Advertisement

Numerical Algorithms

, Volume 78, Issue 3, pp 673–700 | Cite as

Extended explicit pseudo two-step RKN methods for oscillatory systems y + M y = f(y)

  • Jiyong Li
  • Shuo Deng
  • Xianfen Wang
Original Paper
  • 44 Downloads

Abstract

In this paper, extended explicit pseudo two-step Runge-Kutta-Nyström (EEPTSRKN) methods for the numerical integration of oscillatory system y + M y = f(y) are proposed and studied. These methods inherit the framework of explicit pseudo two-step Runge-Kutta-Nyström (EPTSRKN) methods (Cong et al. Comput. Math. Appl. 38, 17–39, 1999) and they are parallel methods. Furthermore, these new methods take into account the special feature of the oscillatory problem so that they integrate exactly unperturbed problem y + M y = 0. The study of corresponding order conditions shows that an s-stage EEPTSRKN method can obtain maximum step order s + 2 and maximum stage order s + 1. We make a global error analysis and present the global error bound for the new method, which is proved to be independent of ∥M∥ under suitable assumptions. Four two-stage methods with step order two, three, three, and four, respectively, are constructed and the corresponding stability regions are given. Numerical results show that our new methods are more efficient in comparison with other well-known high quality methods proposed in the scientific literature.

Keywords

Runge-Kutta-Nyström methods Order conditions Explicit methods Multi-frequency oscillatory systems Parallel methods 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors are grateful to the two anonymous reviewers for their valuable suggestions, which help improve this paper significantly.

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, by the Natural Science Foundation of China under Grant No: 11201113 and and by the Specialized Research Foundation for the Doctoral Program of Higher Education under Grant No: 20121303120001.

References

  1. 1.
    Nyström, E.J.: Ueber die numerische Integration von Differentialgleichungen. Acta Soc. Sci. Fenn. 50, 1–54 (1925)Google Scholar
  2. 2.
    Hairer, E., Wanner, G.: Solving ordinary differential equations II: stiff and differential algebraic problems, 2nd edn. Springer, Berlin (2002)zbMATHGoogle Scholar
  3. 3.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving ordinary differential equations I: nonstiff problems, 2nd edn. Springer, Berlin (2002)zbMATHGoogle Scholar
  4. 4.
    Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration, structure-preserving algorithms for ordinary differential equations, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  5. 5.
    Gautschi, W.: Numerical integration of ordinary differential equation based on trigonometric polynomials. Numer. Math. 3, 381–397 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ixaru, L.Gr.: Operations on oscillatory functions. Comput. Phys. Comm. 105, 1–19 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ixaru, L.Gr.: Numerical methods for differential equations and applications. Reidel, Lancaster, Dordrecht, Boston (1984)zbMATHGoogle Scholar
  8. 8.
    Ixaru, L.Gr.: Exponential Fitting. Kluwer Academic Publishers, Dordrecht, Boston, London (2004)CrossRefzbMATHGoogle Scholar
  9. 9.
    Wu, X., You, X., Xia, J.: Order conditions for ARKN methods solving oscillatory systems. Comput. Phys. Commun. 180, 2250–2257 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Wu, X., You, X., Shi, W., Wang, B.: ERKN integrators for systems of oscillatory second-order differential equations. Comput. Phys. Commun. 181, 1873–1887 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration illustrated by the Störmer-Verlet method. Acta. Numer. 12, 399–450 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hairer, E., Lubich, C.: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM. J. Numer. Anal. 38, 414–441 (2001)CrossRefzbMATHGoogle Scholar
  13. 13.
    Cong, N.H., Strehmel, K., Weiner, R.: Runge-Kutta-Nyström-type parallel block predictor-corrector methods. Adv. Comput. Math. 10, 115–133 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    van der Houwen, P.J., Sommeijer, B.P., Cong, N.H.: Stability of collocation-based Runge-Kutta-Nyström methods. BIT 31, 469–481 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cong, N.H.: Explicit pseudo two-step RKN methods with stepsize control. Appl. Numer. Math. 38, 135–144 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. National Bureau of Standards Applied Mathematics Series 55, Dover (1970)zbMATHGoogle Scholar
  17. 17.
    Cong, N.H., Strehmel, K., Weiner, R.: A general class of explicit pseudo two-step RKN methods on parallel computers. Comput. Math. Appl. 38, 17–39 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Li, J.Y., Wu, X.Y.: Error analysis of explicit TSERKN methods for highly oscillatory systems. Numer. Algor. 65, 465–483 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hayes, L.J.: Galerkin alternating-direction methods for nonrectangular regions using patch approximations. SIAM J. Numer. Anal. 18, 627–643 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dahlquist, G.: On accuracy and unconditional stability of linear multistep methods for second order differential equations. BIT 18, 133–136 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lambert, J.D., Watson, I.A.: Symmetric multistep methods for periodic initial value problems. J. Inst. Math. Appl. 18, 189–202 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Wu, X.Y., Wang, B.: Multidimensional adapted Runge–Kutta–Nyström methods for oscillatory systems. Comput. Phys. Commun. 181, 1955–1962 (2010)CrossRefzbMATHGoogle Scholar
  24. 24.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Li, J.Y., Wu, X.Y.: Adapted Falkner-type methods solving oscillatory second-order differential equations. Numer. Algorithms 62, 355–381 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Franco, J.M.: New methods for oscillatory systems based on ARKN methods. Appl. Numer. Math. 56, 1040–1053 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

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

Personalised recommendations