, Volume 54, Issue 1, pp 117–140 | Cite as

Sixth-order symplectic and symmetric explicit ERKN schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations

  • Bin WangEmail author
  • Hongli Yang
  • Fanwei Meng


This paper is devoted to the analysis of the sixth-order symplectic and symmetric explicit extended Runge–Kutta–Nyström (ERKN) schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations. Fourteen practical sixth-order symplectic and symmetric explicit ERKN schemes are constructed, and their phase properties are investigated. The paper is accompanied by five numerical experiments, including a nonlinear two-dimensional wave equation. The numerical results in comparison with the sixth-order symplectic and symmetric Runge–Kutta–Nyström methods and a Gautschi-type method demonstrate the efficiency and robustness of the new explicit schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations.


Symplectic and symmetric explicit schemes Sixth-order extended Runge–Kutta–Nyström schemes Multi-frequency oscillatory nonlinear Hamiltonian equations Structure-preserving algorithms 

Mathematics Subject Classification

65L05 65L20 65N40 65P10 34C15 



The authors are sincerely thankful to two anonymous reviewers for their valuable suggestions, which help improve the presentation of the manuscript significantly.


  1. 1.
    Alolyan, I., Anastassi, Z.A., Simos, T.E.: A new family of symmetric linear four-step methods for the efficient integration of the Schrödinger equation and related oscillatory problems. Appl. Math. Comput. 218, 5370–5382 (2012)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Celledoni, E., Grimm, V., McLachlan, R.I., McLaren, D.I., ONeale, D., Owren, B., Quispel, G.R.W.: Preserving energy resp. dissipation in numerical PDEs using the Average Vector Field method. J. Comput. Phys. 231, 6770–6789 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cohen, D., Hairer, E., Lubich, C.: Numerical Energy Conservation for Multi-Frequency Oscillatory Differential Equations. BIT 45, 287–305 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Franco, J.M.: Runge-Kutta-Nyström methods adapted to the numerical integration of perturbed oscillators. Comput. Phys. Comm. 147, 770–787 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    García, A., Martín, P., González, A.B.: New methods for oscillatory problems based on classical codes. Appl. Numer. Math. 42, 141–157 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    García-Archilla, B., Sanz-Serna, J.M., Skeel, R.D.: Long-time-step methods for oscillatory differential equations. SIAM J. Sci. Comput. 20, 930–963 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hairer, E., Lubich, C.: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38, 414–441 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer-Verlag, Berlin, Heidelberg (2006)zbMATHGoogle Scholar
  10. 10.
    Hochbruck, M., Lubich, C.: A Gautschi-type method for oscillatory second-order differential equations. Numer. Math. 83, 403–426 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Okunbor, D., Skeel, R.D.: Canonical Runge-Kutta-Nyström methods of order 5 and 6. J. Comput. Appl. Math. 51, 375–382 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Panopoulos, G. A., Simos, T. E.: An eight-step semi-embedded predictor-corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown. J. Comput. Appl. Math. 1–15 (2015)Google Scholar
  13. 13.
    Stavroyiannis, S., Simos, T.E.: Optimization as a function of the phase-lag order of two-step P-stable method for linear periodic IVPs. App. Numer. Math. 59, 2467–2474 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Stiefel, E.L., Scheifele, G.: Linear and regular celestial mechanics. Springer-Verlag, New York (1971)CrossRefzbMATHGoogle Scholar
  15. 15.
    Tocino, A., Vigo-Aguiar, J.: Symplectic conditions for exponential fitting Runge-Kutta-Nyström methods. Math. Comput. Modell. 42, 873–876 (2005)CrossRefzbMATHGoogle Scholar
  16. 16.
    Van de Vyver, H.: Stability and phase-lag analysis of explicit Runge-Kutta methods with variable coefficients for oscillatory problems. Comput. Phys. Comm. 173, 115–130 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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. Roy. Soc. London Ser. A 460, 561–567 (2004)CrossRefzbMATHGoogle Scholar
  18. 18.
    Wang, B., Iserles, A., Wu, X.: Arbitrary-order trigonometric Fourier collocation methods for multi-frequency oscillatory systems. Found. Comput. Math. 16, 151–181 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wang, B., Li, G.: Bounds on asymptotic-numerical solvers for ordinary differential equations with extrinsic oscillation. Appl. Math. Modell. 39, 2528–2538 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wang, B., Liu, K., Wu, X.: A Filon-type asymptotic approach to solving highly oscillatory second-order initial value problems. J. Comput. Phys. 243, 210–223 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Wang, B., Wu, X.: A new high precision energy-preserving integrator for system of oscillatory second-order differential equations. Phys. Lett. A 376, 1185–1190 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wang, B., Wu, X.: A highly accurate explicit symplectic ERKN method for multi-frequency and multidimensional oscillatory Hamiltonian systems. Numer. Algo. 65, 705–721 (2014)CrossRefzbMATHGoogle Scholar
  23. 23.
    Wang, B., Wu, X.: Explicit multi-frequency symmetric extended RKN integrators for solving multi-frequency and multidimensional oscillatory reversible systems. CALCOLO 52, 207–231 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wang, B., Wu, X., Xia, J.: Error bounds for explicit ERKN integrators for systems of multi-frequency oscillatory second-order differential equations. Appl. Numer. Math. 74, 17–34 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wang, B., Wu, X., Zhao, H.: Novel improved multidimensional Strömer-Verlet formulas with applications to four aspects in scientific computation. Math. Comput. Modell. 57, 857–872 (2013)CrossRefzbMATHGoogle Scholar
  26. 26.
    Wu, X.: A note on stability of multidimensional adapted Runge-Kutta-Nyström methods for oscillatory systems. Appl. Math. Modell. 36, 6331–6337 (2012)CrossRefzbMATHGoogle Scholar
  27. 27.
    Wu, X., Wang, B., Liu, K., Zhao, H.: ERKN methods for long-term integration of multidimensional orbital problems. Appl. Math. Modell. 37, 2327–2336 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wu, X., Wang, B., Shi, W.: Efficient energy-preserving integrators for oscillatory Hamiltonian systems. J. Comput. Phys. 235, 587–605 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wu, X., You, X., Shi, W., Wang, B.: ERKN integrators for systems of oscillatory second-order differential equations. Comput. Phys. Comm. 181, 1873–1887 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wu, X., Wang, B., Xia, J.: Explicit symplectic multidimensional exponential fitting modified Runge-Kutta-Nyström methods. BIT 52, 773–795 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Wu, X., You, X., Wang, B.: Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer-Verlag, Berlin, Heidelberg (2013)CrossRefzbMATHGoogle Scholar
  32. 32.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Yang, H., Zeng, X., Wu, X., Ru, Z.: A simplified Nyström-tree theory for extended Runge-Kutta-Nyström integrators solving multi-frequency oscillatory systems. Comput. Phys. Comm. 185, 2841–2850 (2014)CrossRefzbMATHGoogle Scholar
  34. 34.
    Yoshida, H.: Construction of higher order symplectic integrators. Phys. Lett. A 150, 262–268 (1990)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesQufu Normal UniversityQufuChina
  2. 2.Department of Mathematics and PhysicsNanjing Institute of TechnologyNanjingChina

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