Butcher series: A story of rooted trees and numerical methods for evolution equations

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 abstract algebra, and finally to the resolution of the characterization problem. This resolution introduces geometric tools and perspectives to an area traditionally explored using analysis and combinatorics.