B-series methods are exactly the affine equivariant methods
Journal article, 2016

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.


R. I. McLachlan

Massey University

Klas Modin

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

H. Munthe-Kaas

University of Bergen

Olivier Verdier

Hogskolen i Bergen

Numerische Mathematik

0029-599X (ISSN) 0945-3245 (eISSN)

Vol. 133 3 599-622

Subject Categories

Computational Mathematics



Basic sciences



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