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

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Shared-face flux telescoping conservation

Statement

Fix $γ > 1$ , an integer $N \geq 2$ , a cell width $Δ x > 0$ , a time step $Δ t \geq 0$ , and periodic cell states $U_{i} = (ρ_{i}, m_{i}, E_{i}) \in G_{γ}$ for $i \in Z / N Z$ . Assign one vector flux ${\hat{F}}_{i + 1 / 2} \in R^{3}$ to each periodic face and use that same value in both neighboring cell updates $U_{i}^{+} = U_{i} - λ ({\hat{F}}_{i + 1 / 2} - {\hat{F}}_{i - 1 / 2}), λ = \frac{Δ t}{Δ x} .$ Define the total conserved vector before and after the step by $M = Δ x \sum_{i \in Z / N Z} U_{i}, M^{+} = Δ x \sum_{i \in Z / N Z} U_{i}^{+} .$ Then $M^{+} = M$ . Thus the total density, momentum, and total energy components are unchanged by this periodic shared-face update. This algebraic conservation claim alone does not assert admissibility, entropy stability, or convergence.

Statement references

Euler state, flux, positive set and numerical roles — fixes the Euler state, the shared-face update, and the meaning of one vector flux per face.

De Sterck and Ullrich, §§3.4.1 and 3.4.3 — full course notes treatment of finite-volume updates and interior-flux cancellation.

Seibold, MIT OCW Lecture 19 — full lecture-note treatment of cell averages and conservative face fluxes.

Facts & Assumptions

Given: $γ > 1$ , $N \geq 2$ , $Δ x > 0$ , $Δ t \geq 0$ , periodic states $U_{i} \in G_{γ}$ , one shared vector flux per face, and the stated update.

[A1]

There is exactly one vector ${\hat{F}}_{i + 1 / 2}$ at each periodic face, and the update uses it with opposite signs in the two adjacent cells.

[F1]

Euler state, flux, positive set and numerical roles: the one-step update is $U_{i}^{+} = U_{i} - λ ({\hat{F}}_{i + 1 / 2} - {\hat{F}}_{i - 1 / 2})$ with periodic indexing.

Proof

1.1givenA1F1algebra

Sum the stated update over the finite periodic index set and multiply by $Δ x$ . By [F1], the difference of totals is $M^{+} - M = - Δ x λ \sum_{i \in Z / N Z} ({\hat{F}}_{i + 1 / 2} - {\hat{F}}_{i - 1 / 2}) .$ [A1] ensures that the same face vector is used in each occurrence.

2.1step 1.1A1algebra∎

Periodic reindexing sends the list of faces $i - 1 / 2$ bijectively to the list $i + 1 / 2$ , so $\sum_{i} {\hat{F}}_{i - 1 / 2} = \sum_{i} {\hat{F}}_{i + 1 / 2}$ componentwise. The sum in step 1.1 is therefore zero and $M^{+} = M$ .

Depends on

Euler state, flux, positive set and numerical roles