The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators
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Date
2023-08-08
Open Access Location
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Publisher
Cambridge University Press
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(c) 2023 The Author/s
CC BY 4.0
CC BY 4.0
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.
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Keywords
58J10: Differential complexes ; elliptic complexes, 37M15: Symplectic integrators, 41A58: Series expansions (e.g. Taylor, Lidstone series, but not Fourier series), 05C05: Trees, 65L06: Multistep, Runge-Kutta and extrapolation methods, 58A12: de Rham theory
Citation
Laurent 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.