TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Conservation and entropy dissipation of the periodic SBP scheme

Statement

Let $U_{i} (t) \in G$ , $i \in I$ , be any differentiable solution of Periodic diagonal-norm SBP compressible viscous scheme. Then every component of the weighted conservative state $\sum_{i} H_{i i} U_{i} (t)$ is constant, and

$\frac{d}{d t} \sum_{i \in I} H_{i i} η (U_{i} (t)) = - \sum_{i \in I} H_{i i} \sum_{j, k = 1}^{d} (D_{j} v_{i})^{T} K_{j k} (U_{i}) (D_{k} v_{i}) \leq 0.$

The exact entropy production is the finite nodal sum displayed above.

Facts & Assumptions

Given: The finite periodic node set, diagonal matrix $H$ , derivative matrices $D_{j}$ , and semidiscrete equation of Periodic diagonal-norm SBP compressible viscous scheme, with all nodal states remaining in $G$ .

[F1]

Periodic diagonal-norm SBP compressible viscous scheme: $H$ is positive diagonal, $H D_{j} + D_{j}^{T} H = 0$ , $D_{j} 1 = 0$ , and the semidiscrete residual is the sum of the symmetric two-point Euler flux differencing term and the divergence of $K_{j k} (U_{i}) (D_{k} v)_{i}$ .

[F2]

Properties of the explicit entropy-conservative flux: $F_{j}^{e c} (U_{i}, U_{ℓ}) = F_{j}^{e c} (U_{ℓ}, U_{i})$ and $(v_{ℓ} - v_{i})^{T} F_{j}^{e c} (U_{i}, U_{ℓ}) = ψ_{j} (U_{ℓ}) - ψ_{j} (U_{i})$ .

[F3]

Positive-semidefinite entropy-variable factorization of viscous flux: the full spatial-state block matrix $K (U)$ is symmetric positive semidefinite.

Proof

1.1F1algebra

For each direction set $Q_{j} = H D_{j}$ . Then [F1] gives $Q_{j}^{T} = - Q_{j}$ and $Q_{j} 1 = H D_{j} 1 = 0$ . Skew-symmetry also gives $1^{T} Q_{j} = - (Q_{j} 1)^{T} = 0$ .

2.1F1F2step 1.1algebra

Multiply the semidiscrete equation by $H_{i i}$ and sum in $i$ . For the Euler term, each component of $\sum_{i, ℓ} (Q_{j})_{i ℓ} F_{j}^{e c} (U_{i}, U_{ℓ})$ is zero: swapping $i$ and $ℓ$ changes the sign of $Q_{j}$ by step 1.1 and leaves the flux unchanged by [F2]. The viscous term sums to $1^{T} H D_{j} W = 1^{T} Q_{j} W = 0$ for each nodal flux array $W$ . Thus the weighted sum of each conservative component has derivative zero.

2.2F1F2step 1.1algebra

By the chain rule, the Euler contribution to the entropy derivative is $- 2 \sum_{j, i, ℓ} (Q_{j})_{i ℓ} v_{i}^{T} F_{j}^{e c} (U_{i}, U_{ℓ})$ . Skew-symmetry and flux symmetry [F2], after swapping the two indices in one copy of the sum, rewrite this as $- \sum_{j, i, ℓ} (Q_{j})_{i ℓ} (v_{i} - v_{ℓ})^{T} F_{j}^{e c} (U_{i}, U_{ℓ})$ . The entropy-potential identity [F2] changes the summand to $(Q_{j})_{i ℓ} (ψ_{j} (U_{i}) - ψ_{j} (U_{ℓ}))$ . Both its row sum and column sum vanish by step 1.1, so the total Euler contribution is zero.

3.1F1F3algebra∎

For each viscous direction, $v^{T} H D_{j} W = - (D_{j} v)^{T} H W$ by $D_{j}^{T} H = - H D_{j}$ . Applying this to $W_{i} = \sum_{k} K_{j k} (U_{i}) (D_{k} v)_{i}$ shows that the remaining entropy derivative is exactly the negative sum in the Statement. Each nodal quadratic form is nonnegative by [F3], and each weight $H_{i i}$ is positive; hence the derivative is nonpositive.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Eitan Tadmor, Entropy Stable Schemes, Chapter 18, Handbook of Numerical Methods for Hyperbolic Problems: Basic and Fundamental Issues, 2016, complete chapter (standard reference, not scraped)
Pablo Fernandez, Ngoc-Cuong Nguyen, and Jaime Peraire, Entropy-stable hybridized discontinuous Galerkin methods for the compressible Euler and Navier-Stokes equations, complete author preprint (standard reference, not scraped)