TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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CFL energy stability and finite termination

Physical scope

Pure mathematics; no physical premise.

Statement

For the matrices and algorithm of the finite compatible matrix system, set $K : = M_{b}^{1 / 2} C M_{e}^{- 1 / 2} : R^{n_{e}} \to R^{n_{b}}$ . If $n_{e} = 0$ , define $∥ K ∥_{2} : = 0$ ; if $n_{e} > 0$ , define $∥ K ∥_{2} : = \max_{∥ x ∥_{2} = 1} ∥ K x ∥_{2}$ , where $∥ x ∥_{2} : = (x^{T} x)^{1 / 2}$ . For a zero-dimensional symmetric positive-definite matrix, define its square root and inverse square root to be its unique $0 \times 0$ matrix. For a positive-dimensional real symmetric positive-definite matrix $M$ , define $M^{1 / 2}$ by applying the positive scalar square root to its eigenvalues in an orthonormal eigenbasis, and define $M^{- 1 / 2} : = (M^{1 / 2})^{- 1}$ . Call $h ∥ K ∥_{2} < 2$ the strict CFL condition for this finite matrix system. With $x_{n} : = M_{e}^{1 / 2} e_{n}$ and $y_{n - 1 / 2} : = M_{b}^{1 / 2} b_{n - 1 / 2}$ , define

$E_{h} (x_{n}, y_{n - 1 / 2}) : = \frac{1}{2} (∥ x_{n} ∥_{2}^{2} + ∥ y_{n - 1 / 2} ∥_{2}^{2} - h y_{n - 1 / 2}^{T} K x_{n}) .$

For every $N \geq 0$ , the algorithm terminates after $N$ updates. If $h ∥ K ∥_{2} < 2$ , then each step preserves $E_{h}$ , and

$\frac{1 - h ∥ K ∥_{2} / 2}{2} (∥ x_{n} ∥_{2}^{2} + ∥ y_{n - 1 / 2} ∥_{2}^{2}) \leq E_{h} (x_{n}, y_{n - 1 / 2}) \leq \frac{1 + h ∥ K ∥_{2} / 2}{2} (∥ x_{n} ∥_{2}^{2} + ∥ y_{n - 1 / 2} ∥_{2}^{2}) .$

Consequently this modified energy is positive definite under the strict CFL condition. The constraint $D b_{n - 1 / 2}$ is preserved. If $h ∥ K ∥_{2} > 2$ , set $σ : = ∥ K ∥_{2}$ . There are unit vectors $v \in R^{n_{e}}$ and $u \in R^{n_{b}}$ with $K v = σ u$ and $K^{T} u = σ v$ . For a normalized state with $x_{n} = a v$ and $y_{n - 1 / 2} = b u$ , one update sends the coefficient vector by

$(\begin{array}{c} a \\ b \end{array}) ⟼ T_{σ} (\begin{array}{c} a \\ b \end{array}), T_{σ} : = (\begin{array}{cc} 1 - h^{2} σ^{2} & h σ \\ - h σ & 1 \end{array}) .$

The matrix $T_{σ}$ has a real eigenvalue $λ$ with $∣ λ ∣ > 1$ . Starting from a nonzero coefficient eigenvector, the Euclidean norm of the corresponding normalized state is multiplied by $∣ λ ∣$ at each update. Equality at $h ∥ K ∥_{2} = 2$ is excluded from both assertions; no stability claim is made there.

Statement references

Finite compatible matrix system

Staggered compatible leapfrog

Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis

Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||

The operator norm is 0 on the zero domain and otherwise equals the largest singular value, attained at a right-singular vector

Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors

Facts & Assumptions

Given: The finite compatible matrix system with $S = 0$ , $j = 0$ , $M_{e}, M_{b}$ symmetric positive definite, $D C = 0$ , nonnegative state dimensions as in its definition, and a step size $h > 0$ and finite horizon $N \geq 0$ for the staggered leapfrog algorithm.

[F1]

Every self-adjoint endomorphism of a finite-dimensional real inner-product space has an orthonormal eigenbasis (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis).

[F2]

Every linear map $T : V \to W$ between finite-dimensional real or complex inner-product spaces has orthonormal right and left singular bases and singular values $s_{1} \geq \dots \geq s_{r} > 0$ (Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition).

[F3]

For a map between finite-dimensional inner-product spaces, its operator norm is the maximum of $∥ T v ∥$ over unit vectors, with norm zero on the zero domain (The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||).

[F4]

On a nonzero finite-dimensional domain, the operator norm equals the largest singular value and is attained at a right singular vector (The operator norm is 0 on the zero domain and otherwise equals the largest singular value, attained at a right-singular vector).

[F5]

In a real or complex inner-product space, $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

[F6]

The matrices $M_{e}, M_{b}$ are symmetric positive definite and $D C = C G = 0$ (Finite compatible matrix system).

[F7]

The algorithm's exact updates are $b_{n + 1 / 2} = b_{n - 1 / 2} - h C e_{n}$ and $e_{n + 1} = e_{n} + h M_{e}^{- 1} C^{T} M_{b} b_{n + 1 / 2}$ (Staggered compatible leapfrog).

