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.

The Lie derivative and Noether's theorem on the aromatic bicomplex for the study of volume-preserving numerical integrators

“…For the sake of simplicity, we refer to [28,Sect. 4.1] for the precise definition of I : Ω 0,p → Ω 0,p .…”

“…The augmented aromatic bicomplex is the diagram drawn in Figure 1 that displays the interactions between the different spaces of aromatic forms. The aromatic bicomplex can be seen as a generalised subcomplex of the variational bicomplex [2,3] that focuses on specific classes of homogeneous forms and is fully independent of the dimension of the problem (see [28,Rk. 2…”

“…The recent work [28] introduces new tools from the calculus of variations, such as the aromatic bicomplex (see also [2,3,41,35] and references therein), and these tools yield valuable insight of the form of the Taylor expansion of a volume-preserving methods. In particular, it shows that aromatic Runge-Kutta methods do not preserve volume in general, while aromatic exponential methods are promising starting points.…”

“…The notations and vocabulary are chosen to match with the literature of variational calculus, as we shall draw bridges between numerical analysis and variational calculus in Section 3. We skip the technical details on the Euler and homotopy operators and refer the reader to [28] for more details.…”

“…The horizontal derivative is defined using grafting operations, while the vertical derivative uses the replacing operation. The sign change in the definition of the total derivative is explained in [28,Rk. 2.8].…”