LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck pass

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.

Second-order face-flux consistency for the one-dimensional viscous operator

Statement

Let $U \in C^{3} ([0, L], G)$ have image in a compact subset of $G$ . On the uniform mesh in One-dimensional orthogonal finite-volume viscous scheme, sample $U_{i} = U (x_{i})$ . For every interior face $x_{i + 1 / 2}$ , the defined viscous face flux satisfies

$∣ G_{i + 1 / 2} - K_{11} (U (x_{i + 1 / 2})) \partial_{x} v (U (x_{i + 1 / 2})) ∣ \leq C h^{2},$

for all sufficiently small $h$ , where $C$ depends on $U$ , the fixed physical parameters, and the compact positive-state set, but not on $i$ or $h$ . This is a face-flux consistency estimate only; it asserts no mesh-uniform solution error bound.

Facts & Assumptions

Given: $L > 0$ , a $C^{3}$ field $U : [0, L] \to G$ whose image lies in a compact subset of $G$ , and the uniform cell centers and face flux formula of One-dimensional orthogonal finite-volume viscous scheme.

[F1]

One-dimensional orthogonal finite-volume viscous scheme: $U_{i + 1 / 2} = (U_{i} + U_{i + 1}) / 2$ and $G_{i + 1 / 2} = K_{11} (U_{i + 1 / 2}) (v (U_{i + 1}) - v (U_{i})) / h$ .

[F2]

Positive-semidefinite entropy-variable factorization of viscous flux: $K_{11} (U)$ is the smooth state block representing the one-direction viscous flux $K_{11} (U) \partial_{x} v$ on $G$ .

[F3]

Strict convexity and entropy variables for the ideal-gas Euler flux: $v = (c_{v} γ - s - β ∣ u ∣^{2} / 2, β u, - β)$ , where $β = c_{v} ρ / ε = 1 / T$ .

Proof

1.1F1givenalgebra

Fix an interior face $x_{f} = x_{i + 1 / 2}$ and put $U_{f} = U (x_{f})$ . Taylor's theorem at $x_{f}$ gives $U (x_{f} \pm h / 2) = U_{f} \pm (h / 2) U^{'} (x_{f}) + (h^{2} / 8) U^{''} (x_{f}) + O (h^{3})$ , uniformly in the face index because $U^{'''}$ is bounded. Averaging the two expansions and using [F1] yields $U_{i + 1 / 2} = U_{f} + O (h^{2})$ .

1.2F3givenalgebra

In [F3], $u = m / ρ$ , $ε = E - ∣ m ∣^{2} / (2 ρ)$ , and $s = c_{v} \log ((γ - 1) ε / ρ^{γ}) + s_{0}$ . These functions and $β$ are smooth on $G$ , where $ρ, ε > 0$ , so $v \circ U$ is $C^{3}$ on $[0, L]$ with bounded third derivative. Its centered Taylor expansion gives $(v (U (x_{f} + h / 2)) - v (U (x_{f} - h / 2))) / h = \partial_{x} v (U_{f}) + O (h^{2})$ uniformly in $i$ .

2.1F1F2step 1.1step 1.2algebra∎

The block $K_{11} (U)$ is smooth on $G$ by its explicit finite-dimensional factorization [F2], hence has bounded first derivatives on a compact neighborhood of the image of $U$ . Step 1.1 gives $K_{11} (U_{i + 1 / 2}) = K_{11} (U_{f}) + O (h^{2})$ . Multiplying this by the bounded difference quotient in step 1.2 and using [F1] proves the stated $O (h^{2})$ face-flux estimate.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

R. Eymard, T. Gallouët, and R. Herbin, Finite Volume Methods, January 2019 updated author text, complete manuscript (standard reference, not scraped)
Jérôme Droniou, Finite volume schemes for diffusion equations: introduction to and review of modern methods, 2014, complete preprint (standard reference, not scraped)