–1053 (2006).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory for Novel Software Technology at Nanjing University, Nanjing University, Nanjing, 210093, People’s Republic of China

    Xiong You & Jinxi Zhao

  2. Department of Applied Mathematics, Nanjing Agricultural University, Nanjing, 210095, People’s Republic of China

    Xiong You

  3. Institute of Mathematics, Nanjing Institute of Technology, Nanjing, 211167, People’s Republic of China

    Hongli Yang

  4. Department of Mathematics and Information Science, Zaozhuang University, Zaozhuang, 277160, People’s Republic of China

    Yonglei Fang

  5. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Xinyuan Wu

Authors
  1. Xiong You
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinxi Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hongli Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yonglei Fang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xinyuan Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xinyuan Wu.

Additional information

This research was in part supported by NSF of China (No. 11171155, No. 11271186, No. 11101357), the Fundamental Research Fund for the Central Universities (No. Y0201100265), the Research Fund for the Doctoral Program of Higher Education (No. 20100091110033), the foundation of Shangdong Outstanding Young Scientists Award Project (No. BS2010SF031), the foundation of Scientific Research Project of Shangdong Universities (No. J11LG69), NSF of Shandong Province, China (No. ZR2011AL006) and Talented Faculty Fund of Nanjing Institute of Technology (No. YKJ201114).

Rights and permissions

Reprints and permissions

About this article

Cite this article

You, X., Zhao, J., Yang, H. et al. Order conditions for RKN methods solving general second-order oscillatory systems. Numer Algor 66, 147–176 (2014). https://doi.org/10.1007/s11075-013-9728-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-013-9728-5

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation