LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck pass

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Hydrostatic pressure corrections converge to the smooth-bed source

Statement

Let $b \in C^{2} (R)$ , let the uniform cell width be $d > 0$ , let $x_{i}$ be the center of cell $i$ , and let $0 \leq h_{i}^{n} \leq H$ on a fixed compact space-time cylinder. Set $b_{i} = b (x_{i})$ , $δ_{i, +} = (b (x_{i} + d) - b (x_{i}))_{+}$ , and $δ_{i, -} = (b (x_{i} - d) - b (x_{i}))_{+}$ . For the source term $S_{i}^{n}$ in Hydrostatic reconstruction with a dry-safe HLL flux and one-sided pressure corrections, $S_{i}^{n} = (0, \frac{g}{2 d} ((h_{i}^{n} - δ_{i, +})_{+}^{2} - (h_{i}^{n} - δ_{i, -})_{+}^{2})),$ there is a constant $C$ depending only on the compact interval, $b$ , $g$ , and $H$ such that $∣ S_{i}^{(2)} + g h_{i} b^{'} (x_{i}) ∣ \leq C d .$ For time slabs $J_{n} = [n τ, (n + 1) τ)$ , define the piecewise-constant source reconstruction $S_{d} (x, t) = S_{i}^{n}$ on $[x_{i} - d / 2, x_{i} + d / 2) \times J_{n}$ . Consequently, as $d \to 0$ , if the corresponding piecewise-constant depth reconstructions $h_{d}$ converge strongly in $L_{l o c}^{1} (R \times (0, T))$ to $h$ and are locally uniformly bounded, then $S_{d} \to (0, - g h b^{'})$ strongly in $L_{l o c}^{1}$ .

Facts & Assumptions

Given: $g > 0$ , $b \in C^{2}$ , the hydrostatic source formula, a uniform mesh of width $d$ , and bounded cell depths. Derive a uniform first-order source consistency estimate.

[F1]

Hydrostatic reconstruction with a dry-safe HLL flux and one-sided pressure corrections: the source momentum is $g ((h_{i, +}^{*})^{2} - (h_{i, -}^{*})^{2}) / (2 d)$ , and hydrostatic reconstructed depths are positive parts after subtracting the increase to the maximum neighboring bed.

[F2]

One-dimensional shallow-water balance law with a dry-state convention: for a smooth bed the momentum source in the balance law is $- g h b^{'}$ .

Proof

1.1givenconstructalgebra

Fix a compact spatial interval $I$ containing the cell centers under consideration and let $I^{+}$ be its closed unit enlargement. For $0 < d \leq 1$ , Taylor's formula at $x_{i} \in I$ gives $b (x_{i} + d) - b (x_{i}) = a + r_{+}$ and $b (x_{i} - d) - b (x_{i}) = - a + r_{-}$ , where $a = b^{'} (x_{i}) d$ and $∣ r_{\pm} ∣ \leq \frac{1}{2} \sup_{I^{+}} ∣ b^{''} ∣ d^{2}$ . Since $s \mapsto s_{+}$ is $1$ -Lipschitz and $a_{+} - (- a)_{+} = a$ , this yields $∣ δ_{i, +} - δ_{i, -} - b^{'} (x_{i}) d ∣ \leq C_{0} d^{2}$ . The same local derivative bounds give $0 \leq δ_{i, \pm} \leq C_{1} d$ , with constants depending only on $I$ and $b$ .

2.1F1step 1.1algebra

For $h \geq 0$ and $δ \geq 0$ , write $(h - δ)_{+}^{2} = h^{2} - 2 h δ + R (h, δ)$ . If $δ \leq h$ , then $R = δ^{2}$ ; if $δ > h$ , then $R = 2 h δ - h^{2}$ , which lies between $0$ and $δ^{2}$ because $δ^{2} - (2 h δ - h^{2}) = (δ - h)^{2}$ . For $0 < d \leq 1$ , steps 1.1 and [F1] therefore give $∣ (S_{i}^{n})_{2} + g h_{i}^{n} b^{'} (x_{i}) ∣ \leq g H C_{0} d + \frac{g}{2 d} (δ_{i, +}^{2} + δ_{i, -}^{2}) \leq C_{2} d .$ For $d \geq 1$ , both squared positive parts in [F1] lie in $[0, H^{2}]$ , so $∣ (S_{i}^{n})_{2} ∣ \leq g H^{2} / (2 d)$ . Since $∣ b^{'} (x_{i}) ∣ \leq \sup_{I} ∣ b^{'} ∣$ , it follows that $∣ (S_{i}^{n})_{2} + g h_{i}^{n} b^{'} (x_{i}) ∣ \leq \frac{g H^{2}}{2 d} + g H \sup_{I} ∣ b^{'} ∣ \leq C_{3} d .$ Taking $C = \max (C_{2}, C_{3})$ proves the stated estimate for every $d > 0$ .

3.1F2step 2.1algebra∎

Now let $d \to 0$ and fix a compact space-time rectangle. For all sufficiently small $d$ , the centers of cells meeting this rectangle lie in a fixed compact interval, so the estimate in step 2.1 and $∣ b^{'} (x) - b^{'} (x_{i}) ∣ \leq \frac{1}{2} d \sup ∣ b^{''} ∣$ on a fixed enlargement show that $S_{d} - (0, - g h_{d} b^{'} (x)) \to 0$ uniformly there. Since $b^{'}$ is bounded on compact intervals and $h_{d} \to h$ strongly in local $L^{1}$ , $(0, - g h_{d} b^{'} (x)) \to (0, - g h b^{'} (x))$ strongly in local $L^{1}$ , proving the asserted source convergence.

Depends on

Used by

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

Dependency tree · one level

2 results within one dependency step 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

E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein, and B. Perthame, A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows (standard reference, not scraped)
F. Bouchut, III. Shallow water problem – Hydrostatic reconstruction scheme (standard reference, not scraped)