Lie–Butcher characters of 2N commutator-free methods #
The Bazavov lift of a Williamson 2N low-storage scheme advances by
composing exponentials exp(V_e) with V_e = Σ_{i} β_{e,i} K_i, where
K_i = ξ(Y_i) is the vector field frozen at the i-th stage point. Its
Lie–Butcher character on planar forests is built by Owren's pseudo-stage
substitution recursion (arXiv:2509.20599, Appendix E.2; Owren, Order
conditions for commutator-free Lie group methods, Theorem 2.5):
the flow character after
e + 1exponentials splits over deconcatenations of the forest — the frozen exponential composes by operator concatenation —u_{e+1}(ω) = Σ_{ω = ω₁ω₂} u_e(ω₁) ⋅ exp(V_e)(ω₂),with the shuffle-symmetric single-exponential values
exp(V)(ω₂) = (1/|ω₂|!) ∏_{τ ∈ ω₂} V(τ);the slope characters carry the stage dependence by substitution:
K_i(τ) = u_i(children τ), thei-th stage flow character applied to the branches ofτ.
The method character is ϕ = u_s. We formalise the planar order and
antisymmetric order conditions of arXiv:2509.20599, Theorem E.1, and
machine-check them for the concrete CF-EES(2,5;1/4) integrator,
cross-validating the character values against Table 6 of the paper.
An LB character has planar order p if it matches the exact-flow
coefficients 1/τ! on every planar tree of order at most p
(arXiv:2509.20599, Theorem E.1(1)).
Equations
- φ.HasPlanarOrder p = ∀ (τ : HopfAlgebras.PTree), τ.order ≤ p → φ [τ] = (↑τ.treeFactorial)⁻¹
Instances For
The symmetric defect of an LB character: the Grossman–Larson
composition of the grading-signed character with the character itself —
the LB character of the roundtrip Φ_{-h} ∘ Φ_h
(arXiv:2509.20599, Theorem E.1(2)).
Equations
- φ.symmetricDefect = HopfAlgebras.PlanarForest.mkwConvolution (fun (ω : HopfAlgebras.PlanarForest) => (-1) ^ HopfAlgebras.PTree.orderList ω * φ ω) φ
Instances For
An LB character has antisymmetric order m if its symmetric
defect vanishes on every planar tree of order at most m: the roundtrip
Φ_{-h} ∘ Φ_h recovers the identity to order m.
Equations
- φ.HasPlanarAntisymOrder m = ∀ (τ : HopfAlgebras.PTree), τ.order ≤ m → φ.symmetricDefect [τ] = 0
Instances For
The Lie–Butcher flow character u_e after e exponentials of the
Bazavov commutator-free lift (Owren's substitution recursion;
arXiv:2509.20599, Appendix E.2, equation (18)): u_0 = ε and
u_{e+1}(ω) = Σ_{ω = ω₁ω₂} u_e(ω₁) ⋅ (1/|ω₂|!) ∏_{τ ∈ ω₂} V_e(τ),
where V_e(τ) = Σ_i β_{e,i} u_i(children τ) combines the vector fields
frozen at the stage points.
Equations
Instances For
The Lie–Butcher character of the commutator-free method induced by a
low-storage scheme: the flow character after all s exponentials
(arXiv:2509.20599, Appendix E.2, equation (19)).
Equations
- ls.methodChar = ls.flowChar s
Instances For
The flow character on the single-node forest accumulates the row sums
of the slope weights: u_e([•]) = Σ_{m<e} Σ_i β_{m,i}.
Consistency of the Lie–Butcher character: on the single-node tree
the method character of the commutator-free lift equals the sum of the
induced quadrature weights Σⱼ bⱼ.
The flow character on trees of order at most two #
In general the method character reproduces the Runge–Kutta elementary weights of the induced tableau on the bullet and the two-chain, which yields the planar order-two condition for a whole family at once.
The flow character on the single bullet is the partial abscissa: for
i < s, u_i([•]) = c_i, the i-th row sum of the induced tableau.
The flow character on the two-chain accumulates the slope weights against the bullet values.
The Lie–Butcher character matches the Runge–Kutta weight on the
two-chain: ϕ(chain₂) = Σᵢ bᵢ cᵢ for the induced tableau.
A memoised evaluator #
flowChar recomputes stage characters exponentially often along deep
trees. For machine-checked order conditions we use an equivalent
evaluator that annotates each tree once with its slope values
[u_0(children τ), …, u_{s-1}(children τ)] and never re-descends into
trees, making character evaluation polynomial in the forest order.
The slope values of a tree, as data:
trueSlopes τ = [u_0(children τ), …, u_{s-1}(children τ)].
Equations
- ls.trueSlopes τ = List.map (fun (i : ℕ) => ls.flowChar i τ.children) (List.range s)
Instances For
The flow character evaluated on a forest of precomputed slope-value lists: the recursion never re-enters the trees.
Equations
Instances For
The memoised slope annotation: each subtree's slope list is computed exactly once.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The memoised method character.
Equations
- ls.fastMethodChar ω = ls.flowCharAnn s (List.map ls.slopeList ω)
Instances For
The memoised evaluator computes the method character. All
machine-checked order conditions evaluate fastMethodChar instead of
the exponential-time methodChar recursion.
Planar order two from the induced tableau #
Planar order two from the induced tableau weights: if the induced
Butcher tableau of a low-storage scheme is consistent (Σⱼ bⱼ = 1) and
satisfies the classical order-two condition (Σᵢ bᵢcᵢ = 1/2), then the
commutator-free lift has planar order two on any homogeneous space.
CF-EES(2,5;x) has planar order two for every admissible parameter
(arXiv:2509.20599, Theorem E.1(1) in full generality): the Lie–Butcher
character of the commutator-free lift of EES(2,5;x) matches the exact
flow on all planar trees of order at most two, over any field of
characteristic ≠ 2 and any x ∉ {1, ±1/2}.