The adjoint of a Runge–Kutta scheme at tableau level

The adjoint method of a Runge–Kutta scheme (A, b) is again a Runge–Kutta scheme, with tableau (𝟙bᵀ - A, b), i.e. a*ᵢⱼ = bⱼ - aᵢⱼ (Hairer–Nørsett–Wanner, Theorem II.8.3; arXiv:2507.21006, Section 3).

The proof factors through two facts:

• negating the tableau corresponds to the canonical grading involution ψ̄(τ) = (-1)^{|τ|} ψ(τ) of the character (step reversal h ↦ -h), and
• the tableau (A - 𝟙bᵀ, -b) gives the convolution inverse of the character, proven by composing with the original scheme and checking the second-block stages collapse onto the first-block stages.

Combining, adjointScheme = negScheme ∘ inverseScheme realizes ψ* = (ψ⁻¹)̄ at the level of tableaux.

def BSeries.RungeKutta.negScheme {ι : Type u} {R : Type v} [Neg R] (rk : RungeKutta ι R) : RungeKutta ι R

The scheme with negated tableau: one step of size -h.

Equations

• rk.negScheme = { A := fun (i j : ι) => -rk.A i j, b := fun (j : ι) => -rk.b j }

def BSeries.RungeKutta.inverseScheme {ι : Type u} {R : Type v} [Sub R] [Neg R] (rk : RungeKutta ι R) : RungeKutta ι R

The scheme inverting one step of rk: tableau (A - 𝟙bᵀ, -b).

Equations

• rk.inverseScheme = { A := fun (i j : ι) => rk.A i j - rk.b j, b := fun (j : ι) => -rk.b j }

def BSeries.RungeKutta.adjointScheme {ι : Type u} {R : Type v} [Sub R] (rk : RungeKutta ι R) : RungeKutta ι R

The adjoint scheme: tableau (𝟙bᵀ - A, b), i.e. a*ᵢⱼ = bⱼ - aᵢⱼ (Hairer–Nørsett–Wanner, Theorem II.8.3).

Equations

• rk.adjointScheme = { A := fun (i j : ι) => rk.b j - rk.A i j, b := fun (j : ι) => rk.b j }

@[simp]

theorem BSeries.RungeKutta.negScheme_A {ι : Type u} {R : Type v} [Neg R] (rk : RungeKutta ι R) (i j : ι) : rk.negScheme.A i j = -rk.A i j

@[simp]

theorem BSeries.RungeKutta.negScheme_b {ι : Type u} {R : Type v} [Neg R] (rk : RungeKutta ι R) (j : ι) : rk.negScheme.b j = -rk.b j

@[simp]

theorem BSeries.RungeKutta.inverseScheme_A {ι : Type u} {R : Type v} [Neg R] [Sub R] (rk : RungeKutta ι R) (i j : ι) : rk.inverseScheme.A i j = rk.A i j - rk.b j

@[simp]

theorem BSeries.RungeKutta.inverseScheme_b {ι : Type u} {R : Type v} [Neg R] [Sub R] (rk : RungeKutta ι R) (j : ι) : rk.inverseScheme.b j = -rk.b j

@[simp]

theorem BSeries.RungeKutta.adjointScheme_A {ι : Type u} {R : Type v} [Sub R] (rk : RungeKutta ι R) (i j : ι) : rk.adjointScheme.A i j = rk.b j - rk.A i j

@[simp]

theorem BSeries.RungeKutta.adjointScheme_b {ι : Type u} {R : Type v} [Sub R] (rk : RungeKutta ι R) (j : ι) : rk.adjointScheme.b j = rk.b j

theorem BSeries.RungeKutta.negScheme_inverseScheme {ι : Type u} {R : Type v} [SubtractionMonoid R] (rk : RungeKutta ι R) : rk.inverseScheme.negScheme = rk.adjointScheme

The adjoint tableau is the negation of the inverting tableau.

theorem BSeries.RungeKutta.stageWeight_negScheme {ι : Type u} {R : Type v} [Fintype ι] [CommRing R] (rk : RungeKutta ι R) (t : HopfAlgebras.PTree) (i : ι) : rk.negScheme.stageWeight t i = (-1) ^ (t.order + 1) * rk.stageWeight t i

Stage weights change sign according to the subtree order under tableau negation.

theorem BSeries.RungeKutta.stageWeightList_negScheme {ι : Type u} {R : Type v} [Fintype ι] [CommRing R] (rk : RungeKutta ι R) (ts : List HopfAlgebras.PTree) (i : ι) : rk.negScheme.stageWeightList ts i = (-1) ^ HopfAlgebras.PTree.orderList ts * rk.stageWeightList ts i

theorem BSeries.RungeKutta.weight_negScheme {ι : Type u} {R : Type v} [Fintype ι] [CommRing R] (rk : RungeKutta ι R) (t : HopfAlgebras.PTree) : rk.negScheme.weight t = (-1) ^ t.order * rk.weight t

Elementary weights change sign according to the tree order under tableau negation: negating the tableau is a step of size -h.

theorem BSeries.RungeKutta.toCharacter_series_negScheme {ι : Type u} {R : Type v} [Fintype ι] [CommRing R] (rk : RungeKutta ι R) : rk.negScheme.series.toCharacter = ForestAlgebra.Character.involution rk.series.toCharacter

Negating the tableau realizes the grading involution of the character.

theorem BSeries.RungeKutta.stageWeight_compose_inverseScheme {ι : Type u} {R : Type v} [Fintype ι] [CommRing R] (rk : RungeKutta ι R) (t : HopfAlgebras.PTree) (i : ι) : (rk.compose rk.inverseScheme).stageWeight t (Sum.inr i) = rk.stageWeight t i

In the composition of a scheme with its inverting scheme, the second-block stages collapse onto the first-block stages.

theorem BSeries.RungeKutta.stageWeightList_compose_inverseScheme {ι : Type u} {R : Type v} [Fintype ι] [CommRing R] (rk : RungeKutta ι R) (ts : List HopfAlgebras.PTree) (i : ι) : (rk.compose rk.inverseScheme).stageWeightList ts (Sum.inr i) = rk.stageWeightList ts i

theorem BSeries.RungeKutta.weight_compose_inverseScheme {ι : Type u} {R : Type v} [Fintype ι] [CommRing R] (rk : RungeKutta ι R) (t : HopfAlgebras.PTree) : (rk.compose rk.inverseScheme).weight t = 0

All elementary weights of the composed scheme rk ∘ rk⁻¹ vanish: the composition is the identity method.

theorem BSeries.RungeKutta.convolution_toCharacter_series_inverseScheme {ι : Type u} {R : Type v} [Fintype ι] [CommRing R] (rk : RungeKutta ι R) : rk.series.toCharacter.convolution rk.inverseScheme.series.toCharacter = HopfAlgebras.ForestAlgebra.Character.unit R

The inverting scheme is a right convolution inverse of the scheme.

theorem BSeries.RungeKutta.toCharacter_series_inverseScheme {ι : Type u} {R : Type v} [Fintype ι] [CommRing R] (rk : RungeKutta ι R) : rk.inverseScheme.series.toCharacter = ForestAlgebra.Character.inverseCharacter rk.series.toCharacter

The inverse method is a Runge–Kutta method: the tableau (A - 𝟙bᵀ, -b) realizes the convolution inverse of the character.

theorem BSeries.RungeKutta.toCharacter_series_adjointScheme {ι : Type u} {R : Type v} [Fintype ι] [CommRing R] (rk : RungeKutta ι R) : rk.adjointScheme.series.toCharacter = ForestAlgebra.Character.adjointCharacter rk.series.toCharacter

The adjoint method is a Runge–Kutta method (Hairer–Nørsett–Wanner, Theorem II.8.3; arXiv:2507.21006, Section 3): the tableau a*ᵢⱼ = bⱼ - aᵢⱼ, b* = b realizes the adjoint character ψ*.

theorem BSeries.RungeKutta.isOdd_toCharacter_series_compose_adjointScheme {ι : Type u} {R : Type v} [Fintype ι] [CommRing R] (rk : RungeKutta ι R) : ForestAlgebra.Character.IsOdd (rk.adjointScheme.compose rk).series.toCharacter

Composing a scheme with its adjoint yields a symmetric method: the character of rk* ∘ rk is odd (arXiv:2507.21006, Theorem thm:main, the constructive direction, at tableau level).

theorem BSeries.RungeKutta.hasOrder_adjointScheme_iff {ι : Type u} {R : Type v} [Fintype ι] [Field R] [CharZero R] (rk : RungeKutta ι R) (n : ℕ) : rk.adjointScheme.HasOrder n ↔ rk.HasOrder n

A scheme and its adjoint have the same order (Hairer–Nørsett–Wanner, Theorem II.8.3; arXiv:2507.21006, Proposition 7.1(1) at tableau level).