Runge-Kutta Solvers for Ordinary Differential Equations

  • Liviu Gr. Ixaru
  • Guido Vanden Berghe
Part of the Mathematics and Its Applications book series (MAIA, volume 568)


Since the original papers of Runge [24] and Kutta [17] a great number of papers and books have been devoted to the properties of Runge-Kutta methods. Reviews of this material can be found in [4], [5], [12], [18]. Kutta [17] formulated the general scheme of what is now called a Runge-Kutta method.




Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Albrecht P. (1987). The extension of the theory of A-methods to RK methods, in: K. Strehmel, ed. Numerical Treatment of Differential Equations, Proc. 4th Seminar NUMDIFF-4, Tuebner-Texte zur Mathematik (Tuebner, Leipzig):8–18.Google Scholar
  2. [2]
    Albrecht, P. (1987) A new theoretical approach to RK methods. SIAM J. Numer AnaL 24: 391–406.CrossRefMATHMathSciNetGoogle Scholar
  3. [3]
    Butcher, J. C. (1964). On Runge-Kutta processes of high order. Math. Comput., 18: 50–64.CrossRefMATHMathSciNetGoogle Scholar
  4. [4]
    Butcher, J. C. (1987). The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods. Chichester John Wiley and Sons.Google Scholar
  5. [5]
    Butcher, J. C. (2003). Numerical Methods for Ordinary Differential Equations. Chichester John Wiley and Sons.Google Scholar
  6. [6]
    Coleman, J. P. (1998). Mixed interpolation methods with arbitrary nodes. J. Comp. Appl. Math., 92: 69–83.CrossRefMATHGoogle Scholar
  7. [7]
    Coleman, J. P. and Duxbury, S. C. (2000). Mixed collocation methods for yn = (x, y). J. Comp. AppL Math., 126: 47–75.CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    De Meyer, H., Vanthournout, J. and Vanden Berghe, G. (1990). On a new type of mixed interpolation. J. Comp. Appl. Math., 30: 55–69.CrossRefMATHGoogle Scholar
  9. [9]
    England, R. (1969). Error estimates for Runge-Kutta type solutions to systems of ordinary differential equations. Comput. J., 12: 166–170.CrossRefMATHMathSciNetGoogle Scholar
  10. [10]
    Franco, J. M. (2002). An embedded pair of exponentially fitted explicit Runge-Kutta methods. J. Comp. Appl. Math., 149: 407–414.CrossRefMATHMathSciNetGoogle Scholar
  11. [11]
    Gautschi, W. (1962). Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer. Math., 3: 381–397.CrossRefMathSciNetGoogle Scholar
  12. [12]
    Hairer, E., NOrsett, S. R and Wanner G. (1993). Solving Ordinary Differential Equations I, Nonstiff Problems. Berlin Springer-Verlag.Google Scholar
  13. [13]
    Henrici, R. (1962). Discrete Variable Methods in Ordinary Differential Equations. John Wiley and Sons,Inc., New-York - London.MATHGoogle Scholar
  14. [14]
    Ixaru, L. Gr. (1997). Operations on oscillatory functions. Comput. Phys. Comm., 105: 1–19.CrossRefMATHMathSciNetGoogle Scholar
  15. [15]
    Ixaru, L. Gr., De Meyer, H. and Vanden Berghe, G. (2002). Frequency evaluation in exponential fitting multistep algorithms. J. Comp. Appl. Math., 140: 423–433.CrossRefMATHGoogle Scholar
  16. [16]
    Ixaru, L. Gr. and Paternoster, B. (2001). A Gauss quadrature rule for oscillatory integrands. Comput. Phys. Comm., 133: 177–188.CrossRefMATHMathSciNetGoogle Scholar
  17. [17]
    Kutta, W. (1901). Beitrag zur näherungsweisen Integration totaler Differentialgleichungen. Zeitschr. für Math. u. Phys., 46: 435–453.MATHGoogle Scholar
  18. [18]
    Lambert, J. D. (1991). Numerical Methods for Ordinary Differential Systems, The Initial Value Problem. Chichester John Wiley and Sons.Google Scholar
  19. [19]
    Lyche, T. (1972). Chebyshevian multistep methods for ordinary differential equations. Numer. Math., 19: 65–75.CrossRefMATHMathSciNetGoogle Scholar
  20. [20]
    Oliver, J. (1975). A Curiosity of Low-Order Explicit Runge-Kutta Methods. Math. Comp., 29: 1032–1036.CrossRefMATHMathSciNetGoogle Scholar
