ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-23 (gpt-6.1-sol)

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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.
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Constant Euler state preserved by consistent shared flux

Example

Fix $γ > 1$ , $N \geq 2$ , $Δ x > 0$ , and $Δ t \geq 0$ on a periodic uniform grid. Let $G_{γ} = {(ρ, m, E) \in R^{3} : ρ > 0, E - \frac{m^{2}}{2 ρ} > 0}$ and choose a constant state $U = (ρ, m, E) \in G_{γ}$ . Its ideal-gas Euler flux is $F (U) = (m, \frac{m^{2}}{ρ} + (γ - 1) (E - \frac{m^{2}}{2 ρ}), \frac{m}{ρ} (E + (γ - 1) (E - \frac{m^{2}}{2 ρ}))) .$ Set $U_{i} = U$ in every cell and use one consistent shared face flux at each interface, meaning ${\hat{F}}_{i + 1 / 2} (U, U) = F (U)$ . For the update $U_{i}^{+} = U_{i} - λ ({\hat{F}}_{i + 1 / 2} - {\hat{F}}_{i - 1 / 2}), λ = \frac{Δ t}{Δ x},$ every cell remains exactly $U_{i}^{+} = U$ . This is a one-step constant-state fixed-point calculation; it does not assert any other stability or convergence property.

Example references

Euler state, flux, positive set and numerical roles — supplies the shared-face update and consistency convention.

Entropy, invariant-domain and weak-limit hypotheses — separates this fixed-point identity from admissibility, entropy, and weak-limit claims.

De Sterck and Ullrich, §§3.4.1 and 3.4.3 — full course notes treatment of finite-volume cell averages and shared face fluxes.

Ketcheson, LeVeque, and del Razo, Euler approximate-solver chapter — full chapter treatment of the Euler flux and approximate finite-volume solvers.

Facts & Assumptions

Given: $γ > 1$ , a periodic grid with $N \geq 2$ , $Δ x > 0$ , $Δ t \geq 0$ , a constant admissible Euler state $U$ , and consistent shared fluxes.

[F1]

Euler state, flux, positive set and numerical roles: a two-state flux is consistent when $\hat{F} (U, U) = F (U)$ for every $U \in G_{γ}$ .

[F2]

Euler state, flux, positive set and numerical roles: the update is $U_{i}^{+} = U_{i} - λ ({\hat{F}}_{i + 1 / 2} - {\hat{F}}_{i - 1 / 2})$ .

[F3]

Entropy, invariant-domain and weak-limit hypotheses: shared-face conservation, admissibility, entropy stability, and convergence are distinct properties with separate hypotheses; the algebraic flux cancellation alone proves none of the latter three.

Verification

1.1givenF1construct

At every face the adjacent states are $(U, U)$ . By consistency in [F1], each face flux equals $F (U)$ , so the right and left fluxes in a cell update are equal.

2.1givenF2F3step 1.1algebra∎

Substitute this equality in [F2]: $U_{i}^{+} = U - λ (F (U) - F (U)) = U$ . Since the supplied state lies in $G_{γ}$ , the update preserves this constant state. This exact identity supplies no separate entropy or weak-limit conclusion, as [F3] states.

Constant Euler state preserved by consistent shared flux

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources