Skip to main content
Log in

Adapted Falkner-type methods solving oscillatory second-order differential equations

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

Abstract

The classical Falkner methods (Falkner, Phil Mag S 7:621, 1936) are well-known for solving second-order initial-value problems u′′(t) = f(t, u(t), u′(t)). In this paper, we propose the adapted Falkner-type methods for the systems of oscillatory second-order differential equations u′′(t) + Mu(t) = g(t, u(t)) and make a rigorous error analysis. The error bounds for the global errors on the solution and the derivative are presented. In particular, the error bound for the global error of the solution is shown to be independent of ||M||. We also give a stability analysis and plot the regions of stability for our new methods. Numerical examples are included to show that our new methods are very competitive compared with the reformed Falkner methods in the scientific literature.

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

Access this article

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. Franco, J.M., Palacian, J.F.: High order adaptive methods of Nyström-Cowell type. J. Comput. Appl. Math. 81, 115–134 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. López, D.J., Martín, P., Farto, J.M.: Generalization of the Störmer method for perturbed oscillators without explicit firrst derivatives. J. Comput. Appl. Math. 111, 123–132 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Van de Vyver, H.: Scheifele two-step methods for perturbed oscillators. J. Comput. Appl. Math. 224, 415–432 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Li, J.Y., Wang, B., You, X., Wu, X.Y.: Two-step extended RKN methods for oscillatory systems. Comput. Phys. Commun. 182, 2486–2507 (2011)

    Article  MathSciNet  Google Scholar 

  5. Falkner, V.M.: A method of numerical solution of differential equations. Phil. Mag. S. 7, 621 (1936)

    Google Scholar 

  6. Collatz, L.: The Numerical Treatment of Differential Equations. Springer, Berlin (1966)

    Google Scholar 

  7. Vigo-Aguiar, J., Ramos, H.: Variable stepsize implementation of multistep methods for y″ = f(x, y, y′). J. Comput. Appl. Math. 192, 114–131 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ramos, H., Lorenzo, C.: Review of explicit Falkner methods and its modifications for solving special second-order I.V.P.s. Comput. Phys. Commun. 181, 1833–1841 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Wu, X.Y., You, X., Shi, W., Wang, B.: ERKN integrators for systems of oscillatory second-order differential equations. Comput. Phys. Commun. 181, 1873–1887 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Vigo-Aguiar, J., Ferrandiz, J.M.: A general procedure for the adaptation of multistep algorithms to the integration of oscillatory problems. SIAM J. Numer. Anal. 35(4), 1684–1708 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Swope, C., Andersen, H.C., Berens, P.H., Wilson, K.R.: A computer simulation method for the calculation of equilibrium constants for the formation of physical clusters of molecules: application to small water clusters. J. Chem. Phys. 76, 637 (1982)

    Article  Google Scholar 

  13. Hayes, L.J.: Galerkin alternating-direction methods for nonrectangular regions using patch approximations. SIAM J. Numer. Anal. 18, 627–643 (1987)

    Article  MathSciNet  Google Scholar 

  14. Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Springer, Berlin (1987)

    MATH  Google Scholar 

  15. Wu, X.Y.: A note on stability of multidimensional adapted Runge-Kutta-Nyström methods for oscillatory systems. Appl. Math. Model. (2012). doi:10.1016/j.apm.2012.01.053

    Google Scholar 

  16. Coleman, J.P., Ixaru, L. Gr.: P-stability and exponential-fitting methods for y″ = f(x, y). IMA J. Numer. Anal. 16, 179–199 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Vigo-Aguiar, J., Simos, T.E., Ferrándiz, J.M.: Controlling the error growth in long-term numerical integration of perturbed oscillations in one or more frequencies. Proc. R. Soc. Lond., A 460, 561–567 (2004)

    Article  MATH  Google Scholar 

  18. Hairer, E., Nøsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd revised edn. Springer, Berlin (1993)

    MATH  Google Scholar 

  19. Tocino, A., Vigo-Aguiar, J.: Symplectic conditions for exponential fitting Runge–Kutta–Nyström methods. Math. Comput. Model. 42, 873–876 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wu, X.Y., Wang, B.: Multidimensional adapted Runge–Kutta–Nyström methods for oscillatory systems. Comput. Phys. Commun. 181, 1955–1962 (2010)

    Article  MATH  Google Scholar 

  21. Shi, W., Wu, X.Y.: On symplectic and symmetric ARKN methods. Comput. Phys. Commun. 183, 1250–1258 (2012)

    Article  MathSciNet  Google Scholar 

  22. Wang, B., Wu, X.Y.: A new high precision energy-preserving integrator for system of oscillatory second-order differential equations. Phys. Lett. A. 376, 1185–1190 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wu, X.Y., Wang, B., Xia, J.L.: Explicit symplectic multidimensional exponential fitting modified Runge–Kutta–Nystrom methods. BIT Numer. Math. (2012). doi:10.1007/s10543-012-0379-z

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xinyuan Wu.

Additional information

The research was supported in part by the Specialized Research Foundation for the Doctoral Program of Higher Education under Grant 20100091110033, by A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, by the 985 Project at Nanjing University under Grant 9112020301 and by the Natural Science Foundation of China under Grant 10771099.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, J., Wu, X. Adapted Falkner-type methods solving oscillatory second-order differential equations. Numer Algor 62, 355–381 (2013). https://doi.org/10.1007/s11075-012-9583-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-012-9583-9

Keywords

Navigation