Backward error analysis and the substitution law for Lie group integrators
For researchers in geometric numerical integration, this work formalizes backward error analysis for Lie group methods, which is an incremental theoretical extension of existing Butcher series theory.
This paper extends backward error analysis to Lie group integrators using Lie-Butcher series, providing a rigorous framework for analyzing numerical methods on manifolds.
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 Lie--Butcher series. This paper presents the theory of backward error analysis for methods based on Lie--Butcher series.