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Modified velocity-pressure energy identity for periodic pressure correction

Statement

For one step of Backward-Euler incremental pressure correction on a periodic staggered grid on Periodic staggered-grid velocity and pressure spaces, with $u^{n}$ divergence- free and $p^{n}, p^{n + 1} \in Q_{h}$ , the following identity holds exactly: $\begin{aligned} ∥ u^{n + 1} ∥_{M_{h}}^{2} - ∥ u^{n} ∥_{M_{h}}^{2} & + ∥ {\tilde{u}}^{n + 1} - u^{n} ∥_{M_{h}}^{2} + 2 ν k ∥ G_{h} {\tilde{u}}^{n + 1} ∥_{M_{h}}^{2} \\ + k^{2} (∥ G_{h} p^{n + 1} ∥_{M_{h}}^{2} - ∥ G_{h} p^{n} ∥_{M_{h}}^{2}) = 2 k (f^{n + 1}, {\tilde{u}}^{n + 1})_{M_{h}} . \end{aligned}$ This is a modified velocity-pressure energy identity for the stated periodic discretization; it makes no bounded-wall claim.

Facts & Assumptions

Given: The backward-Euler predictor, pressure increment and correction on the periodic staggered grid, with $ν > 0$ , $k > 0$ , $D_{h} u^{n} = 0$ , and $p^{n}, p^{n + 1} \in Q_{h}$ .

[F1]

The predictor equation, pressure equation, and pressure update are $({\tilde{u}}^{n + 1} - u^{n}) / k + ν A_{h} {\tilde{u}}^{n + 1} + G_{h} p^{n} = f^{n + 1}$ , $Δ_{h} ϕ^{n + 1} = k^{- 1} D_{h} {\tilde{u}}^{n + 1}$ , $u^{n + 1} = {\tilde{u}}^{n + 1} - k G_{h} ϕ^{n + 1}$ , and $p^{n + 1} = p^{n} + ϕ^{n + 1}$ (Backward-Euler incremental pressure correction on a periodic staggered grid).

[F2]

The discrete adjoint relation is $(G_{h} q, v)_{M_{h}} = - (q, D_{h} v)_{Q_{h}}$ , and the pressure Laplacian satisfies $(q, Δ_{h} r)_{Q_{h}} = - (G_{h} q, G_{h} r)_{M_{h}}$ (Mass-orthogonal periodic discrete Helmholtz projection and exact incompressibility).

[F3]

$P_{h}$ is a mass-orthogonal projector and $u^{n + 1} = P_{h} {\tilde{u}}^{n + 1}$ (Mass-orthogonal periodic discrete Helmholtz projection and exact incompressibility).

[F4]

$(A_{h} v, v)_{M_{h}} = ∥ G_{h} v ∥_{M_{h}}^{2}$ by the periodic componentwise Laplacian definition (Periodic staggered-grid velocity and pressure spaces).

Proof

1.1givenF1F4algebra

By finite periodic reindexing in each component and coordinate, $(A_{h} v, v)_{M_{h}} = h^{d - 2} \sum_{i, ℓ, j} ∣ v_{i, j + e_{ℓ}} - v_{i, j} ∣^{2} = ∥ G_{h} v ∥_{M_{h}}^{2}$ . Take the $M_{h}$ inner product of the predictor equation [F1] with $2 k {\tilde{u}}^{n + 1}$ . The identity $2 ({\tilde{u}}^{n + 1} - u^{n}, {\tilde{u}}^{n + 1})_{M_{h}} = ∥ {\tilde{u}}^{n + 1} ∥_{M_{h}}^{2} - ∥ u^{n} ∥_{M_{h}}^{2} + ∥ {\tilde{u}}^{n + 1} - u^{n} ∥_{M_{h}}^{2}$ and [F4] give $∥ {\tilde{u}}^{n + 1} ∥_{M_{h}}^{2} - ∥ u^{n} ∥_{M_{h}}^{2} + ∥ {\tilde{u}}^{n + 1} - u^{n} ∥_{M_{h}}^{2} + 2 ν k ∥ G_{h} {\tilde{u}}^{n + 1} ∥_{M_{h}}^{2} + 2 k (G_{h} p^{n}, {\tilde{u}}^{n + 1})_{M_{h}} = 2 k (f^{n + 1}, {\tilde{u}}^{n + 1})_{M_{h}} .$

2.1F1F2algebrastep 1.1

By [F2] and the pressure equation in [F1], $2 k (G_{h} p^{n}, {\tilde{u}}^{n + 1})_{M_{h}} = - 2 k (p^{n}, D_{h} {\tilde{u}}^{n + 1})_{Q_{h}} = - 2 k^{2} (p^{n}, Δ_{h} ϕ^{n + 1})_{Q_{h}} = 2 k^{2} (G_{h} p^{n}, G_{h} ϕ^{n + 1})_{M_{h}} .$ The last equality follows from the second identity in [F2].

3.1F1F3step 2.1algebra

Since $P_{h}$ is orthogonal and $u^{n + 1} = P_{h} {\tilde{u}}^{n + 1}$ by [F3], the projection residual ${\tilde{u}}^{n + 1} - u^{n + 1} = k G_{h} ϕ^{n + 1}$ is orthogonal to $u^{n + 1}$ . Pythagoras gives $∥ {\tilde{u}}^{n + 1} ∥_{M_{h}}^{2} = ∥ u^{n + 1} ∥_{M_{h}}^{2} + k^{2} ∥ G_{h} ϕ^{n + 1} ∥_{M_{h}}^{2} .$ The pressure update $p^{n + 1} = p^{n} + ϕ^{n + 1}$ then gives $k^{2} (∥ G_{h} p^{n + 1} ∥_{M_{h}}^{2} - ∥ G_{h} p^{n} ∥_{M_{h}}^{2}) = k^{2} ∥ G_{h} ϕ^{n + 1} ∥_{M_{h}}^{2} + 2 k^{2} (G_{h} p^{n}, G_{h} ϕ^{n + 1})_{M_{h}} .$

4.1step 1.1step 3.1algebra∎

Substitute the two identities from step 3.1 into step 1.1, replacing $∥ {\tilde{u}}^{n + 1} ∥_{M_{h}}^{2}$ and the pressure cross term. The result is exactly the displayed modified energy identity.

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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)
NPTEL, Lecture 5.4: Methods based on Coupling of Pressure and Velocity (standard reference, not scraped)