Aromatic Butcher Series

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.