Numerische Mathematik

, Volume 52, Issue 6, pp 621–638 | Cite as

Partitioned adaptive Runge-Kutta methods for the solution of nonstiff and stiff systems

  • Jürgen Bruder
  • Karl Strehmel
  • Rüdiger Weiner


For the numerical solution of initial value problems of ordinary differential equations partitioned adaptive Runge-Kutta methods are studied. These methods consist of an adaptive Runge-Kutta methods for the treatment of a stiff system and a corresponding explicit Runge-Kutta method for a nonstiff system. First we modify the theory of Butcher series for partitioned adaptive Runge-Kutta methods. We show that for any explicit Runge-Kutta method there exists a translation invariant partitoned adaptive Runge-Kutta method of the same order. Secondly we derive a special translaton invariant partitioned adaptive Runge-Kutta method of order 3. An automatic stiffness detection and a stepsize control basing on Richardson-extrapolation are performed. Extensive tests and comparisons with the partitioned RKF4RW-algorithm from Rentrop [16] and the partitioned algorithm LSODA from Hindmarsh [9] and Petzold [15] show that the partitoned adaptive Runge-Kutta algorithm works reliable and gives good numericals results. Furthermore these tests show that the automatic stiffness detection in this algorithm is effective.

Subject Classifications

AMS(MOS): 65L05 CR: G1.7 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Day J.D.: Run time estimation of the spectral radius of Jacobians. J. Comput. Appl. Math.11, 315–323, (1984)Google Scholar
  2. 2.
    Enright, W.H., Hull, T.E., Lindberg, B.: Comparing numerical methods for stiff systems of ordinary differential equations. BIT15, 10–48 (1975)Google Scholar
  3. 3.
    Gottwald, B.A., Wanner, G.: A reliable Rosenbrock integrator for stiff differential equations. Computing26, 355–360 (1981)Google Scholar
  4. 4.
    Griepentrog, E.: Gemischte Runge-Kutta-Verfahren für steife Systeme. Seminarbericht Nr. 11. Berlin: Humboldt-Universität 1978Google Scholar
  5. 5.
    Grigorieff, R.D.: Numerik gewöhnlicher Differentialgleichungen. Bd. 1: Einschrittverfahren. Stuttgart: Teubner 1972Google Scholar
  6. 6.
    Hairer, E., Wanner, G.: On the Butcher group and general multivalue methods. Computing13, 1–15 (1974)Google Scholar
  7. 7.
    Hairer, E.: Order Conditions for Numerical Methods for Partitioned Ordinary Differential Equations. Numer. Math.36, 431–445 (1981)Google Scholar
  8. 8.
    Hofer, E.: A partially implicit method for large stiff systems of ODE'S with only a few equations introducing small time-constants. SIAM J. Numer. Anal.13, 645–663 (1976)Google Scholar
  9. 9.
    Hindmarsh, A.C.: ODEPACK, a systematized collections of ODE solvers. Lawrence Livermore National Laboratory, Rept. UCRL-88007 (1982)Google Scholar
  10. 10.
    Hull, T.E., Enright, W.H., Fellen, B.M., Sedgwick A.E.: Comparing numerical methods for ordinar differential equations. SIAM J. Numer. Anal.9, 603–637 (1972)Google Scholar
  11. 11.
    Hundsdorfer, W.: The numerical solution of nonlinear stiff initial value problems. CWI-Tract 12, Amsterdam 1985Google Scholar
  12. 12.
    Kaps, P.: Modifizierte Rosenbrockmethoden der Ordnung 4,5 und 6 zur numerischen Integration steifer Differentialgleichungen. Dissertation, Technische Universität Innsbruck, 1977Google Scholar
  13. 13.
    Lapidus, L., Seinfeld, J.H.: Numerical solution of ordinary differential equations. New York 1971Google Scholar
  14. 14.
    Norsett, S.P.:C-polynomials for rational approximation to the exponential function. Numer. Math.25, 39–56 (1975)Google Scholar
  15. 15.
    Petzold, L.: Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations. SIAM J. Sci. Stat. Comput.4, 136–148 (1983)Google Scholar
  16. 16.
    Rentrop, P.: Partitioned Runge-Kutta methods with stiffness detection and stepsize control. Numer. Math.47, 545–564 (1985)Google Scholar
  17. 17.
    Shampine, L.: Implementation of Rosenbrock Methods. ACM Trans. Math. Softw.8, 93–113 (1982)Google Scholar
  18. 18.
    Shampine, L.: Type-insensitive ODE-codes based on extrapolation methods. SIAM J. Sci. Stat. Comput.4, 635–644 (1983)Google Scholar
  19. 19.
    Sottas, G.: Dynamic adaptive selection between explicit and implicit methods when solving ODE'S. University of Geneva, Report 1984Google Scholar
  20. 20.
    Strehmel, K., Weiner, R.: Behandlung steifer Anfangswertprobleme gewöhnlicher Differentialgleichungen mit adaptiven Runge-Kutta-Methoden. Computing29, 153–165 (1982)Google Scholar
  21. 21.
    Strehmel, K., Weiner, R.: Partitioned adaptive Runge-Kutta methods and Their Stability. Numer. Math.11, 315–323 (1984)Google Scholar
  22. 22.
    Veldhuizen, M., van:D-Stability and Kaps-Rentrop Methods. Computing32, 229–237 (1984)Google Scholar
  23. 23.
    Veldhuizen, M., van:D-Stability. SIAM J. Numer. Anal.18, 45–64 (1981)Google Scholar
  24. 24.
    Bruder, J.: Numerische Lösung steifer und nichtsteifer Differentialgleichungssysteme mit partitionierten adaptiven Runge-Kutta-Methoden. Dissertation, Halle 1985Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Jürgen Bruder
    • 1
  • Karl Strehmel
    • 1
  • Rüdiger Weiner
    • 1
  1. 1.Sektion MathematikUniversität HalleHalleGerman Democratic Republic

Personalised recommendations