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arXiv · 1409.1019

B-series methods are exactly the affine equivariant methods

Robert I. McLachlan·Klas Modin·Hans Munthe-Kaas·Olivier Verdier
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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.

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Robert I. McLachlan, Klas Modin, Hans Munthe-Kaas, Olivier Verdier. 2014-09-03. B-series methods are exactly the affine equivariant methods. https://doi.org/10.1007/s00211-015-0753-2

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