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.
CITATION STYLE
McLachlan, R. I., Modin, K., Munthe-Kaas, H., & Verdier, O. (2016). B-series methods are exactly the affine equivariant methods. Numerische Mathematik, 133(3), 599–622. https://doi.org/10.1007/s00211-015-0753-2
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