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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.

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AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Upwind damping versus downwind growth on a periodic grid

Example

On a four-point periodic grid take $ν = 1 / 2$ and the Fourier mode $z = i$ . Its upwind multiplier has modulus $1 / \sqrt{2}$ , while its downwind multiplier has modulus $\sqrt{5 / 2}$ .

Facts & Assumptions

Given: $N = 4$ , $z = \exp (2 π i / 4) = i$ , and $ν = 1 / 2$ .

[F1]

The upwind and downwind factors are $1 - ν + ν z^{- 1}$ and $1 + ν - ν z$ (Amplification symbols for periodic scalar advection stencils).

[F2]

A Fourier multiplier of modulus less than one decays under powers, while a multiplier of modulus greater than one grows under powers (Scheme-specific periodic power stability for scalar advection).

Verification

1.1F1algebra

Since $z^{- 1} = - i$ , the upwind factor is $(1 - i) / 2$ , whose squared modulus is $((1)^{2} + (- 1)^{2}) / 4 = 1 / 2$ .

1.2F1algebra

The downwind factor is $3 / 2 - i / 2$ , whose squared modulus is $(9 + 1) / 4 = 5 / 2$ .

2.1F2step 1.1step 1.2algebra∎

Taking powers multiplies this mode by the corresponding factor to the same power. Thus the upwind component decays and the downwind component grows geometrically.

Depends on

Used by

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Dependency tree · two levels

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Sources

Benjamin Seibold, MIT 18.336 Lecture 15, Finite Difference Methods for the One-Way Wave Equation (standard reference, not scraped)