  21. [21]
    Ozawa, K. (1999). A Four-stage Implicit Runge-Kutta-Nyström Methods with Variable Coefficients for Solving Periodic Initial Value Problems. Japan Journal of Industrial and Applied Mathematics, 16: 25–46.CrossRefMathSciNetGoogle Scholar
  22. [22]
    Paternoster, B. (1998). Runge-Kutta(-Nyström) methods for ODEs with periodic solutions based in trigonometric polynomials, Appl. Num. Math., 28: 401–412.CrossRefMATHMathSciNetGoogle Scholar
  23. [23]
    Runge, C. and König, H. (1924). Vorlesungen über numerisches Rechnen, Grundlehren XI, Springer Verlag.CrossRefMATHGoogle Scholar
  24. [24]
    Runge, C. (1895). Über die numerische Auflösung von Differentialgleichungen. Math. Ann., 46: 167–192.CrossRefMATHMathSciNetGoogle Scholar
  25. [25]
    Simos, T. E., Dimas, E. and Sideridis,A. B. (1994). A Runge-Kutta-Nyström method for the numerical integration of special second-order periodic initial-value problems, J. Comp. Appl. Math., 51: 317–326.CrossRefMATHMathSciNetGoogle Scholar
  26. [26]
    Simos, T. E. (1998). An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions. Comput. Phys. Comm., 115: 1–8.CrossRefMATHMathSciNetGoogle Scholar
  27. [27]
    Simos, T. E. (2001). A fourth algebraic order exponentially-fitted Runge-Kutta method for the numerical solution of the Schrödinger equation. IMA Journ. of Numerical Analysis, 21: 919–931.CrossRefMATHMathSciNetGoogle Scholar
  28. [28]
    Vanden Berghe,G., De Meyer, H. Van Daele, M. and Van Hecke, T. (1999). Exponentially-fitted explicit Runge-Kutta methods. Computer Phys. Comm., 123: 7–15.CrossRefGoogle Scholar
  29. [29]
    Vanden Berghe, G., De Meyer, H., Van Daele, M. and Van Hecke, T. (2000). Exponentially-fitted Runge-Kutta methods. J. Comp. Appl. Math., 125: 107–115.CrossRefMATHGoogle Scholar
  30. [30]
    Vanden Berghe, G., Ixaru, L. Gr. and Van Daele, M. (2001). Optimal implicit exponentially-fitted Runge-Kutta methods. Comp. Phys. Commun., 140: 346–357.CrossRefMATHGoogle Scholar
  31. [31]
    Vanden Berghe, G., Ixaru, L. Gr. and De Meyer, H. (2001). Frequency determination and step—length control for exponentially-fitted Runge—Kutta methods. J. Comp. Appl. Math., 132: 95–105.CrossRefMATHGoogle Scholar
  32. [32]
    Vanden Berghe, G., Van Daele, M. and Vande Vyver, H. (2003). Exponential-fitted Runge—Kutta methods of collocation type: fixed or variable knot points? J. Comp. AppL Math., 159: 217–239.CrossRefMATHGoogle Scholar
  33. [33]
    Van der Houwen, P. J. and Sommeijer, B. P. (1987). Explicit Runge-Kutta(-Nyström) methods with reduced phase errors for computing oscillating solution. SIAM J. Numer AnaL, 24: 595–617.CrossRefMATHMathSciNetGoogle Scholar
  34. [34]
    Van der Houwen, P. J. and Sommeijer, B. P. (1987). Phase-lag analysis of implicit Runge-Kutta methods. SIAM J. Numer Anal., 26: 214–228.CrossRefGoogle Scholar
  35. [35]
    Van der Houwen,P. J., Sommeijer, B. P., Strehmel, K. and Weiner, R. (1986). On the numerical integration of second order initial value problems with a periodic forcing force. Computing, 37: 195–218.CrossRefMATHMathSciNetGoogle Scholar
  36. [36]
    Wright, K. (1970). Some relationships between implicit Runge-Kutta, collocation and Lanczos r-methods, and their stability properties. BIT, 10: 217–227.CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2004

Authors and Affiliations

  • Liviu Gr. Ixaru
    • 1
  • Guido Vanden Berghe
    • 2
  1. 1.“Horia Hulubei”, Department of Theoretical PhysicsNational Institute for Research and Development for Physics and Nuclear EngineeringBucharestRomania
  2. 2.Department of Applied Mathematics and Computer ScienceUniversity of GentGentBelgium

Personalised recommendations