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Amplification symbols for periodic scalar advection stencils

Statement

On an $N$ -point periodic grid let $z_{ℓ} = \exp (2 π i ℓ / N)$ , $C_{ℓ} = Re z_{ℓ}$ , $S_{ℓ} = Im z_{ℓ}$ , and $ν = a k / h$ with $a > 0$ . The Fourier vector $j \mapsto z_{ℓ}^{j}$ is multiplied in one step by $g_{u p} = 1 - ν + ν z_{ℓ}^{- 1}, g_{d o w n} = 1 + ν - ν z_{ℓ}, g_{c e n t e r e d} = 1 - i ν S_{ℓ}, g_{L F} = C_{ℓ} - i ν S_{ℓ}, g_{L W} = 1 - ν^{2} (1 - C_{ℓ}) - i ν S_{ℓ} .$ Their squared moduli, in the same order, are $1 - 2 ν (1 - ν) (1 - C_{ℓ}), 1 + 2 ν (1 + ν) (1 - C_{ℓ}), 1 + ν^{2} S_{ℓ}^{2}, 1 - (1 - ν^{2}) S_{ℓ}^{2}, 1 - ν^{2} (1 - ν^{2}) (1 - C_{ℓ})^{2} .$ The $N$ Fourier vectors form an orthogonal basis under the periodic inner product of Uniform periodic, infinite, and Dirichlet grid conventions, and each has squared norm $N h$ .

Facts & Assumptions

Given: $N \geq 2$ , $a, h, k > 0$ , and the periodic stencils of Periodic advection schemes and the finite-interval upwind row.

[F1]

The five one-step formulas and periodic wrap are defined in Periodic advection schemes and the finite-interval upwind row.

[F2]

The finite root-of-unity sum $\sum_{j = 0}^{N - 1} \exp (2 π i (ℓ - r) j / N)$ equals $N$ if $ℓ \equiv r (m o d N)$ and $0$ otherwise (Orthogonality of the characters $x \mapsto e^{2 π i k x / N}$ on $Z / N Z$ ).

[F3]

The displayed periodic Fourier modes form a basis of the complexified grid space (Periodic heat-stencil eigenvalues and θ amplification factor).

[F4]

For complex numbers, $\overset{ˉ}{z} = Re z - i Im z$ and $z \overset{ˉ}{z} = ∣ z ∣^{2}$ ; conjugation is multiplicative and modulus obeys the field laws (Real and imaginary parts, complex conjugation, and modulus, Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F5]

The periodic grid inner product and norm are $⟨ p, q ⟩_{2, h} = h \sum_{j} p_{j} \overline{q_{j}}$ and $∥ p ∥_{2, h}^{2} = h \sum_{j} ∣ p_{j} ∣^{2}$ (Uniform periodic, infinite, and Dirichlet grid conventions).

Proof

1.1F1F4algebra

Write $ϕ_{j} = z_{ℓ}^{j}$ . Then $ϕ_{j + 1} = z_{ℓ} ϕ_{j}$ , $ϕ_{j - 1} = z_{ℓ}^{- 1} ϕ_{j}$ , and $z_{ℓ}^{- 1} = {\overset{ˉ}{z}}_{ℓ}$ . Substitution into each update in [F1] gives, respectively, $1 - ν + ν z_{ℓ}^{- 1}, 1 + ν - ν z_{ℓ}, 1 - \frac{ν}{2} (z_{ℓ} - z_{ℓ}^{- 1}), \frac{z_{ℓ} + z_{ℓ}^{- 1}}{2} - \frac{ν}{2} (z_{ℓ} - z_{ℓ}^{- 1}), 1 - \frac{ν}{2} (z_{ℓ} - z_{ℓ}^{- 1}) + \frac{ν^{2}}{2} (z_{ℓ} + z_{ℓ}^{- 1} - 2) .$ Since $z_{ℓ}^{- 1} = C_{ℓ} - i S_{ℓ}$ , these are the displayed factors.

2.1F4step 1.1algebra

The identity $∣ z_{ℓ} ∣ = 1$ gives $C_{ℓ}^{2} + S_{ℓ}^{2} = 1$ . Multiplying each factor in step 1.1 by its conjugate and simplifying with this identity gives the five displayed squared-modulus formulas.

3.1F2F3F5step 1.1∎

By [F2] the distinct Fourier modes are mutually orthogonal; by [F3] they form a basis. By [F5], each has squared norm $N h$ , so the normalized modes are an orthonormal basis. Every update is diagonal in this basis with the factors from step 1.1, which therefore describe the full periodic updates.

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Sources

Endre Süli, B6.1 Numerical Solution of Partial Differential Equations (standard reference, not scraped)
Benjamin Seibold, MIT 18.336 Lecture 15, Finite Difference Methods for the One-Way Wave Equation (standard reference, not scraped)