# Aromatic Butcher Series

@article{MuntheKaas2016AromaticBS, title={Aromatic Butcher Series}, author={Hans Z. Munthe-Kaas and Olivier Verdier}, journal={Foundations of Computational Mathematics}, year={2016}, volume={16}, pages={183-215} }

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…

## 18 Citations

### B-series methods are exactly the affine equivariant methods

- MathematicsNumerische Mathematik
- 2016

It is proved that a sequence of smooth maps between vector fields on affine spaces has a B-series expansion if and only if it is affine equivariant, meaning it respects all affine maps between affine Spaces.

### What are Butcher series, really? The story of rooted trees and numerical methods for evolution equations

- Mathematics
- 2015

A review of early work in Runge-Kutta methods and Butcher (B) series, especially Robin Merson's 1957 paper and John Butcher's early work; and a discussion of work in the past decade on algebraic and…

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

- MathematicsMath. Comput.
- 2020

We introduce a new algebraic framework based on a modification (called exotic) of aromatic Butcher-series for the systematic study of the accuracy of numerical integrators for the invariant measure…

### Full affine equivariance and weak natural transformations in numerical analysis - the case of B-Series

- Mathematics
- 2016

Many algorithms in numerical analysis are affine equivariant: they are immune to changes of affine coordinates. This is because those algorithms are defined using affine invariant constructions.…

### Butcher series: A story of rooted trees and numerical methods for evolution equations

- Mathematics
- 2015

Butcher series appear when Runge-Kutta methods for ordinary differential equations are expanded in power series of the step size parameter. Each term in a Butcher series consists of a weighted…

### Constructing general rough differential equations through flow approximations

- MathematicsElectronic Journal of Probability
- 2022

The non-linear sewing lemma constructs flows of rough differential equations from a braod class of approximations called almost flows. We consider a class of almost flows that could be approximated…

### Using aromas to search for preserved measures and integrals in Kahan's method

- Mathematics
- 2022

. The numerical method of Kahan applied to quadratic diﬀerential equations is known to often generate integrable maps in low dimensions and can in more general situations exhibit preserved measures…

### Functional equivariance and conservation laws in numerical integration

- Mathematics, PhysicsFoundations of Computational Mathematics
- 2022

The concept of functional equivariance is introduced, a natural sense in which a numerical integrator may preserve the dynamics satisfied by certain classes of observables, whether or not they are invariant, to obtain results on methods preserving local conservation laws in PDEs.

### Integrators on Homogeneous Spaces: Isotropy Choice and Connections

- MathematicsFound. Comput. Math.
- 2016

It is demonstrated that the RKMK, Crouch–Grossman, or commutator-free methods are equivariant, and it is shown that the space of matrices of fixed rank possesses no connection.

### Order conditions for sampling the invariant measure of ergodic stochastic differential equations on manifolds

- MathematicsFound. Comput. Math.
- 2022

A new methodology for the construction of high order integrators for sampling the invariant measure of ergodic stochastic differential equations with dynamics constrained on a manifold is derived using an extension of the exotic aromatic Butcher-series formalism.

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