The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators

Abstract

Aromatic B-series were introduced as an extension of standard Butcher-series for the study of volume-preserving integrators. It was proven with their help that the only volume-preserving B-series method is the exact flow of the differential equation. The question was raised whether there exists a volume-preserving integrator that can be expanded as an aromatic B-series. In this work, we introduce a new algebraic tool, called the aromatic bicomplex, similar to the variational bicomplex in variational calculus. We prove the exactness of this bicomplex and use it to describe explicitly the key object in the study of volume-preserving integrators: the aromatic forms of vanishing divergence. The analysis provides us with a handful of new tools to study aromatic B-series, gives insights on the process of integration by parts of trees, and allows to describe explicitly the aromatic B-series of a volume-preserving integrator. In particular, we conclude that an aromatic Runge-Kutta method cannot preserve volume.

arXiv comment41 pages

References in corpus (10)

• Two interacting Hopf algebras of trees
• Geometric properties of Kahan's method
• On integrability of Hirota-Kimura type discretizations
• Stochastic Taylor expansions for the expectation of functionals of diffusion processes
• Aromatic Butcher Series
• B-series methods are exactly the affine equivariant methods
• Order conditions for sampling the invariant measure of ergodic stochastic differential equations on manifolds
• Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs
• The universal equivariance properties of exotic aromatic B-series
• Exotic B-series and S-series: algebraic structures and order conditions for invariant measure sampling

Cited by in corpus (6)

• The Lie derivative and Noether's theorem on the aromatic bicomplex for the study of volume-preserving numerical integrators
• The universal equivariance properties of exotic aromatic B-series
• Hopf algebra structures for the backward error analysis of ergodic stochastic differential equations
• Post-Hopf algebroids, post-Lie-Rinehart algebras and geometric numerical integration
• Volume preservation of Butcher series methods from the operad viewpoint
• Pseudo-Energy-Preserving Explicit Runge-Kutta Methods