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A collocated central gradient misses an alternating pressure mode

Statement

Use the periodic index set $J_{N}$ and $h = 1 / N$ from Periodic staggered-grid velocity and pressure spaces, but put pressure and velocity components at the same cell centres and define the collocated centred pressure gradient by $(G_{c} q)_{i, j} = \frac{q_{j + e_{i}} - q_{j - e_{i}}}{2 h} .$ If $N$ is even, then the nonconstant, zero-mean array $q_{j} = (- 1)^{j_{1}}$ satisfies $G_{c} q = 0$ . Thus this collocated central gradient cannot distinguish this pressure from a constant; the staggered gradient defined on the A page is a different operator.

Facts & Assumptions

Given: The periodic index set $J_{N}$ for $N \geq 2$ , and the separate collocated centred-gradient convention displayed in the statement.

[F1]

$J_{N} = (Z / N Z)^{d}$ with all index additions taken modulo $N$ (Periodic staggered-grid velocity and pressure spaces).

[F2]

On an even cycle, $(- 1)^{j_{1} + 1} = (- 1)^{j_{1} - 1} = - (- 1)^{j_{1}}$ , with parity interpreted modulo the even integer $N$ .

Proof

1.1F1givenalgebra

Since $N$ is even, summing $(- 1)^{j_{1}}$ over its $N$ values gives zero; the other coordinates simply repeat this sum. Hence $q$ has zero discrete mean. It is nonconstant because its values at $j_{1} = 0$ and $j_{1} = 1$ differ.

2.1F1F2step 1.1algebra∎

For $i = 1$ , [F2] gives $q_{j + e_{1}} = q_{j - e_{1}}$ , so the first component of $G_{c} q$ is zero. For every $i > 1$ , the array is independent of coordinate $i$ , hence again $q_{j + e_{i}} = q_{j - e_{i}}$ . The defining centred difference therefore gives $G_{c} q = 0$ in every component, proving the nonconstant zero-mean pressure mode is invisible.

Depends on

Periodic staggered-grid velocity and pressure spaces

Used by

Nothing in the library uses this result yet.

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

NPTEL, Lecture 5.4: Methods based on Coupling of Pressure and Velocity (standard reference, not scraped)
J. L. Guermond, P. Minev, and J. Shen, An overview of projection methods for incompressible flows (standard reference, not scraped)