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

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Positive first-order compressible model update

Physical scope

One one-dimensional periodic first-order symmetric HLL/LLF step for calorically perfect ideal-gas Euler states. Under the face-speed and CFL hypotheses, every updated density and thermodynamic pressure is positive. No fully discrete entropy inequality, convergence result, exact wave-speed enclosure, or empirical claim follows.

Statement

Use the calorically perfect ideal-gas state relation adopted earlier for numerical states on $v > 0$ , $s \in R$ , $e (v, s) = c_{v} T_{*} {(\frac{v_{*}}{v})}^{γ - 1} \exp (\frac{s - s_{*}}{c_{v}}),$ where $c_{v} > 0$ and $R > 0$ are model parameters in joules per kilogram kelvin, $v_{*} > 0$ and $T_{*} > 0$ are chosen reference data in cubic metres per kilogram and kelvins, $s_{*} \in R$ is chosen reference data in joules per kilogram kelvin, and $γ : = 1 + R / c_{v} > 1$ is dimensionless. Its thermodynamic pressure map, for $ρ > 0$ and $s \in R$ , is $p_{t h} (ρ, s) : = R T_{*} v_{*}^{γ - 1} ρ^{γ} \exp (\frac{s - s_{*}}{c_{v}}) .$ Its admissible state set is $G_{γ} = {(ρ, m, E) \in R^{3} : ρ > 0, E - \frac{m^{2}}{2 ρ} > 0} .$ For every $U = (ρ, m, E) \in G_{γ}$ , define velocity, specific internal energy, specific volume, and specific entropy by $u (U) : = \frac{m}{ρ}, e_{i n t} (U) : = \frac{E}{ρ} - \frac{u (U)^{2}}{2}, v (U) : = \frac{1}{ρ}, s_{t h} (U) : = s_{*} + c_{v} \log (\frac{e_{i n t} (U)}{c_{v} T_{*}} {(\frac{v (U)}{v_{*}})}^{γ - 1}) .$ Here $u (U)$ is velocity in metres per second, $e_{i n t} (U) > 0$ is specific internal energy in joules per kilogram, and $v (U)$ is specific volume in cubic metres per kilogram. The logarithm is natural with positive dimensionless argument, and $s_{t h} (U)$ has units of specific entropy, joules per kilogram kelvin. Define the numerical pressure and sound speed by $p (U) : = (γ - 1) (E - \frac{m^{2}}{2 ρ}), c (U) : = \sqrt{\frac{γ p (U)}{ρ}} .$ Here $p (U)$ is pressure in pascals and $c (U)$ is speed in metres per second. The constitutive relation gives $e (v (U), s_{t h} (U)) = e_{i n t} (U)$ and $p_{t h} (ρ, s_{t h} (U)) = p (U) > 0$ . At fixed specific entropy, differentiation of the stated pressure map gives ${\partial_{ϱ} p_{t h} (ϱ, s) ∣}_{(ϱ, s) = (ρ, s_{t h} (U))} = \frac{γ p_{t h} (ρ, s_{t h} (U))}{ρ} = \frac{γ p (U)}{ρ} = c (U)^{2} .$ Thus $c (U)$ is the positive isentropic sound speed of the adopted model, defined by the fixed-specific-entropy pressure derivative.

On a periodic uniform mesh of $N \geq 2$ cells, each of width $Δ x > 0$ metres, let $U_{i} = (ρ_{i}, m_{i}, E_{i}) \in G_{γ}$ for $i \in Z / N Z$ , where $ρ_{i}$ is mass density in kilograms per cubic metre, $m_{i}$ is momentum density in kilograms per square metre second, and $E_{i}$ is total energy density in joules per cubic metre. Write $u_{i} : = u (U_{i})$ , $e_{i} : = e_{i n t} (U_{i})$ , $v_{i} : = v (U_{i})$ , $s_{i} : = s_{t h} (U_{i})$ , $p_{i} : = p (U_{i})$ , and $c_{i} : = c (U_{i})$ ; then $m_{i} = ρ_{i} u_{i}$ and $E_{i} = ρ_{i} (e_{i} + u_{i}^{2} / 2)$ . The flux is $F_{i} = (m_{i}, m_{i}^{2} / ρ_{i} + p_{i}, u_{i} (E_{i} + p_{i}))$ , whose components have units of kilograms per square metre second, pascals, and watts per square metre. For a time step $Δ t \geq 0$ seconds, put $λ = Δ t / Δ x$ in seconds per metre. At every periodic face choose a supplied speed $a_{i + 1 / 2} > 0$ in metres per second satisfying $a_{i + 1 / 2} \geq \max (∣ u_{i} ∣ + c_{i}, ∣ u_{i + 1} ∣ + c_{i + 1}), 2 λ a_{i + 1 / 2} \leq 1.$ Use the symmetric HLL, or local Lax–Friedrichs, face flux and the shared-face update ${\hat{F}}_{i + 1 / 2} = \frac{1}{2} (F_{i} + F_{i + 1}) - \frac{a_{i + 1 / 2}}{2} (U_{i + 1} - U_{i}), U_{i}^{+} = U_{i} - λ ({\hat{F}}_{i + 1 / 2} - {\hat{F}}_{i - 1 / 2}) .$ Then $U_{i}^{+} \in G_{γ}$ for every cell $i$ , so each updated density and thermodynamic pressure is strictly positive. The speed bounds are algorithmic hypotheses on supplied numerical speeds; this result does not establish that they enclose the waves of an exact nonlinear Riemann solution. It is a conditional deduction about computed states in the adopted model and asserts no fully discrete entropy inequality, convergence, or agreement with physical measurements.

