Foundations of Computational Mathematics

, Volume 13, Issue 2, pp 161–186

Backward Error Analysis and the Substitution Law for Lie Group Integrators


DOI: 10.1007/s10208-012-9130-z

Cite this article as:
Lundervold, A. & Munthe-Kaas, H. Found Comput Math (2013) 13: 161. doi:10.1007/s10208-012-9130-z


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 analysisButcher seriesHopf algebrasLie group integratorsLie–Butcher seriesRooted treesSubstitution law

Mathematics Subject Classification (2010)


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