Butcher series: A story of rooted trees and numerical methods for evolution equations
Robert I McLachlan, Klas Modin, Hans Munthe-Kaas, and Olivier Verdier

TL;DR
This paper explores the historical development and mathematical structure of Butcher series, rooted trees, and B-series methods in numerical solutions of evolution equations, culminating in a new geometric characterization of these methods.
Contribution
It provides a comprehensive story linking Cayley's rooted trees, Butcher's group, and modern algebraic and geometric tools to characterize B-series methods.
Findings
Connected rooted trees to numerical methods for differential equations.
Resolved the characterization problem of B-series methods.
Introduced geometric perspectives to the analysis of numerical methods.
Abstract
Butcher series appear when Runge-Kutta methods for ordinary differential equations are expanded in power series of the step size parameter. Each term in a Butcher series consists of a weighted elementary differential, and the set of all such differentials is isomorphic to the set of rooted trees, as noted by Cayley in the mid 19th century. A century later Butcher discovered that rooted trees can also be used to obtain the order conditions of Runge-Kutta methods, and he found a natural group structure, today known as the Butcher group. It is now known that many numerical methods also can be expanded in Butcher series; these are called B-series methods. A long-standing problem has been to characterize, in terms of qualitative features, all B-series methods. Here we tell the story of Butcher series, stretching from the early work of Cayley, to modern developments and connections to…
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Taxonomy
TopicsNumerical methods for differential equations · Polynomial and algebraic computation
