Runge-Kutta Methods with Equation Dependent Coefficients

  • L. Gr. Ixaru
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)


The simplest two and three stage explicit Runge-Kutta methods are examined by a conveniently adapted form of the exponential fitting approach. The unusual feature is that the coefficients of the new versions are no longer constant, as in standard versions, but depend on the equation to be solved. Some valuable properties emerge from this. Thus, in the case of three-stage versions, although in general the order is three, that is the same as for the standard method, this is easily increased to four by a suitable choice of the position of the stage abscissas. Also, the stability properties are massively enhanced. Two particular versions of order four are A-stable, a fact which is quite unusual for explicit methods. This recommends them as efficient tools for solving stiff differential equations.


Runge-Kutta method Exponential fitting A-stability 


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  1. 1.
    Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, Chichester (2008)CrossRefzbMATHGoogle Scholar
  2. 2.
    D’Ambrosio, R., Ixaru, L.G., Paternoster, B.: Construction of the ef-based Runge-Kutta methods revisited. Comput. Phys. Commun. 182, 322–329 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ixaru, L.G.: Runge-Kutta method with equation dependent coefficients. Comput. Phys. Commun. 183, 63–69 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Calvo, M., Franco, J.M., Montijano, J.I., et al.: Sixth-order symmetric and symplectic exponentially fitted Runge-Kutta methods of the Gauss type. J. Comput. Appl. Math. 223, 387–398 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Franco, J.M.: An embedded pair of exponentially fitted explicit Runge-Kutta methods. J. Comput. Appl. Math. 149, 407–414 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Franco, J.M.: Exponentially fitted explicit Runge-Kutta-Nystrom methods. J. Comput. Appl. Math. 167, 1–19 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ixaru, L.G., Vanden Berghe, G.: Exponential Fitting. Kluwer Academic Publishers (2004)Google Scholar
  8. 8.
    Martin-Vaquero, J., Janssen, B.: Second-order stabilized explicit Runge-Kutta methods for stiff problems. Comput. Phys. Commun. 180, 1802–1810 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ozawa, K., Japan, J.: Indust. Appl. Math. 18, 107 (2001)zbMATHGoogle Scholar
  10. 10.
    Paternoster, B.: Runge-Kutta(-Nystrom) methods for ODEs with periodic solutions based on trigonometric polynomials. Appl. Numer. Math. 28, 401–412 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Simos, T.: An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions. Comput. Phys. Commun. 115, 1–8 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Vanden Berghe, G., De Meyer, H., Van Daele, M., et al.: Exponentially-fitted explicit Runge-Kutta methods. Comput. Phys. Commun. 123, 7–15 (1999)CrossRefzbMATHGoogle Scholar
  13. 13.
    Vanden Berghe, G., De Meyer, H., Van Daele, M., et al.: Exponentially fitted Runge-Kutta methods. J. Comput. Appl. Math. 125, 107–115 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Vanden Berghe, G., Ixaru, L.G., De Meyer, H.: Frequency determination and step-length control for exponentially-fitted Runge-Kutta methods. J. Comput. Appl. Math. 132, 95–105 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ixaru, L.G., Vanden Berghe, G., De Meyer, H.: Exponentially fitted variable two-step BDF algorithm for first order ODEs. Comput. Phys. Commun. 150, 116–128 (2003)CrossRefzbMATHGoogle Scholar
  16. 16.
    Vanden Berghe, G., Van Daele, M., Vyver, H.V.: Exponential fitted Runge-Kutta methods of collocation type: fixed or variable knot points? J. Comput. Appl. Math. 159, 217–239 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Van De Vyver, H.: Frequency evaluation for exponentially fitted Runge-Kutta methods. J. Comput. Appl. Math. 184, 442–463 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Van de Vyver, H.: On the generation of P-stable exponentially fitted Runge-Kutta-Nystrom methods by exponentially fitted Runge-Kutta methods. J. Comput. Appl. Math. 188, 309–318 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Vigo-Aguiar, J., Martin-Vaquero, J., Ramos, H.: Exponential fitting BDF-Runge-Kutta algorithms. Comput. Phys. Commun. 178, 15–34 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jesus, V.-A., Higinio, R.: A family of A-stable Runge-Kutta collocation methods of higher order for initial-value problems. IMA J. Numerical Analysis 27, 798–817 (2007)CrossRefzbMATHGoogle Scholar
  21. 21.
    Lambert, J.D.: Numerical methods for ordinary differential systems: The initial value problem. Wiley, Chichester (1991)zbMATHGoogle Scholar
  22. 22.
    Ixaru, L.G.: Operations on oscillatory functions. Comput. Phys. Commun. 105, 1–19 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • L. Gr. Ixaru
    • 1
    • 2
  1. 1.Department of Theoretical Physics“Horia Hulubei” National Institute of Physics and Nuclear EngineeringBucharestRomania
  2. 2.Academy of Romanian ScientistsBucharestRomania

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