## Abstract

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.

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## Notes

\({\mathcal{N}}\) with concatenation product can equivalently be defined as the linear space spanned by trees, \(V=k\{\operatorname {OT}\}\), equipped with a tensor product. Hence \({\mathcal{N}}\) can be defined as the tensor algebra on

*V*. However, because we shall need other tensor products later we prefer the definition via concatenation of words.The vector fields are interpreted as differential operators acting on functions.

This is true when \(\mathfrak {g}^{M}\) is replaced by

*Ξ*^{M}for any vector space*Ξ*.We note that the two operations

*f*,*g*↦*f*[*g*] and*f*,*g*↦*f*⋄*g*gives \(U(\mathfrak {g})^{M}\) the structure of a unital dipterous algebra (as defined in [21]).Coproducts will occasionally be written using the

*Sweedler*notation*Δ*(*ω*)=∑*ω*_{(1)}⊗*ω*_{(2)}.Note that the concatenation deshuffling Hopf algebra is dual to the shuffle deconcatenation Hopf algebra.

Being a graded and connected bialgebra \({\mathcal{H}}_{N}\) is automatically a Hopf algebra. A more direct argument, and formulas for the antipode, can be found in [28].

In most applications we want to substitute infinite series and extend

*a*⋆ to a homomorphism \(a\star\colon {\mathcal{N}}^{*}\rightarrow {\mathcal{N}}^{*}\). The extension to infinite substitution is straightforward because of the grading, we omit details. We write*a*⋆ also for infinite substitution.Lie series are formal series whose homogeneous components are Lie polynomials [33].

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## Acknowledgements

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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Communicated by Elizabeth Mansfield.

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Lundervold, A., Munthe-Kaas, H. Backward Error Analysis and the Substitution Law for Lie Group Integrators.
*Found Comput Math* **13**, 161–186 (2013). https://doi.org/10.1007/s10208-012-9130-z

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DOI: https://doi.org/10.1007/s10208-012-9130-z

### Keywords

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