Foundations of Computational Mathematics

, Volume 16, Issue 1, pp 183–215 | Cite as

Aromatic Butcher Series

  • Hans Munthe-Kaas
  • Olivier VerdierEmail author


We show that without other further assumption than affine equivariance and locality, a numerical integrator has an expansion in a generalized form of Butcher series (B-series), which we call aromatic B-series. We obtain an explicit description of aromatic B-series in terms of elementary differentials associated to aromatic trees, which are directed graphs generalizing trees. We also define a new class of integrators, the class of aromatic Runge–Kutta methods, that extends the class of Runge–Kutta methods and have aromatic B-series expansion but are not B-series methods. Finally, those results are partially extended to the case of more general affine group equivariance.


B-Series Butcher series Equivariance Aromatic series Aromatic trees Functional graph Directed pseudo-forest 

Mathematics Subject Classification

37C80 37C10 41A58 15A72 



The authors would like to thank Robert McLachlan for many comments and discussions. This research was supported by the Spade Ace Project, by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme, and by the J.C. Kempe memorial fund.


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© SFoCM 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BergenBergenNorway

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