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Convex scaling preserves the cell mean and positive-margin node states

Statement

Fix $γ > 1$ , $ε > 0$ , a finite nonempty node set $J$ , positive weights $w_{j}$ summing to one, and candidate states $V_{j} \in R^{3}$ with mean $\overset{ˉ}{V} = \sum_{j} w_{j} V_{j} \in K_{γ, ε}$ . Apply the maximal convex scaling limiter of Maximal convex scaling toward an admissible cell mean with $K = K_{γ, ε}$ . Its output is well-defined, every limited node lies in $K_{γ, ε}$ , and the weighted cell mean remains $\overset{ˉ}{V}$ . Hence every limited node has positive density and positive pressure. If the unscaled nodes already lie in $K_{γ, ε}$ , the maximal parameter is $θ_{*} = 1$ . In particular, if smooth target states stay a distance at least $δ > 0$ from $R^{3} ∖ K_{γ, ε}$ and each reconstruction node has error less than $δ$ , the limiter is inactive and preserves that reconstruction's accuracy. No positivity between finitely many nodes is claimed.

Facts & Assumptions

Given: The finite weighted nodes and admissible mean, the closed convex set $K_{γ, ε}$ , and the maximal scaling procedure. Prove existence, conservation, nodewise admissibility, and the strict-margin accuracy implication.

[F1]

Maximal convex scaling toward an admissible cell mean: the limiter selects the largest $θ \in [0, 1]$ such that $\overset{ˉ}{V} + θ (V_{j} - \overset{ˉ}{V}) \in K$ for all nodes.

[F2]

Positive-margin admissible states for ideal-gas Euler: $K_{γ, ε}$ imposes $ρ \geq ε$ and $E - m^{2} / (2 ρ) \geq ε$ .

[F3]

Euler positive states are convex and positive-margin sets are closed convex: $K_{γ, ε}$ is closed and convex.

Proof

1.1givenF1F3algebra

The feasible parameter set $S = {θ \in [0, 1] : \overset{ˉ}{V} + θ (V_{j} - \overset{ˉ}{V}) \in K_{γ, ε} for every j}$ contains $0$ . Each node map is continuous and $K_{γ, ε}$ is closed by [F3], so each inverse image is closed; the finite intersection $S$ is closed in compact $[0, 1]$ . Thus $S$ is nonempty compact and has a maximum $θ_{*}$ . By its definition, every limited node lies in $K_{γ, ε}$ .

2.1givenF1F2step 1.1algebra

Using $\sum_{j} w_{j} = 1$ and $\sum_{j} w_{j} V_{j} = \overset{ˉ}{V}$ , the limited weighted mean is $\sum_{j} w_{j} (\overset{ˉ}{V} + θ_{*} (V_{j} - \overset{ˉ}{V})) = \overset{ˉ}{V} + θ_{*} (\overset{ˉ}{V} - \overset{ˉ}{V}) = \overset{ˉ}{V}$ . By [F2], each limited node has density at least $ε > 0$ and pressure at least $(γ - 1) ε > 0$ .

3.1givenF1F3step 2.1algebra∎

If every original $V_{j} \in K_{γ, ε}$ , then $1 \in S$ , so $θ_{*} = 1$ . If a target state has distance at least $δ$ from the complement and the candidate differs from it by less than $δ$ , the candidate belongs to $K_{γ, ε}$ ; applying this at every node makes $1$ feasible. Thus the limiter leaves all candidates unchanged under the stated strict margin, and their reconstruction error is unchanged. The conclusion concerns only the supplied nodes.

Depends on

Used by

Two Euler quadrature nodes rescaled to a positive margin Example

Dependency tree · two levels

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