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

Adrien Laurent

doi:10.3934/jcd.2023011

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2024, Volume 11, Issue 1: 10-22. Doi: 10.3934/jcd.2023011

This issue Previous Article Convergence of the vertical gradient flow for the Gaussian Monge problem Next Article Euler–Maruyama approximations of the stochastic heat equation on the sphere

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

Adrien Laurent

Univ Rennes, INRIA (MINGuS), IRMAR (CNRS UMR 6625) and ENS Rennes, France

Received: July 2023

Revised: November 2023

Early access: November 2023

Published: January 2024

The work of the author was supported by the Research Council of Norway through project 302831 "Computational Dynamics and Stochastics on Manifolds" (CODYSMA).

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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.

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Mathematics Subject Classification: Primary: 58E30, 58J10, 05C05; Secondary: 41A58, 37M15, 58A12.

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Figure 1. The augmented aromatic bicomplex

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