Backward error analysis and the substitution law for Lie group integrators

TL;DR

This paper extends backward error analysis and the substitution law to Lie group integrators using Lie-Butcher series, generalizing classical results from vector spaces to manifolds.

Contribution

It develops the backward error analysis framework for Lie--Butcher series, enabling analysis of numerical methods on manifolds beyond vector spaces.

Findings

Backward error analysis is formulated for Lie--Butcher series.

The substitution law is established for Lie group integrators.

The framework applies to numerical methods on manifolds.

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 Lie--Butcher series. This paper presents the theory of backward error analysis for methods based on Lie--Butcher series.