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      B-series methods are exactly the affine equivariant methods

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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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          A new class of energy-preserving numerical integration methods

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            On the Butcher group and general multi-value methods

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              The Hopf Algebra of Rooted Trees, Free Lie Algebras, and Lie Series

              A Murua (2006)
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                Author and article information

                Journal
                2014-09-03
                2015-04-27
                Article
                10.1007/s00211-015-0753-2
                1409.1019
                7f980f4e-7c75-4d93-a4d8-71b2ee006225

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                37C80, 37C10, 41A58
                math.NA math.CA

                Numerical & Computational mathematics
                Numerical & Computational mathematics

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