Foundations of Computational Mathematics

, Volume 13, Issue 2, pp 161–186 | Cite as

Backward Error Analysis and the Substitution Law for Lie Group Integrators

  • Alexander Lundervold
  • Hans Munthe-Kaas


Butcher series are combinatorial devices used in the study of numerical methods for differential equations evolving on vector spaces. More precisely, they are formal series developments of differential operators indexed over rooted trees, and can be used to represent a large class of numerical methods. The theory of backward error analysis for differential equations has a particularly nice description when applied to methods represented by Butcher series. For the study of differential equations evolving on more general manifolds, a generalization of Butcher series has been introduced, called the Lie–Butcher series. This paper presents the theory of backward error analysis for methods based on Lie–Butcher series.


Backward error analysis Butcher series Hopf algebras Lie group integrators Lie–Butcher series Rooted trees Substitution law 

Mathematics Subject Classification (2010)

65L05 65L06 37C10 



We are grateful to Kurusch Ebrahimi-Fard, Dominique Manchon and Jon-Eivind Vatne for interesting and enlightening discussions, and to the anonymous referees for their valuable comments. We would also like to acknowledge support from the Aurora Program, project 205042/V11.


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Copyright information

© SFoCM 2012

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway
  2. 2.Department of MathematicsUniversity of BergenBergenNorway

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