# Lie–Butcher series

### What and why

Alexander Lundervold

ICT Engineering, Bergen University College, Norway

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## The what

Lie–Butcher series is a mathematical tool for studying numerical methods for dynamical systems evolving on homogeneous manifolds

## The why

Lie–Butcher series are interesting from several points of view:

• Algebra (pre- and post-Lie algebras, Hopf algebras)
• Geometry (homogeneous manifolds, connections)
• Applications (numerical methods)

## Butcher series

Lie–Butcher series is a generalization of Butcher series

B-series gives a way to study differential equations on vector spaces.
(References: Butcher. 1963, Hairer, Lubich, Wanner. 2006)

Consider a differential equation on $\mathbb{R}^n$:

$$y' = f(y), \quad y_0 = y(0)$$

By iteratively differentiating the expression, we can write the solution as a formal power series

$$y' = f(y), \quad y_0 = y(0)$$

\begin{align*} y'' &= f'(y)\, y' = f'(y) f(y) \\ y''' &= f''(y) y' f(y) + f'(y) f'(y)y' = f''(y) \left(f(y)^2\right) + \left(f'(y)\right)^2f(y)\\ y^{(4)} &= f'''(y)(f(y))^3 + 4 f''(y)f'(y)(f(y))^2 + (f'(y))^2 f'(y)f(y)\\ &\vdots \end{align*}

Obtain a (messy!) Taylor series expansion of the solution in derivatives of $\,f$

Cayley, 1852 (Merson, 1957): The derivatives of $f$ can be associated to rooted trees.

Elementary differentials $\,\mathcal{F}$: vector fields defined recursively on trees.

We can rewrite the series expansion of the solution as a series of elementary differentials indexed by trees.

$$y = \sum_{\tau \in T} \alpha(\tau) \mathcal{F}(\tau).$$

### Numerical methods as B-series

Many numerical methods can be represented as B-series

Example: Midpoint method

\begin{align*}y_{k+1} &= y_k + hf(y_k) + \frac{h^2}{2} f'(y_k) f(y_k) + \mathcal{O}(h^3) \end{align*}

Consider a numerical time-stepping method

$$\phi_{h,f}: \mathbb{R}^n \rightarrow \mathbb{R}^n,$$
with $\,y_{k+1} = \phi_{h,f}(y_k)\,$ and $\,y_k \approx y(kh)$.

If $\,\phi\,$ is for example a Runge–Kutta method, then it can be expanded in a B-series

$$y_{k+1} = y_k + \sum_{t \in T} h^{|t|}\alpha(t) \mathcal{F}_f(t)(y_k).$$

### A question

Which methods are B-series methods?

B-series methods are exactly the local, affine equivariant methods, McLachlan, Modin, Munthe-Kaas, Verdier. 2014. arXiv:1409.1019

## The why

What can we use Butcher series for?

### Order theory

Assume we have expressed a numerical method as a B-series

$$y_{k+1} = y_k + \sum_{t \in T} h^{|t|}\alpha(t) \mathcal{F}_f(t)(y_k).$$ We can find the order by comparing with the B-series for the exact solution.

Example: The coefficients of the midpoint method agree with the coefficients of the exact B-series up to order 2. Therefore: second order method.

In general: order conditions

### Structure-preservation

Structure-preserving properties of the B-series method can be captured in the coefficients.

Example: A vector field $F$ given by a B-series is Hamiltonian if

$$\alpha(\tau_1 \circ \tau_2) + \alpha(\tau_2 \circ \tau_1) = 0$$
for all trees $\tau_1$,$\tau_2$. The product is the Butcher product. It is symplectic if

$$\alpha(\tau_1 \circ \tau_2) + \alpha(\tau_2 \circ \tau_1) = \alpha(\tau_1)\alpha(\tau_2).$$

Calvo, Sanz-Serna. 1994. Hairer, Lubich, Wanner. 2006

### Help us uncover connections to other fields

This is the point that's perhaps most interesting for us today.

One of the most important structures in this regard:

### Pre-Lie algebras

Also: Combinatorial and incidence Hopf algebras, noncommutative Bell polynomials, Faà di Bruno formulas

## Pre-Lie algebras

Weakened associative algebras that still gives rise to Lie algebras

Definition: A vector space $A$ equipped with a bilinear product $\triangleright$ such that

$$[L(a), L(b)] = L([a,b)],$$
where $L(a)$ denotes left multiplication $L(a)\triangleright b = a \triangleright b$.

