B-series methods are exactly the affine equivariant methods

TL;DR

This paper characterizes B-series methods as exactly those numerical methods that are affine equivariant, meaning they respect all affine transformations, providing a complete understanding of their structure.

Contribution

It proves that sequences of smooth maps with B-series expansions are precisely those that are affine equivariant, solving a long-standing characterization problem.

Findings

B-series methods are exactly the affine equivariant methods

Affine equivariance characterizes B-series expansions

Provides a complete classification of B-series methods

Abstract

Butcher series, also called B-series, are a type of expansion, fundamental in the analysis of numerical integration. Numerical methods that can be expanded in B-series are defined in all dimensions, so they correspond to \emph{sequences of maps}---one map for each dimension. A long-standing problem has been to characterise those sequences of maps that arise from B-series. This problem is solved here: we prove that a sequence of smooth maps between vector fields on affine spaces has a B-series expansion if and only if it is \emph{affine equivariant}, meaning it respects all affine maps between affine spaces.