Aromatic Butcher Series
Hans Munthe-Kaas, Olivier Verdier

TL;DR
This paper introduces aromatic B-series, a generalized form of B-series for affine equivariant and local numerical integrators, and defines aromatic Runge-Kutta methods that extend traditional Runge-Kutta methods.
Contribution
It provides an explicit description of aromatic B-series using aromatic trees and introduces aromatic Runge-Kutta methods that expand the class of integrators beyond standard B-series.
Findings
Aromatic B-series generalize traditional B-series for affine equivariant integrators.
Aromatic Runge-Kutta methods have aromatic B-series but are not B-series methods.
Results extend to more general affine group equivariance.
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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