In other words: $$a_{\triangleright}(a,b,c) - a_{\triangleright}(b,a,c) = 0,$$
where $a_{\triangleright} = (a\triangleright b)\triangleright c - a \triangleright (b \triangleright c)$ is the associator.

Note: $\,[a,b] := a \triangleright b - b \triangleright a\,$ defines a Lie algebra on $A$.

### Examples

The pre-Lie algebra of vector fields. Let $\nabla$ be a flat and torsion free (Koszul) connection on the tangent bundle of a manifold $M$.

$$v \triangleright w := \nabla_v w$$ is pre-Lie.

The pre-Lie algebra of trees. The set of rooted nonplanar trees $\mathcal{T}$ equipped with the grafting product:

This is the free pre-Lie algebra on one generator. (Ref: Chapoton–Livernet, 2001)

### Elementary differentials revisited

The vector fields $\mathcal{X} \mathbb{R}^n$ form a pre-Lie algebra. Since $\mathcal{T}$ is the free pre-Lie algebra there is a pre-Lie map $\mathcal{F}$ satisfying

\begin{align*} \mathcal{F}(\bullet) &= f\\ \mathcal{F}(\tau_1 \triangleright \tau_2) &= \mathcal{F}(\tau_1) \triangleright \mathcal{F}(\tau_2). \end{align*}

Elementary differentials

### Pre-Lie B-series

Forget the vector fields and formulate B-series as expansions in pre-Lie algebras:

$$\sum_{\tau \in \mathcal{T}} \alpha(\tau) \tau$$

## Lie–Butcher series

A generalization to differential equations on homogeneous manifolds. I.e. manifolds equipped with transitive actions by Lie groups. Think spheres or $SO(3)$.

Lie group methods aim to approximate differential equations

$$y' = F(y) = f(y) \cdot y, \quad y(0) = y_0,$$
where $F$ is a vector field, represented by $f: M \rightarrow \mathfrak{g}$, where $\mathfrak{g}$ is the Lie algebra of $G$.

Example: Runge–Kutta–Munthe-Kaas methods

Reference: Lie group methods. Iserles, Munthe-Kaas, Nørsett, Zanna. 2000

### Constructing LB-series

Similar story to B-series, but nonplanar trees are replaced by forests of planar trees, and pre-Lie algebras by post-Lie algebras.

A Lie–Butcher series:

$$\sum_{\omega \in F} h^{|\omega|} \alpha(\omega) \mathcal{F}_f(\omega)$$

## Why?

What can Lie–Butcher series be used for?

• Order theory
• Structure-preservation
• Help us discover connections to other fields

## Post-Lie algebras

A Lie-algebra $\,[\cdot, \cdot]\,$ equipped with a bilinear product $\,\triangleright$, satisfying compatibility relations:

\begin{align*} x \triangleright [y,z] &= [x \triangleright y, z] + [y, x \triangleright z] \\ [x,y] \triangleright z &= a_{\triangleright}(x,y,z) - a_{\triangleright}(y,x,z), \end{align*}

where $a_{\triangleright}$ is the associator.

Reference: On Post-Lie algebras, Lie–Butcher series and Moving frames. Lundervold, Munthe-Kaas. 2013.

### Facts:

• A post-Lie bracket with $\,[\cdot,\cdot] = 0$ is a pre-Lie algebra
• A flat and constant torsion connection $\nabla$ on a manifold $M$ gives rise to a post-Lie algebra on the smooth vector fields $\mathcal{X}M$ equipped with the torsion bracket
• The free post-Lie algebra is the free Lie algebra over planar rooted trees, equipped with the left grafting product: $$\mbox{postLie}(\bullet) = \{\mbox{Lie}(\mbox{OT}), [\cdot, \cdot], \triangleright\}$$
• The so-called D-algebras are universal enveloping algebras of post-Lie algebras
• Post-Lie algebras also appear in the theory of operads. Homology of generalized partition posets. Vallette. 2007

## Some future work

• SW implementation of algebraic structures, with the goal of doing order theory etc. Work in progress, together with Munthe-Kaas and K. Føllesdal (a new PhD student of HMK).

• Which methods can be represented as LB-series?