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

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dc.citation.volume11
dc.contributor.authorLaurent A
dc.contributor.authorMcLachlan RI
dc.contributor.authorMunthe-Kaas HZ
dc.contributor.authorVerdier O
dc.date.accessioned2024-10-08T02:30:40Z
dc.date.available2024-10-08T02:30:40Z
dc.date.issued2023-08-08
dc.description.abstractAromatic 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.
dc.description.confidentialfalse
dc.edition.edition2023
dc.identifier.citationLaurent A, Mclachlan RI, Munthe-Kaas HZ, Verdier O. (2023). The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators. Forum of Mathematics, Sigma. 11.
dc.identifier.doi10.1017/fms.2023.63
dc.identifier.eissn2050-5094
dc.identifier.elements-typejournal-article
dc.identifier.issn2050-5094
dc.identifier.numbere69
dc.identifier.urihttps://mro.massey.ac.nz/handle/10179/71621
dc.languageEnglish
dc.publisherCambridge University Press
dc.publisher.urihttps://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/aromatic-bicomplex-for-the-description-of-divergencefree-aromatic-forms-and-volumepreserving-integrators/1EFB90284C2D94EE276E39694A469737
dc.relation.isPartOfForum of Mathematics, Sigma
dc.rights(c) 2023 The Author/s
dc.rightsCC BY 4.0
dc.rights.urihttps://creativecommons.org/licenses/by/4.0/
dc.subject58J10: Differential complexes ; elliptic complexes
dc.subject37M15: Symplectic integrators
dc.subject41A58: Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
dc.subject05C05: Trees
dc.subject65L06: Multistep, Runge-Kutta and extrapolation methods
dc.subject58A12: de Rham theory
dc.titleThe aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators
dc.typeJournal article
pubs.elements-id480392
pubs.organisational-groupOther
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