Numerical Algorithms

, Volume 53, Issue 2, pp 153–170

Trees and numerical methods for ordinary differential equations

Original Paper

DOI: 10.1007/s11075-009-9285-0

Cite this article as:
Butcher, J.C. Numer Algor (2010) 53: 153. doi:10.1007/s11075-009-9285-0


This paper presents a review of the role played by trees in the theory of Runge–Kutta methods. The use of trees is in contrast to early publications on numerical methods, in which a deceptively simpler approach was used. This earlier approach is not only non-rigorous, but also incorrect. It is now known, for example, that methods can have different orders when applied to a single equation and when applied to a system of equations; the earlier approach cannot show this. Trees have a central role in the theory of Runge–Kutta methods and they also have applications to more general methods, involving multiple values and multiple stages.


Runge-Kutta methods Trees Order conditions Taylor expansions 

Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  1. 1.The University of AucklandAucklandNew Zealand

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