PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-23 (gpt-6.1-sol)

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Ideal-gas LLF admissibility specialization

Statement

Fix $γ > 1$ , an integer $N \geq 2$ , a width $Δ x > 0$ , a time step $Δ t \geq 0$ , and a periodic array of one-dimensional ideal-gas Euler states $U_{i} = (ρ_{i}, m_{i}, E_{i}) \in G_{γ} \subset R^{3}$ , indexed by $i \in Z / N Z$ . For each state set $p_{i} = (γ - 1) (E_{i} - \frac{m_{i}^{2}}{2 ρ_{i}}), u_{i} = \frac{m_{i}}{ρ_{i}}, c_{i} = \sqrt{\frac{γ p_{i}}{ρ_{i}}}, F_{i} = (m_{i}, \frac{m_{i}^{2}}{ρ_{i}} + p_{i}, u_{i} (E_{i} + p_{i})),$ where $G_{γ} = {(ρ, m, E) : ρ > 0, E - \frac{m^{2}}{2 ρ} > 0} .$ At every face choose a supplied speed $a_{i + 1 / 2}$ satisfying $a_{i + 1 / 2} \geq \max (∣ u_{i} ∣ + c_{i}, ∣ u_{i + 1} ∣ + c_{i + 1}), 2 λ a_{i + 1 / 2} \leq 1, λ = \frac{Δ t}{Δ x} .$ Use the symmetric HLL speeds $S_{L} = - a_{i + 1 / 2}$ and $S_{R} = a_{i + 1 / 2}$ , so that ${\hat{F}}_{i + 1 / 2} = \frac{1}{2} (F_{i} + F_{i + 1}) - \frac{a_{i + 1 / 2}}{2} (U_{i + 1} - U_{i}), U_{i}^{+} = U_{i} - λ ({\hat{F}}_{i + 1 / 2} - {\hat{F}}_{i - 1 / 2}) .$ Then $U_{i}^{+} \in G_{γ}$ for every cell $i$ ; in particular, each updated density and pressure is strictly positive. This conclusion is for the specified one-step symmetric branch and does not assert positivity for other HLL speed pairs or a fully discrete entropy inequality.

Statement references

Euler state, flux, positive set and numerical roles — fixes the Euler flux, state domain, periodic update convention, and supplied-speed HLL branch.

Symmetric HLL speeds preserve ideal-gas density and internal-energy positivity — proves positivity for the symmetric HLL branch under the stated endpoint speed and CFL bounds.

Zhang and Shu, §2 — full article treatment of positivity-preserving Euler updates and convex combinations.

Ketcheson, LeVeque, and del Razo, Euler approximate-solver chapter — full chapter treatment of the HLL middle state and Euler solver context.

Facts & Assumptions

Given: $γ > 1$ , periodic states $U_{i} \in G_{γ}$ , face speeds $a_{i + 1 / 2}$ satisfying the displayed endpoint bound and CFL condition, and the symmetric HLL update.

[A1]

The input states satisfy $ρ_{i} > 0$ and $E_{i} - m_{i}^{2} / (2 ρ_{i}) > 0$ , and the chosen speeds obey $a_{i + 1 / 2} \geq \max (∣ u_{i} ∣ + c_{i}, ∣ u_{i + 1} ∣ + c_{i + 1})$ and $2 λ a_{i + 1 / 2} \leq 1$ .

[F1]

Euler state, flux, positive set and numerical roles: the Euler state domain is $G_{γ}$ , the update uses $λ = Δ t / Δ x$ , and the symmetric HLL middle flux uses $S_{L} = - a$ , $S_{R} = a$ .

[F2]

Symmetric HLL speeds preserve ideal-gas density and internal-energy positivity: for positive ideal-gas Euler cell states, symmetric speeds with $a$ at least the adjacent endpoint values $∣ u ∣ + c$ , and $2 ν a \leq 1$ , one first-order HLL update preserves $G_{γ}$ .

Proof

1.1givenA1F1algebra

Since every input state belongs to $G_{γ}$ and $γ > 1$ , each $p_{i} > 0$ and $c_{i} > 0$ . Hence every $a_{i + 1 / 2} > 0$ , so $S_{L} = - a_{i + 1 / 2} < 0 < S_{R} = a_{i + 1 / 2}$ is the HLL middle branch. The local update has $h = Δ x$ , so the supplier's ratio $ν = Δ t / h$ equals $λ$ and the local CFL condition gives $2 ν a_{i + 1 / 2} = 2 λ a_{i + 1 / 2} \leq 1$ at every face.

2.1givenA1F2step 1.1∎

The remaining hypotheses of [F2] are exactly the stated input-state and endpoint-speed bounds, and step 1.1 matches its symmetric branch and CFL convention. Applying [F2] to the periodic array yields $U_{i}^{+} \in G_{γ}$ for each $i$ . By the displayed definition of $G_{γ}$ , $ρ_{i}^{+} > 0$ and $E_{i}^{+} - (m_{i}^{+})^{2} / (2 ρ_{i}^{+}) > 0$ ; multiplication by $γ - 1 > 0$ gives $p (U_{i}^{+}) > 0$ .

Ideal-gas LLF admissibility specialization

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources