CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

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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.

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A centered pressure flux and centered bed source move a discrete lake

Statement refuted

On a periodic three-cell grid, the central physical-flux rule ${\hat{F}}_{i + 1 / 2} = (F (U_{i}) + F (U_{i + 1})) / 2$ with centered bed source $(0, - g h_{i} (b_{i + 1} - b_{i - 1}) / (2 Δ x))$ preserves every lake-at-rest state under the update $U_{i}^{+} = U_{i} - \frac{Δ t}{Δ x} ({\hat{F}}_{i + 1 / 2} - {\hat{F}}_{i - 1 / 2}) + Δ t (0, - \frac{g h_{i} (b_{i + 1} - b_{i - 1})}{2 Δ x}) .$

Facts & Assumptions

Given: The periodic central-flux/centered-source update in the refuted statement, $Δ x = g = 1$ , any $Δ t > 0$ , bed values $(b_{0}, b_{1}, b_{2}) = (0, 1 / 2, 0)$ , and states $m_{i} = 0$ , $h_{i} = 1 - b_{i} = (1, 1 / 2, 1)$ . These are admissible lake-at-rest data with level $η_{0} = 1$ under One-dimensional shallow-water balance law with a dry-state convention. Show one updated momentum is nonzero.

[F1]

One-dimensional shallow-water balance law with a dry-state convention: for a wet rest state, $m_{i} = 0$ , $h_{i} + b_{i} = η_{0}$ , and the momentum flux is $g h_{i}^{2} / 2$ .

Counterexample

1.1givenF1algebra

The pressure values are $p_{0} = 1 / 2$ , $p_{1} = 1 / 8$ , and $p_{2} = 1 / 2$ by [F1]. Thus the central right and left momentum fluxes at cell $0$ are ${\hat{F}}_{1 / 2}^{m} = (p_{0} + p_{1}) / 2 = 5 / 16$ and ${\hat{F}}_{- 1 / 2}^{m} = (p_{2} + p_{0}) / 2 = 1 / 2$ .

2.1givenstep 1.1algebra∎

The central-flux contribution to $m_{0}^{+}$ is $- Δ t (5 / 16 - 1 / 2) = 3 Δ t / 16$ . The centered source contributes $- Δ t h_{0} (b_{1} - b_{2}) / 2 = - Δ t / 4 = - 4 Δ t / 16$ . Therefore $m_{0}^{+} = - Δ t / 16 \neq 0$ , so the stated update moves this lake and the universal preservation claim is false.

Depends on

One-dimensional shallow-water balance law with a dry-state convention

Used by

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Sources

E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein, and B. Perthame, A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows (standard reference, not scraped)
F. Bouchut, III. Shallow water problem – Hydrostatic reconstruction scheme (standard reference, not scraped)