ExampleConstruction: AI-generatedVerification: 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.

An expanding cell preserves a constant state

Statement

On a two-cell periodic grid take $Δ ξ = 1$ , $Δ t = 0.1$ , period $L = 2$ , initial faces $x_{1 / 2} = 0$ , $x_{3 / 2} = 1$ , $x_{5 / 2} = 2$ , physical speed $a = 0$ , and periodic face velocities $w_{1 / 2} = 0$ , $w_{3 / 2} = 0.1$ , $w_{5 / 2} = 0$ . If both initial cell averages equal $U \in R$ , the successful ALE upwind step has Jacobians $1.01$ and $0.99$ , preserves both averages at $U$ , and conserves total mass $2 U$ .

Facts & Assumptions

Given: The grid, time step, face velocities, physical speed, and constant initial average in the Statement.

[F1]

Periodic one-dimensional moving grid: $J_{i}^{n} = (x_{i + 1 / 2}^{n} - x_{i - 1 / 2}^{n}) / Δ ξ$ , the face update is $x_{i + 1 / 2}^{n + 1} = x_{i + 1 / 2}^{n} + Δ t w_{i + 1 / 2}^{n}$ , and $c_{i + 1 / 2} = a - w_{i + 1 / 2}$ .

[F2]

Conservative upwind ALE finite-volume step: after checking $J_{i}^{n + 1} > 0$ , the algorithm uses ${\hat{F}}_{i + 1 / 2} = c_{i + 1 / 2}^{+} {\overset{ˉ}{u}}_{i} + c_{i + 1 / 2}^{-} {\overset{ˉ}{u}}_{i + 1}$ and $J_{i}^{n + 1} {\overset{ˉ}{u}}_{i}^{n + 1} = J_{i}^{n} {\overset{ˉ}{u}}_{i}^{n} - (Δ t / Δ ξ) ({\hat{F}}_{i + 1 / 2} - {\hat{F}}_{i - 1 / 2})$ .

[F3]

The upwind ALE step conserves mass and preserves constants: a successful periodic upwind ALE step preserves mass and every constant input state.

Verification

1.1givenF1F2algebra

The moved faces are $0$ , $1.01$ , and $2$ . Hence the two updated Jacobians are $1.01$ and $0.99$ , both positive, so [F2] returns success and performs the state update.

1.2givenF1F2algebra

The relative speeds at these faces are $0$ , $- 0.1$ , and $0$ , so the fluxes are $0$ , $- 0.1 U$ , and $0$ . Formula [F2] gives $J_{1}^{n + 1} {\overset{ˉ}{u}}_{1}^{n + 1} = U - 0.1 (- 0.1 U) = 1.01 U$ and $J_{2}^{n + 1} {\overset{ˉ}{u}}_{2}^{n + 1} = U - 0.1 (0.1 U) = 0.99 U$ . Dividing by the positive Jacobians gives both updated averages equal to $U$ .

2.1F1F3step 1.2algebra∎

The new total mass is $1.01 U + 0.99 U = 2 U$ , equal to the old mass $U + U$ . This is the finite instance of [F3].

Depends on

Used by

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Sources

J. Donea, A. Huerta, J.-Ph. Ponthot, and A. Rodríguez-Ferran, Arbitrary Lagrangian–Eulerian Methods, Encyclopedia of Computational Mechanics, Vol. 1, Chapter 14 (2004) (standard reference, not scraped)