Aromatic Butcher Series

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.