THE LIE DERIVATIVE AND NOETHER'S THEOREM ON THE AROMATIC BICOMPLEX FOR THE STUDY OF VOLUME-PRESERVING NUMERICAL INTEGRATORS

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, dif-ferent concepts of symmetries, and Noether's theory in the context of aro-matic 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.