The Lie derivative and Noether's theorem on the aromatic bicomplex for the study of volume-preserving numerical integrators

TL;DR

This paper extends the aromatic bicomplex framework by integrating Lie derivatives and Noether's theorem, providing new algebraic tools to analyze volume-preserving numerical integrators through symmetries and variational calculus.

Contribution

It introduces Lie derivatives, symmetries, and Noether's theorem into the aromatic bicomplex, linking volume-preserving methods with symmetries in the Euler-Lagrange complex.

Findings

Established a correspondence between aromatic volume-preserving methods and symmetries.

Derived a version of Noether's theorem within the aromatic context.

Explicitly described aromatic B-series of volume-preserving methods using Lie derivatives.

Abstract

The aromatic bicomplex is an algebraic tool based on aromatic Butcher trees and used in particular for the explicit description of volume-preserving affine-equivariant numerical integrators. The present work defines new tools inspired from variational calculus such as the Lie derivative, different concepts of symmetries, and Noether's theory in the context of aromatic forests. The approach allows to draw a correspondence between aromatic volume-preserving methods and symmetries on the Euler-Lagrange complex, to write Noether's theorem in the aromatic context, and to describe the aromatic B-series of volume-preserving methods explicitly with the Lie derivative.