Statement references

Calorically perfect ideal-gas Euler model for numerical states — adopts the calorically perfect ideal-gas Euler branch and identifies the numerical state pressure with thermodynamic pressure.

Euler state, flux, positive set and numerical roles — defines $G_{γ}$ , the Euler flux, shared-face update, supplied HLL speeds, and $λ$ .

Ideal-gas LLF admissibility specialization — proves the exact one-step symmetric-branch positivity implication used here.

Zhang and Shu, §2 — full article treatment of positive Euler states and Lax–Friedrichs positivity conditions.

Ketcheson, LeVeque, and del Razo, Euler approximate-solver chapter — full chapter treatment of the HLL intermediate-state model and supplied-speed convention.

Facts & Assumptions

Given: The adopted calorically perfect ideal-gas Euler model, $γ > 1$ , a periodic uniform mesh with $N \geq 2$ , states $U_{i} \in G_{γ}$ , and the stated face speeds, CFL bounds, symmetric HLL flux, and one-step update.

[A1]

Calorically perfect ideal-gas Euler model for numerical states: the adopted state relation is $e (v, s) = c_{v} T_{*} (v_{*} / v)^{γ - 1} \exp ((s - s_{*}) / c_{v})$ , with $c_{v}, R > 0$ , $γ = 1 + R / c_{v}$ , $T = e_{s}$ , and $p_{t h} (ρ, s) = - (\partial_{v} e) (1 / ρ, s) = (γ - 1) ρ e (1 / ρ, s)$ ; its numerical fields use $m = ρ u$ and $E = ρ (e + u^{2} / 2)$ .

[F1]

Euler state, flux, positive set and numerical roles: $G_{γ}$ is given by $ρ > 0$ and $E - m^{2} / (2 ρ) > 0$ ; $p = (γ - 1) (E - m^{2} / (2 ρ))$ , $u = m / ρ$ , $c = \sqrt{γ p / ρ}$ , and the periodic shared-face update uses $λ = Δ t / Δ x$ . Its supplied speeds do not by themselves certify exact nonlinear wave enclosure.

[F2]

Ideal-gas LLF admissibility specialization: for positive ideal-gas Euler cell states, the symmetric speeds $S_{L} = - a_{i + 1 / 2}$ and $S_{R} = a_{i + 1 / 2}$ , $a_{i + 1 / 2} \geq \max (∣ u_{i} ∣ + c_{i}, ∣ u_{i + 1} ∣ + c_{i + 1})$ , and $2 λ a_{i + 1 / 2} \leq 1$ at every face, one periodic first-order update gives $U_{i}^{+} \in G_{γ}$ and hence positive density and pressure. It asserts no fully discrete entropy inequality.

Proof

1.1givenA1F1algebra

Since $U_{i} \in G_{γ}$ , $ρ_{i} > 0$ and $e_{i} = E_{i} / ρ_{i} - u_{i}^{2} / 2 > 0$ , so the displayed definition of $s_{i}$ is valid and substitution in [A1]'s constitutive relation gives $e (v_{i}, s_{i}) = e_{i}$ . For any $ρ > 0$ and fixed $s$ , that relation yields $p_{t h} (ρ, s) = R T_{*} v_{*}^{γ - 1} ρ^{γ} \exp ((s - s_{*}) / c_{v})$ ; differentiation in $ρ$ gives $\partial_{ρ} p_{t h} (ρ, s) = γ p_{t h} (ρ, s) / ρ$ . Also $E_{i} - m_{i}^{2} / (2 ρ_{i}) = ρ_{i} e_{i}$ , so [F1] gives $p_{i} = (γ - 1) ρ_{i} e_{i} = p_{t h} (ρ_{i}, s_{i}) > 0$ and $c_{i}^{2} = (\partial_{ρ} p_{t h}) (ρ_{i}, s_{i}) > 0$ . Thus the algebraic $c_{i}$ is the positive isentropic sound speed of the adopted model.

2.1givenF1F2step 1.1∎

The stated states, periodic indexing, symmetric HLL flux, facewise speed bounds, and CFL inequalities are exactly the hypotheses of [F2]. Applying [F2] gives $U_{i}^{+} \in G_{γ}$ for every cell. The definition of $G_{γ}$ and $γ > 1$ then give $ρ_{i}^{+} > 0$ and $p_{i}^{+} = (γ - 1) (E_{i}^{+} - (m_{i}^{+})^{2} / (2 ρ_{i}^{+})) > 0$ . This is a model deduction about one numerical step, not an empirical observation or a discrete entropy result.

Positive first-order compressible model update

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources