PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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The shallow-water wet/dry state set is convex

Statement

The admissible wet/dry state set $K$ from One-dimensional shallow-water balance law with a dry-state convention is convex.

Facts & Assumptions

Given: The set $K = {(h, m) : h > 0} \cup {(0, 0)}$ . Prove its convexity using the PE03 convention.

[F1]

One-dimensional shallow-water balance law with a dry-state convention: every state in $K$ has $h > 0$ with arbitrary $m$ , or is exactly $(0, 0)$ .

[F2]

Convex invariant domains for a one-step update: a set is convex when every segment between two of its elements lies in the set.

Proof

1.1givenF1algebra

Let $(h_{1}, m_{1}), (h_{2}, m_{2}) \in K$ and $θ \in [0, 1]$ . The convex combination has depth $h_{θ} = θ h_{1} + (1 - θ) h_{2} \geq 0$ . If $θ = 0$ or $θ = 1$ , it is an endpoint in $K$ by [F1]. If $0 < θ < 1$ and $h_{θ} = 0$ , nonnegativity forces $h_{1} = h_{2} = 0$ ; [F1] then gives $m_{1} = m_{2} = 0$ , so the combination is $(0, 0) \in K$ . If $h_{θ} > 0$ , [F1] places the combination in $K$ for any momentum.

2.1F2step 1.1∎

Thus every segment between two states in $K$ lies in $K$ , which is exactly the convexity condition in [F2].

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Sources

J.-L. Guermond and B. Popov, Invariant Domains and First-Order Continuous Finite Element Approximation for Hyperbolic Systems (standard reference, not scraped)
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)