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The free tracial post-Lie-Rinehart algebra of planar aromatic trees for the design of divergence-free Lie-group methods

Adrien Busnot Laurent, Hans Munthe-Kaas, Venkatesh G. S

Abstract

Aromatic Butcher series were successfully introduced for the study and design of numerical integrators that preserve volume while solving differential equations in Euclidean spaces. They are naturally associated to pre-Lie-Rinehart algebras and pre-Hopf algebroids structures, and aromatic trees were shown to form the free tracial pre-Lie-Rinehart algebra. In this paper, we present the generalisation of aromatic trees for the study of divergence-free integrators on manifolds. We introduce planar aromatic trees, show that they span the free tracial post-Lie-Rinehart algebra, and apply them for deriving new Lie-group methods that preserve geometric divergence-free features up to a high order of accuracy.

The free tracial post-Lie-Rinehart algebra of planar aromatic trees for the design of divergence-free Lie-group methods

Abstract

Aromatic Butcher series were successfully introduced for the study and design of numerical integrators that preserve volume while solving differential equations in Euclidean spaces. They are naturally associated to pre-Lie-Rinehart algebras and pre-Hopf algebroids structures, and aromatic trees were shown to form the free tracial pre-Lie-Rinehart algebra. In this paper, we present the generalisation of aromatic trees for the study of divergence-free integrators on manifolds. We introduce planar aromatic trees, show that they span the free tracial post-Lie-Rinehart algebra, and apply them for deriving new Lie-group methods that preserve geometric divergence-free features up to a high order of accuracy.

Paper Structure

This paper contains 18 sections, 15 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Lie-Rinehart Algebras and their cohomology
  3. Lie-Rinehart Algebras
  4. Chevalley-Eilenberg Cohomology
  5. Reductive Splitting and Torsion
  6. Affine and Isometric Connections
  7. Levi-Civita and Weitzenböck Connection
  8. Tracial Lie-Rinehart Algebras
  9. Post-Lie-Rinehart algebras
  10. Planar aromatic trees and the free tracial post-Lie-Rinehart algebra
  11. Planar rooted trees and free post-Lie-Rinehart algebra
  12. Planar aromatic trees and free tracial post-Lie-Rinehart algebras
  13. Numerical preservation of divergence-free features on manifolds
  14. Post-Lie-Rinehart structure of the connection algebra
  15. Divergence-free vector fields, volume-preservation, and planar aromatic trees
  16. ...and 3 more sections

Key Result

Proposition 3.8

Floystad20tup If $N$ is a Lie-Rinehart module with respect to the Lie-Rinehart pair $(R,L)$, then $\mathop{\mathrm{End}}\nolimits_R(N)$ is also a Lie-Rinehart module over $(R,L)$.

Figures (1)

  • Figure 1:

Theorems & Definitions (44)

  • Definition 2.1
  • Remark 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Definition 3.5: Lie-Rinehart module
  • Example 3.6
  • Definition 3.7: Morphism of Lie-Rinehart modules
  • Proposition 3.8
  • ...and 34 more