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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.
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Euler vacuum detection at and above the pressure-root threshold

Example

Let $γ = 7 / 5$ , $ρ_{L} = ρ_{R} = 1$ , and $p_{L} = p_{R} = 1$ . Then $c_{L} = c_{R} = c = \sqrt{7 / 5}$ , and the vacuum threshold is $u_{R} - u_{L} \geq 2 (c_{L} + c_{R}) / (γ - 1) = 10 c$ .

Facts & Assumptions

Given: The two positive-density, positive-pressure initial states and the vacuum criterion.

[F1]

Admissible states and pressure for one-dimensional ideal-gas Euler: positive density and pressure give admissible states and $c = \sqrt{γ p / ρ}$ .

[F2]

Ideal-gas rarefaction and shock pressure response: $Φ (0 +) = u_{R} - u_{L} - 2 (c_{L} + c_{R}) / (γ - 1)$ .

[F3]

The ideal-gas star-pressure root exists uniquely exactly below vacuum threshold: $Φ (0 +) \geq 0$ means no positive star-pressure root.

[F4]

Finite-budget safeguarded Newton–bisection for Euler star pressure: the procedure returns vacuum when $Φ (0 +) \geq 0$ .

Verification

1.1givenF1F2F3F4algebra

In the equality case take $u_{L} = - 5 c, u_{R} = 5 c$ . Then $u_{R} - u_{L} = 10 c$ , so [F2] gives $Φ (0 +) = 0$ . All thermodynamic states remain admissible by [F1]; [F3] gives no positive pressure root and [F4] returns vacuum.

2.1step 1.1F1F2F3F4algebra∎

In a strict case take $u_{L} = - 6, u_{R} = 6$ . Then $u_{R} - u_{L} = 12 > 10 \sqrt{7 / 5} = 10 c$ , since $144 > 140$ . Thus $Φ (0 +) > 0$ and the same conclusions follow from [F3]–[F4]. These cases show the threshold equality and strict-vacuum branches without using an approximate numerical pressure.

Depends on

Used by

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Dependency tree · two levels

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Sources

Randall J. LeVeque, Numerical Methods for Conservation Laws (standard reference, not scraped)