LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Discrete weak identity for a conservative finite-volume update

Statement

Let $(U_{j}^{m, n})$ be a scheme from Conservative two-point finite-volume schemes for scalar laws, with reconstruction $(U_{m}, Q_{m})$ as in Piecewise-constant finite-volume state and face-flux reconstructions. For a test function $φ \in C_{c}^{\infty} (R \times (- \infty, T))$ , define its space-time cell average $Φ_{j}^{m, n} = \frac{1}{h_{m} τ_{m}} \int_{t_{n}^{m}}^{t_{n + 1}^{m}} \int_{I_{j}^{m}} φ (x, t) d x d t .$ If $N \geq 1$ is such that $φ$ vanishes for $t \geq t_{N - 1}^{m}$ , then $\sum_{j \in Z} h_{m} U_{j}^{m, 0} Φ_{j}^{m, 0} + \sum_{n = 1}^{N - 1} \sum_{j \in Z} h_{m} U_{j}^{m, n} (Φ_{j}^{m, n} - Φ_{j}^{m, n - 1}) + \sum_{n = 0}^{N - 1} \sum_{j \in Z} τ_{m} Q_{j + 1 / 2}^{m, n} (Φ_{j + 1}^{m, n} - Φ_{j}^{m, n}) = 0.$ All sums are finite because $φ$ is compactly supported in space and time.

Facts & Assumptions

Given: The grid, fluxes and update in Conservative two-point finite-volume schemes for scalar laws, the reconstruction convention in Piecewise-constant finite-volume state and face-flux reconstructions, and the test and terminal index in the Statement.

[F1]

The cell update is $h_{m} (U_{j}^{m, n + 1} - U_{j}^{m, n}) + τ_{m} (Q_{j + 1 / 2}^{m, n} - Q_{j - 1 / 2}^{m, n}) = 0$ . (Conservative two-point finite-volume schemes for scalar laws)

[F2]

The test averages $Φ_{j}^{m, n}$ are constant on each cell-time slab, and $Q_{m} = Q_{j + 1 / 2}^{m, n}$ on $I_{j}^{m} \times J_{n}^{m}$ . (Piecewise-constant finite-volume state and face-flux reconstructions)

Proof

1.1F1F2given

Multiply [F1] by $Φ_{j}^{m, n}$ and sum over $j \in Z$ and $n = 0, \dots, N - 1$ . Only finitely many terms are nonzero, so the summed update is an exact finite sum equal to zero.

2.1step 1.1algebra

For each fixed $j$ , telescoping gives $\sum_{n = 0}^{N - 1} (U_{j}^{m, n + 1} - U_{j}^{m, n}) Φ_{j}^{m, n} = - U_{j}^{m, 0} Φ_{j}^{m, 0} + \sum_{n = 1}^{N - 1} U_{j}^{m, n} (Φ_{j}^{m, n - 1} - Φ_{j}^{m, n}),$ since $Φ_{j}^{m, N - 1} = 0$ by the terminal support condition.

3.1step 1.1step 2.1algebra∎

Reindex the spatial flux sum: $\sum_{j} (Q_{j + 1 / 2}^{m, n} - Q_{j - 1 / 2}^{m, n}) Φ_{j}^{m, n} = \sum_{j} Q_{j + 1 / 2}^{m, n} (Φ_{j}^{m, n} - Φ_{j + 1}^{m, n}) .$ Substitute this and step 2.1 into step 1.1, then multiply the zero identity by $- 1$ to obtain exactly the displayed formula.

Depends on

Used by

Conservative finite-volume strong-limit theorem Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

H. De Sterck and P. Ullrich, Introduction to Computational PDEs, Chapter 3, complete chapter, PDF pp. 81–106 (standard reference, not scraped)
Benjamin Seibold, MIT OpenCourseWare 18.336, Lecture 19: Finite Volume Methods, complete six-page note (standard reference, not scraped)