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Algebraic structures of B-series

Published inFoundations of computational mathematics, vol. 10, no. 4, p. 407-427
Publication date2010
Abstract

B-series are a fundamental tool in practical and theoretical aspects of numerical integrators for ordinary differential equations. A composition law for B-series permits an elegant derivation of order conditions, and a substitution law gives much insight into modified differential equations of backward error analysis. These two laws give rise to algebraic structures (groups and Hopf algebras of trees) that have recently received much attention also in the non-numerical literature. This article emphasizes these algebraic structures and presents interesting relationships among them.

Keywords
  • B-series
  • Rooted trees
  • Composition law
  • Substitution law
  • Butcher group
  • Hopf algebra of trees
  • Coproduct
  • Antipode
  • P-series
  • S-series
Citation (ISO format)
CHARTIER, Philippe, HAIRER, Ernst, VILMART, Gilles. Algebraic structures of B-series. In: Foundations of computational mathematics, 2010, vol. 10, n° 4, p. 407–427. doi: 10.1007/s10208-010-9065-1
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ISSN of the journal1615-3375
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