TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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First-order HLL update preserves a convex invariant domain under admissible fan averages

Statement

Let $K$ be a convex set in the state domain, and consider a periodic uniform mesh and one first-order HLL update as defined in First-order finite-volume update using supplied HLL speeds. Assume every cell mean $U_{i} \in K$ . At every face impose the HLL fan-average hypothesis that the two adjacent half-cell averages $A^{-}, A^{+}$ from First-order HLL updates are averages of adjacent half-cell fan averages lie in $K$ , with that lemma's local CFL bounds. Then every updated mean $U_{i}^{+}$ belongs to $K$ . A sufficient condition for the fan-average hypothesis is that every HLL intermediate state $U_{*} \in K$ , because the stated CFL makes each half-cell average a convex combination of a cell state and $U_{*}$ . The result does not assert a bound on exact nonlinear wave speeds.

Facts & Assumptions

Given: A convex $K$ , cell means in $K$ , and the HLL half-cell fan-average hypothesis with its local CFL bounds. Prove one update remains in $K$ .

[F1]

Convex invariant domains for a one-step update: convexity means every segment between two members of $K$ lies in $K$ .

[F2]

First-order HLL updates are averages of adjacent half-cell fan averages: each $U_{i}^{+}$ is the mean of the right half-cell HLL average at the left face and the left half-cell HLL average at the right face.

Proof

1.1givenF1F2algebra

At each face, the stated HLL fan-average hypothesis gives $A^{-}, A^{+} \in K$ . In the sufficient intermediate-state case, the CFL bounds make the coefficients in each formula of [F2] nonnegative and sum to one, so convexity and $U_{L}, U_{R}, U_{*} \in K$ give the same conclusion.

2.1givenF1F2algebra∎

By [F2], $U_{i}^{+} = \frac{1}{2} (A_{i - 1 / 2}^{+} + A_{i + 1 / 2}^{-})$ . This is a convex combination of two members of $K$ , so $U_{i}^{+} \in K$ for every cell. The argument is one-step and uses only the supplied HLL speeds and the explicit CFL bounds.

Depends on

Used by

Symmetric HLL speeds preserve ideal-gas density and internal-energy positivity Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J.-L. Guermond and B. Popov, Invariant Domains and First-Order Continuous Finite Element Approximation for Hyperbolic Systems (standard reference, not scraped)
D. I. Ketcheson, R. J. LeVeque, and M. J. del Razo, Riemann Problems and Jupyter Solutions (standard reference, not scraped)