LemmaStatement: 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.

Moving faces satisfy the one-dimensional discrete GCL

Statement

For the face motion of Periodic one-dimensional moving grid, every cell satisfies

$J_{i}^{n + 1} = J_{i}^{n} + \frac{Δ t}{Δ ξ} (w_{i + 1 / 2}^{n} - w_{i - 1 / 2}^{n}) .$

Facts & Assumptions

Given: Periodic faces with $Δ ξ > 0$ and the update $x_{i + 1 / 2}^{n + 1} = x_{i + 1 / 2}^{n} + Δ t w_{i + 1 / 2}^{n}$ from Periodic one-dimensional moving grid.

[F1]

Periodic one-dimensional moving grid: $J_{i}^{n} = (x_{i + 1 / 2}^{n} - x_{i - 1 / 2}^{n}) / Δ ξ$ and each face moves by $Δ t$ times its prescribed velocity.

Proof

1.1givenF1algebra

Subtract the left-face update from the right-face update for cell $i$ . This gives $x_{i + 1 / 2}^{n + 1} - x_{i - 1 / 2}^{n + 1} = x_{i + 1 / 2}^{n} - x_{i - 1 / 2}^{n} + Δ t (w_{i + 1 / 2}^{n} - w_{i - 1 / 2}^{n})$ .

2.1F1step 1.1algebra∎

Divide by the positive reference width $Δ ξ$ and use the definition of $J_{i}^{n + 1}$ and $J_{i}^{n}$ in [F1]. The claimed identity follows.

Depends on

Periodic one-dimensional moving grid

Used by

The upwind ALE step conserves mass and preserves constants Theorem

Dependency tree · one level

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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)