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

Abstract

The aromatic bicomplex is an algebraic tool based on aromatic Butcher trees and used in particular for the explicit description of volumepreserving 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.