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Numerical Algorithms

, Volume 70, Issue 4, pp 897–927 | Cite as

Two-derivative Runge-Kutta-Nyström methods for second-order ordinary differential equations

  • Zhaoxia Chen
  • Zeyu Qiu
  • Juan Li
  • Xiong You
Original Paper

Abstract

Classical Runge-Kutta-Nyström (RKN) methods for second-order ordinary differential equations are extended to two-derivative Runge-Kutta-Nyström (TDRKN) methods involving the third derivative of the solution. A new version of Nyström tree theory and the corresponding B-series theory are developed, based on which the order conditions for TDRKN methods are derived. A two-stage explicit TDRKN method of order four and a three-stage explicit TDRKN method of order five are constructed. The linear stability of the new methods is analyzed. The results of numerical experiments show that the new TDRKN methods are more efficient than the traditional RKN methods of the same algebraic order.

Keywords

Second-order ordinary differential equations Two-derivative Runge-Kutta-Nystöm methods Nyström tree theory Order conditions 

Mathematics Subject Classification (2010)

65L05 65L06 65M20 65N40 

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References

  1. 1.
    Avdyushev, V.A.: Special perturbation theory methods in celestial mechanics, I. Principles for the construction and substantiation of the application. Russian Phys. J. 49, 1344–1253 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bebernes, J., Eberly, D.: Mathematical Problems from Combustion Theory. Springer, New York (1989)CrossRefzbMATHGoogle Scholar
  3. 3.
    Butcher, J.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, New York (2008)CrossRefzbMATHGoogle Scholar
  4. 4.
    Butcher, J., Podhaisky, H.: On error estimation in general linear methods for stiff ODEs. Appl. Numer. Math. 56, 345–357 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cash, J.R.: Second derivative extended backward differentiation formulas for the numerical integration of stiff systems. SIAM J. Numer. Anal. 18, 21–36 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chan, R.P.K., Tsai, A.Y.J.: On explicit two-derivative Runge-Kutta methods. Numer. Algor. 53, 171–194 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Enright, W.H.: Second derivative multistep methods for stiff ordinary differential equations. SIAM J. Numer. Anal. 11, 321–3311 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fang, Y., You, X.: New optimized two-derivative Runge-Kutta type methods for solving the radial Schrödinger equation. J. Math. Chem. 52, 240–254 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fang, Y., You, X., Ming, Q.: Exponentially fitted two-derivative Runge-Kutta methods for the Schrödinger equation. Int. J. Mod. Phys. C 24, 1–9 (2013). Article ID 1350073MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fang, X., You, X., Ming, Q.: Trigonometrically fitted two-derivative Runge-Kutta methods for solving oscillatory differential equations. Numer. Algor. 65, 651–667 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Franco, J.M.: Exponentially fitted symplectic integrators of RKN type for solving oscillatory problems. Comput. Phys. Comm. 177, 479–492 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Franco, J.M.: New methods for oscillatory systems based on ARKN methods. Appl. Numer. Math. 56, 1040–1053 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Franco, J.M.: Runge-Kutta-Nyström methods adapted to the numerical integration of perturbed oscillators. Comput. Phys. Comm. 147, 770–787 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin (2006)Google Scholar
  15. 15.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, Berlin (1993)zbMATHGoogle Scholar
  16. 16.
    Hojjati, G., Rahimi Ardabili, M.Y., Hosseini, S.M.: New second derivative multistep methods for stiff systems. Appl. Math. Model. 30, 466–476 (2006)CrossRefzbMATHGoogle Scholar
  17. 17.
    Holland, P.R.: The Quantum Theory of Motion. Cambridge University Press, Cambridge (2000)Google Scholar
  18. 18.
    Ismail, G.A.F., Ibrahim, I.H.: New efficient second derivative multistep methods for stiff systems. Appl. Math. Model. 23, 279–288 (1999)CrossRefzbMATHGoogle Scholar
  19. 19.
    Jordan, D.W., Smith, P.: Nonlinear Ordinary Differential Equations, An Introduction for Scientists and Engineers, 4th edn. Oxford University Press, Oxford (2007)zbMATHGoogle Scholar
  20. 20.
    Joseph, D., Sparrow, E.: Nonlinear diffusion induced by nonlinear sources. Q. Appl. Math. 28, 327–342 (1970)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kaplan, D., Glass, L.: Understanding Nonlinear Dynamics. Springer, New York (1995)CrossRefzbMATHGoogle Scholar
  22. 22.
    Kastlunger, K.H., Wanner, G.: On Turan type implicit Runge-Kutta methods. Computing 9, 317–325 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kastlunger, K.H., Wanner, G.: Runge-Kutta processes with multiple nodes. Computing 9, 9–24 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kristensson, G.: Second Order Differential Equations: Special Functions and Their Classification. Springer, New York (2010)CrossRefGoogle Scholar
  25. 25.
    Landau, L.D., Lifshitz, E.M.: Mechanics, 3rd edn. Butterworth-Heinemann (1982)Google Scholar
  26. 26.
    Shintani, H.: On one-step methods utilizing the second derivative. Hiroshima. Math. J. 1, 349–372 (1971)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Shintani, H.: On explicit one-step methods utilizing the second derivative. Hiroshima Math. J. 2, 353–368 (1972)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Simos, T.E., Vigo-Aguiar, J.: Exponentially fitted symplectic integrator. Phys. Rev. E 67, 1–7 (2003)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Tsai, A.Y.J., Chan, R.P.K., Wang, S.: Two-derivative Runge-Kutta methods for PDEs using a novel discretization approach. Numer. Algor. 65, 687–703 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Urabe, M.: An implicit one-step method of high order accuracy for the numerical integration of ordinary differential equations. Numer. Math. 15, 151–164 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Weinberger, H.F.: A First Course in Partial Differential Equations with Complex Variables and Transform Methods. Dover Publications Inc., New York (1965)zbMATHGoogle Scholar
  32. 32.
    Wu, X., You, X., Wang, B.: Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  33. 33.
    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
  34. 34.
    You, X., Fang, Y., Zhao, J.: Special extended Nyström tree theory for ERKN methods. J. Comput. Appl. Math. 263, 478–499 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    You, X., Zhao, J., Yang, H., Fang, Y., Wu, X.: Order conditions for RKN methods solving general second-order oscillatory systems. Numer. Algor. 66, 147–176 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Zhang, Y., Che, H., Fang, Y., You, X.: A new trigonometrically fitted two-derivative Runge-Kutta method for the numerical solution of the Schrödinger equation and related problems. J. Appl. Math. 2013, 1–9 (2013). Article ID 937858MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Applied MathematicsNanjing Agricultural UniversityNanjingPeople’s Republic of China

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