Aromatic and clumped multi-indices: algebraic structure and Hopf embeddings

TL;DR

This paper introduces aromatic and clumped multi-indices, algebraic tools that simplify the study of volume-preserving numerical methods while maintaining complex structural properties.

Contribution

It develops the algebraic structure of these indices and extends the Hopf embedding to the aromatic context, advancing the mathematical framework for volume-preserving methods.

Findings

Algebraic structures include pre-Lie-Rinehart algebra, Hopf algebroid, and Hopf algebra.

Application of indices in numerical analysis of volume-preserving methods.

Generalization of Hopf embedding to aromatic multi-indices.

Abstract

Butcher forests extend naturally into aromatic and clumped forests and play a fundamental role in the numerical analysis of volume-preserving methods. The design of general volume-preserving methods is a challenging open problem, and recent attempts showed progress on specific dynamics. We introduce aromatic and clumped multi-indices, that are algebraic objects that simplify the study of volume-preservation to the one-dimensional setting, while retaining much of the structure (in stark opposition to standard multi-indices). We provide their algebraic structure of pre-Lie-Rinehart algebra, Hopf algebroid, and Hopf algebra, apply them in numerical analysis, and generalise to the aromatic context the Hopf embedding from multi-indices to the BCK Hopf algebra.