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B-series methods are exactly the affine equivariant methods

Journal article
Authors R.I. McLachlan
Klas Modin
H. Munthe-Kaas
O. Verdier
Published in Numerische Mathematik
Volume 133
Issue 3
Pages 599-622
ISSN 0029-599X
Publication year 2016
Published at Department of Mathematical Sciences, Mathematics
Pages 599-622
Language en
Subject categories Geometry, Computational Mathematics


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 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 affine equivariant, meaning it respects all affine maps between affine spaces.

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