TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)

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Well-definedness of the periodic pressure-correction solves

Statement

On the periodic staggered grid of Periodic staggered-grid velocity and pressure spaces, every admissible backward- Euler step of Backward-Euler incremental pressure correction on a periodic staggered grid and every admissible BDF2 step of BDF2 standard and rotational incremental pressure-correction variants has a unique predictor, a unique mean-zero pressure increment, and a unique updated pair. Both algorithms preserve the zero-mean pressure gauge and exact discrete incompressibility.

Facts & Assumptions

Given: $d \geq 1$ , $N \geq 2$ , $h = 1 / N$ , $ν > 0$ , $k > 0$ , admissible mean-zero data, and either stated pressure-correction algorithm.

[F1]

In the backward-Euler algorithm the predictor matrix is $I + k ν A_{h}$ ; in BDF2 it is $(3 / (2 k)) I + ν A_{h}$ . The right-hand sides and pressure updates are exactly those in the two algorithm definitions (Backward-Euler incremental pressure correction on a periodic staggered grid, BDF2 standard and rotational incremental pressure-correction variants).

[F2]

The periodic componentwise Laplacian and the seminorm satisfy $(A_{h} v, v)_{M_{h}} = h^{d - 2} \sum_{i, ℓ, j} ∣ v_{i, j + e_{ℓ}} - v_{i, j} ∣^{2} \geq 0$ by finite periodic reindexing (Periodic staggered-grid velocity and pressure spaces).

[F3]

$D_{h} v$ has zero mean and $Δ_{h} : Q_{h} \to Q_{h}$ is bijective (Periodic pressure Laplacian gauge, solvability, and uniform Poincare bound).

[F4]

A linear endomorphism of a finite-dimensional space with zero kernel is invertible by rank-nullity (Rank-nullity: $\dim_{F} V = nullity T + rank T$ ); $V_{h}$ is finite-dimensional because it is a finite face-array space.

Proof

1.1givenF1F2F4algebra

For any nonzero $v \in V_{h}$ , the componentwise Laplacian definition and finite periodic reindexing give $(A_{h} v, v)_{M_{h}} = h^{d - 2} \sum_{i, ℓ, j} ∣ v_{i, j + e_{ℓ}} - v_{i, j} ∣^{2} \geq 0.$ Therefore $((I + k ν A_{h}) v, v)_{M_{h}} = ∥ v ∥_{M_{h}}^{2} + k ν (A_{h} v, v)_{M_{h}} > 0$ . The predictor matrix has zero kernel and is invertible by [F4], so the backward-Euler predictor has a unique solution.

2.1givenF1F2F4step 1.1algebra

Similarly, $(((3 / (2 k)) I + ν A_{h}) v, v)_{M_{h}} = (3 / (2 k)) ∥ v ∥_{M_{h}}^{2} + ν (A_{h} v, v)_{M_{h}} > 0$ . The BDF2 predictor matrix has zero kernel and is invertible by [F4], so its predictor is unique for either pressure-update branch.

3.1F1F3step 1.1step 2.1

For either predictor, summing $D_{h} \tilde{u}$ over all cells cancels the two occurrences of each periodic face value. Hence the right-hand side of the corresponding pressure-increment equation has zero mean. By [F3], there is exactly one mean-zero $ϕ^{n + 1}$ solving that equation.

4.1F1F2F3step 3.1algebra∎

The velocity and pressure updates are explicit after the unique predictor and pressure increment. Since $D_{h} G_{h} = Δ_{h}$ , the corrected divergence is $D_{h} \tilde{u} - k Δ_{h} ϕ = 0$ in backward Euler and $D_{h} \tilde{u} - (2 k / 3) Δ_{h} ϕ = 0$ in BDF2. The BDF2 rotational pressure correction subtracts $ν D_{h} \tilde{u}$ , which has zero mean by compatibility; thus both pressure updates stay in $Q_{h}$ .

Depends on

Used by

Cited to discharge well-definedness by Backward-Euler incremental pressure correction on a periodic staggered grid and BDF2 standard and rotational incremental pressure-correction variants.

Dependency tree · two levels

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Sources

J. L. Guermond, P. Minev, and J. Shen, An overview of projection methods for incompressible flows (standard reference, not scraped)