Numerische Mathematik

, Volume 133, Issue 3, pp 599–622 | Cite as

B-series methods are exactly the affine equivariant methods

  • Robert I. McLachlan
  • Klas Modin
  • Hans Munthe-Kaas
  • Olivier VerdierEmail author


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.

Mathematics Subject Classification

37C80 37C10 41A58 



Klas Modin is supported by the Swedish Research Council (Contract VR-2012-335). Olivier Verdier is supported by the J.C. Kempe memorial fund (Grant No. SMK-1238). Robert McLachlan is supported by the Marsden Fund of the Royal Society of New Zealand. We express our greatest gratitude to the referees for their numerous useful remarks, in particular for noticing that one can avoid the assumption of locality. We are also grateful to Alexander Schmeding for giving us the right functional analytic framework of compactly supported vector fields in the definition of an integrator.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Robert I. McLachlan
    • 1
  • Klas Modin
    • 2
  • Hans Munthe-Kaas
    • 3
  • Olivier Verdier
    • 4
    Email author
  1. 1.Department of Mathematical SciencesMassey UniversityPalmerston NorthNew Zealand
  2. 2.Mathematical SciencesChalmers University of Technology and the University of GothenburgGöteborgSweden
  3. 3.Department of MathematicsUniversity of BergenBergenNorway
  4. 4.Department of MathematicsBergen University CollegeBergenNorway

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