Proof

1.1givenF1F6algebra

For a zero-dimensional mass space, the unique square matrix is the identity, its own inverse, and its root. In positive dimension, $M_{e}$ and $M_{b}$ are symmetric and therefore self-adjoint in the standard Euclidean inner product; by [F1] each has an orthonormal eigenbasis. If $v$ is a unit eigenvector of either matrix with eigenvalue $λ$ , positive definiteness in [F6] gives $λ = v^{T} M v > 0$ . Define the square root to act by $\sqrt{λ}$ on the full $λ$ -eigenspace, and its inverse to act by $1 / \sqrt{λ}$ there. These definitions are independent of a basis choice within an eigenspace, satisfy $(M^{1 / 2})^{2} = M$ and $M^{- 1 / 2} M^{1 / 2} = I$ , and agree with the Statement's root convention.

2.1givenF7step 1.1algebra

Let $x_{n} = M_{e}^{1 / 2} e_{n}$ and $y_{n - 1 / 2} = M_{b}^{1 / 2} b_{n - 1 / 2}$ . Multiplying the first update in [F7] by $M_{b}^{1 / 2}$ gives $y_{n + 1 / 2} = y_{n - 1 / 2} - h K x_{n}$ . Multiplying the second update by $M_{e}^{1 / 2}$ and using the root identities from step 1.1 gives $x_{n + 1} = x_{n} + h K^{T} y_{n + 1 / 2}$ .

3.1step 2.1algebra

Write $x = x_{n}$ , $y = y_{n - 1 / 2}$ , $y^{'} = y_{n + 1 / 2}$ , and $x^{'} = x_{n + 1}$ . From step 2.1, $x^{'} = x + h K^{T} y^{'}$ and $y^{'} = y - h K x$ . Expanding gives $∥ x^{'} ∥_{2}^{2} = ∥ x ∥_{2}^{2} + 2 h (y^{'})^{T} K x + h^{2} ∥ K^{T} y^{'} ∥_{2}^{2}$ , $∥ y^{'} ∥_{2}^{2} = ∥ y ∥_{2}^{2} - 2 h y^{T} K x + h^{2} ∥ K x ∥_{2}^{2}$ , and $- h (y^{'})^{T} K x^{'} = - h (y^{'})^{T} K x - h^{2} ∥ K^{T} y^{'} ∥_{2}^{2}$ .

3.2givenF2F3F4step 2.1algebra

Suppose $h ∥ K ∥_{2} > 2$ . By the zero-domain clause of [F3], the domain is nonzero; [F2]–[F4] then give unit vectors $v, u$ for $σ = ∥ K ∥_{2}$ , with $K v = σ u$ and $K^{T} u = σ v$ . The recurrence in step 2.1 sends states $x = a v$ , $y = b u$ to states of the same form, with coefficient matrix $T_{σ} : = (\begin{array}{cc} 1 - h^{2} σ^{2} & h σ \\ - h σ & 1 \end{array})$ . Its trace is $2 - h^{2} σ^{2} < - 2$ and determinant is $1$ , so its characteristic polynomial has two distinct real negative reciprocal roots, one with modulus greater than $1$ . A nonzero eigenvector for that root therefore has its coefficient norm multiplied by a factor greater than $1$ at every update, proving the supercritical growth claim.

4.1step 3.1algebra

Adding the identities in step 3.1 cancels the $∥ K^{T} y^{'} ∥_{2}^{2}$ terms; substituting $y^{'} = y - h K x$ in the remainder gives $∥ x^{'} ∥_{2}^{2} + ∥ y^{'} ∥_{2}^{2} - h (y^{'})^{T} K x^{'} = ∥ x ∥_{2}^{2} + ∥ y ∥_{2}^{2} - h y^{T} K x$ . Therefore $E_{h} (x^{'}, y^{'}) = E_{h} (x, y)$ at each update.

4.2givenF3F4F5step 3.1algebra

By [F5] and [F3]–[F4], $∣ y^{T} K x ∣ \leq ∥ y ∥_{2} ∥ K x ∥_{2} \leq ∥ K ∥_{2} ∥ x ∥_{2} ∥ y ∥_{2}$ . Since $2 ∥ x ∥_{2} ∥ y ∥_{2} \leq ∥ x ∥_{2}^{2} + ∥ y ∥_{2}^{2}$ , the cross term in $E_{h}$ has absolute value at most $\frac{1}{2} h ∥ K ∥_{2} (∥ x ∥_{2}^{2} + ∥ y ∥_{2}^{2})$ . Substitution gives both displayed energy bounds, whose lower coefficient is strictly positive when $h ∥ K ∥_{2} < 2$ .

5.1givenF6F7step 1.1algebra∎

For every update, [F7] and [F6] give $D b_{n + 1 / 2} = D b_{n - 1 / 2} - h D C e_{n} = D b_{n - 1 / 2}$ , so induction preserves $D b$ . The loop in [F7] has exactly the finite index set $0, \dots, N - 1$ ; each exact mass solve is unique by [F6] and step 1.1. It therefore terminates after $N$ updates, including the case $N = 0$ .

CFL energy stability and finite termination

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources