Aromatic Butcher Series

Abstract

We show that without other further assumption than affine equivariance and locality, a numerical integrator has an expansion in a generalized form of Butcher series (B-series) which we call aromatic B-series. We obtain an explicit description of aromatic B-series in terms of elementary differentials associated to aromatic trees, which are directed graphs generalizing trees. We also define a new class of integrators, the class of aromatic Runge-Kutta methods, that extends the class of Runge-Kutta methods, and have aromatic B-series expansion but are not B-series methods. Finally, those results are partially extended to the case of more general affine group equivariance.

Full Affine Equivariance and Weak Natural Transformations in Numerical Analysis—The Case of B-Series

“…A particularly acute question is the characterization of numerical integrators on homogeneous spaces: as [10] shows, group equivariance is not sufficient. So what is the nonlinear equivalent of the affine category?…”

“…It has therefore been conjectured that Runge-Kutta methods, or more precisely, B-Series methods, were the only integrators enjoying that property. A recent result shows that this is not the case [10]. An example of an integrator which is affine equivariant but not a B-Series method is…”

Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs

“…In this section, we adapt the formalism of aromatic B-series of [39] to grafted exotic aromatic forests, in order to use it as a numerical tool for weak Taylor expansions in the next sections. We define the order |γ| of a tree γ P EAT g .…”

“…Remark 3.7. It is proved in [39] that standard B-series methods are exactly the affine equivariant methods. Analogously, it would be interesting to characterize the isometric equivariant maps.…”

Algebraic structure of aromatic B-series

“…The algebraic structure of Bseries has been thoroughly studied [CHV10;Mur99] and others. In [MV15], a generalization of B-series, called aromatic B-series, was introduced as a classification of numerical methods that are affine equivariant, that is not affected by linear changes of coordinates. This paper derives counterparts to some of the results concerning algebraic structure of B-series to aromatic B-series.…”