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

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Exact space-time weak identity for the hydrostatic update

Statement

Let $d, τ > 0$ , $I_{i} = [i d, (i + 1) d)$ , $t_{n} = n τ$ , and $J_{n} = [t_{n}, t_{n + 1})$ . Suppose vector states and shared face fluxes satisfy $d (U_{i}^{n + 1} - U_{i}^{n}) + τ ({\hat{F}}_{i + 1 / 2}^{n} - {\hat{F}}_{i - 1 / 2}^{n}) = d τ S_{i}^{n}$ as in Hydrostatic reconstruction with a dry-safe HLL flux and one-sided pressure corrections. For $φ \in C_{c}^{\infty} (R \times (- \infty, T); R^{2})$ define the space-time cell average $Φ_{i}^{n} = \frac{1}{d τ} \int_{J_{n}} \int_{I_{i}} φ (x, t) d x d t .$ Choose $N$ such that $φ$ vanishes for $t \geq t_{N - 1}$ . Then $\sum_{i} d U_{i}^{0} \cdot Φ_{i}^{0} + \sum_{n = 1}^{N - 1} \sum_{i} d U_{i}^{n} \cdot (Φ_{i}^{n} - Φ_{i}^{n - 1}) + \sum_{n = 0}^{N - 1} \sum_{i} τ {\hat{F}}_{i + 1 / 2}^{n} \cdot (Φ_{i + 1}^{n} - Φ_{i}^{n}) + \sum_{n = 0}^{N - 1} \sum_{i} d τ S_{i}^{n} \cdot Φ_{i}^{n} = 0.$ Every sum is finite because the test is compactly supported.

Facts & Assumptions

Given: The vector finite-volume recurrence with shared face fluxes and cell source, the half-open space-time grid, and the cell-average test values above.

[F1]

Hydrostatic reconstruction with a dry-safe HLL flux and one-sided pressure corrections: the update is equivalent to $d (U_{i}^{n + 1} - U_{i}^{n}) + τ ({\hat{F}}_{i + 1 / 2}^{n} - {\hat{F}}_{i - 1 / 2}^{n}) = d τ S_{i}^{n}$ with one shared HLL flux at every face.

Proof

1.1F1given

Multiply the recurrence in [F1] by $Φ_{i}^{n}$ using the Euclidean dot product and sum over cells $i$ and time levels $0 \leq n < N$ . Compact support leaves only finitely many nonzero terms, and the resulting exact sum is zero.

2.1step 1.1algebra

For each cell, summing in time and using $Φ_{i}^{N - 1} = 0$ gives $\sum_{n = 0}^{N - 1} (U_{i}^{n + 1} - U_{i}^{n}) \cdot Φ_{i}^{n} = - U_{i}^{0} \cdot Φ_{i}^{0} + \sum_{n = 1}^{N - 1} U_{i}^{n} \cdot (Φ_{i}^{n - 1} - Φ_{i}^{n})$ . This is finite summation by parts; no limit or infinite rearrangement is used.

3.1F1step 1.1step 2.1algebra∎

Reindexing the spatial face sum gives $\sum_{i} ({\hat{F}}_{i + 1 / 2}^{n} - {\hat{F}}_{i - 1 / 2}^{n}) \cdot Φ_{i}^{n} = \sum_{i} {\hat{F}}_{i + 1 / 2}^{n} \cdot (Φ_{i}^{n} - Φ_{i + 1}^{n})$ . Insert this and step 2.1 into step 1.1, retain the source sum with its minus sign from [F1], and multiply the zero identity by $- 1$ . The displayed identity follows.

Depends on

Used by

A strong limit of hydrostatic HLL updates solves the smooth-bed balance law weakly Theorem

Dependency tree · one level

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Sources

H. De Sterck and P. Ullrich, Introduction to Computational PDEs, Chapter 3 (standard reference, not scraped)
Benjamin Seibold, MIT OpenCourseWare 18.336, Lecture 19: Finite Volume Methods (standard reference, not scraped)