Numerische Mathematik

, Volume 25, Issue 4, pp 383–400 | Cite as

A theory for Nyström methods

  • E. Hairer
  • G. Wanner


For the numerical solution of differential equations of thesecond order (and systems of ...) there are two possibilities: 1. To transform it into a system of the first order (of doubled dimension) and to integrate by a standard routine. 2. To apply a “direct” method as those invented by Nyström. The benefit of these direct methods is not generally accepted, a historical reason for them was surely the fact that at that time the theories did not consider systems, but single equations only. In any case the second approach is more general, since the class of methods defined in this paper contains the first approach as a special case. So there is more freedom for extending stability or accuracy.

This paper begins with the development of a theory, which extends our theory for first order equations [1] to equations of the second order, and which is applicable to the study of possibly all numerical methods for problems of this type. As an application, we obtain Butcher-type results for Nyström-methods, we characterize numerical methods as applications of a certain set of trees, give formulas for a group-structure (expressing the composition of methods) etc.

Recently in [2] the equations of conditions for Nyström methods have been tabulated up to order 7 (containing errors). Our approach yields not only the correct equations of conditions in a straight-forward way, but also an insight in the structure of methods that is useful for example in choosing good formulas.


Differential Equation Mathematical Method Single Equation Order Equation Correct Equation 
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  1. 1.
    Hairer, E., Wanner, G.: On the Butcher Group and General Multi-Value Methods. Computing13, 1–15 (1974)Google Scholar
  2. 2.
    Bettis, D. G.: Equations of condition for High Order Runge-Kutta-Nyström formulae. Lecture Notes in Mathematics No. 362, p. 76–91, Springer 1972Google Scholar
  3. 3.
    Fehlberg, E.: Classical Eight- and Lower-Order Runge-Kutta-Nyström Formulas with Stepsize Control for Special Second-Order Differential Equations. NASA Technical Report R-381, 1972Google Scholar
  4. 4.
    Albrecht, J.: Beiträge zum Runge-Kutta-Verfahren. Z. Angew. Math. Mech.35, 100–110 (1955)Google Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • E. Hairer
    • 1
  • G. Wanner
    • 1
  1. 1.Section de MathématiqueUniversité de GenèveGenève 24Switzerland

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