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Axioms 2018, 7(3), 52; https://doi.org/10.3390/axioms7030052

Trees, Stumps, and Applications

Department of Mathematics, University of Auckland, Auckland 92019, New Zealand
Received: 23 May 2018 / Revised: 16 July 2018 / Accepted: 16 July 2018 / Published: 1 August 2018
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Abstract

The traditional derivation of Runge–Kutta methods is based on the use of the scalar test problem y(x)=f(x,y(x)). However, above order 4, this gives less restrictive order conditions than those obtained from a vector test problem using a tree-based theory. In this paper, stumps, or incomplete trees, are introduced to explain the discrepancy between the two alternative theories. Atomic stumps can be combined multiplicatively to generate all trees. For the scalar test problem, these quantities commute, and certain sets of trees form isomeric classes. There is a single order condition for each class, whereas for the general vector-based problem, for which commutation of atomic stumps does not occur, there is exactly one order condition for each tree. In the case of order 5, the only nontrivial isomeric class contains two trees, and the number of order conditions reduces from 17 to 16 for scalar problems. A method is derived that satisfies the 16 conditions for scalar problems but not the complete set based on 17 trees. Hence, as a practical numerical method, it has order 4 for a general initial value problem, but this increases to order 5 for a scalar problem. View Full-Text
Keywords: ordinary differential equations; Runge–Kutta; tree; stump; order; elementary differential ordinary differential equations; Runge–Kutta; tree; stump; order; elementary differential
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Butcher, J.C. Trees, Stumps, and Applications. Axioms 2018, 7, 52.

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