AlgorithmAlgorithm: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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.

Maximal convex scaling toward an admissible cell mean

Definition

Let $K \subseteq R^{m}$ be closed and convex (convex in the sense of Convex invariant domains for a one-step update), let $J$ be a finite nonempty index set, let $w_{j} > 0$ with $\sum_{j \in J} w_{j} = 1$ , and let input states $V_{j} \in R^{m}$ have mean $\overset{ˉ}{V} = \sum_{j} w_{j} V_{j} \in K$ . The maximal convex scaling limiter returns the largest $θ \in [0, 1]$ for which every $V_{j}^{θ} = \overset{ˉ}{V} + θ (V_{j} - \overset{ˉ}{V})$ belongs to $K$ , together with all the states $V_{j}^{θ}$ . Equivalently, form $S = {θ \in [0, 1] : V_{j}^{θ} \in K for every j \in J},$ and return $θ_{*} : = \max S$ and $(V_{j}^{θ_{*}})_{j \in J}$ . The mean is assumed admissible; implementation, cost, and any pointwise-in-cell property are not part of this definition.

Depends on

Convex invariant domains for a one-step update

Used by

Dependency tree · one level

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Sources

X. Zhang and C.-W. Shu, On Positivity-Preserving High Order Discontinuous Galerkin Schemes for Compressible Euler Equations on Rectangular Meshes (standard reference, not scraped)