B-series methods are exactly the affine equivariant methods
Numerical Analysis
2017-03-22 v2 Classical Analysis and ODEs
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.
Cite
@article{arxiv.1409.1019,
title = {B-series methods are exactly the affine equivariant methods},
author = {Robert I. McLachlan and Klas Modin and Hans Munthe-Kaas and Olivier Verdier},
journal= {arXiv preprint arXiv:1409.1019},
year = {2017}
}