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

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Scheme-specific periodic power stability for scalar advection

Statement

For the periodic schemes of Periodic advection schemes and the finite-interval upwind row with $a > 0$ , upwind powers are contractions in $∥ \cdot ∥_{2, h}$ for $0 \leq ν \leq 1$ and grow for $ν > 1$ . Lax--Friedrichs and Lax--Wendroff powers are contractions for $∣ ν ∣ \leq 1$ ; when $∣ ν ∣ > 1$ , Lax--Wendroff grows for $N \geq 2$ and Lax--Friedrichs grows for $N \geq 3$ (the $N = 2$ Lax--Friedrichs stencil aliases its nonconstant mode). Downwind powers grow for every $ν > 0$ and $N \geq 2$ . Centered powers grow for every $ν \neq 0$ when $N \geq 3$ , while the $N = 2$ centered stencil aliases to the identity. Reflection gives the same assertions for $a < 0$ with $∣ ν ∣$ .

Facts & Assumptions

Given: A finite periodic grid with $N \geq 2$ , fixed $a > 0$ , $h, k > 0$ , and the exact amplification formulas.

[F1]

The $N$ periodic Fourier vectors form an orthogonal basis (Amplification symbols for periodic scalar advection stencils).

[F2]

The five amplification factors and squared moduli are those of Amplification symbols for periodic scalar advection stencils.

[F3]

The five periodic updates and their indexing are defined in Periodic advection schemes and the finite-interval upwind row.

[F4]

For a bounded linear map, its operator norm is the supremum of $∥ T x ∥$ over the unit ball (The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

[F5]

For $z = C + i S$ with $∣ z ∣ = 1$ , one has $C^{2} + S^{2} = 1$ and hence $1 - C \geq 0$ , $0 \leq S^{2} \leq 1$ ; complex conjugation and modulus obey their 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).

Proof

1.1F2F3F5algebra

By [F5], $1 - C_{ℓ} \geq 0$ and $0 \leq S_{ℓ}^{2} \leq 1$ . Thus for $0 \leq ν \leq 1$ , the upwind formula gives $∣ g_{u p} ∣^{2} \leq 1$ ; for $∣ ν ∣ \leq 1$ , the Lax--Friedrichs formula gives $∣ g_{L F} ∣^{2} \leq 1$ , and the Lax--Wendroff formula gives $∣ g_{L W} ∣^{2} \leq 1$ .

1.2F2F3F5algebra

If $ν > 1$ , every nonconstant mode has $C_{ℓ} < 1$ , so $∣ g_{u p} ∣^{2} = 1 + 2 ν (ν - 1) (1 - C_{ℓ}) > 1$ . If $∣ ν ∣ > 1$ and $N \geq 2$ , the Lax--Wendroff mode $ℓ = 1$ has $C_{ℓ} < 1$ and $∣ g_{L W} ∣^{2} = 1 + ν^{2} (ν^{2} - 1) (1 - C_{ℓ})^{2} > 1$ . For Lax--Friedrichs with $∣ ν ∣ > 1$ and $N \geq 3$ , some mode has $S_{ℓ} \neq 0$ , so $∣ g_{L F} ∣^{2} = 1 + (ν^{2} - 1) S_{ℓ}^{2} > 1$ . Downwind has $∣ g_{d o w n} ∣^{2} > 1$ for each nonconstant mode whenever $ν > 0$ . Centered has $∣ g_{c e n t e r e d} ∣^{2} > 1$ on any mode with $S_{ℓ} \neq 0$ , which exists exactly when $N \geq 3$ .

2.1F1F2F3F4F5step 1.1step 1.2∎

Write an arbitrary vector as $v = \sum_{ℓ} c_{ℓ} ψ_{ℓ}$ in the orthonormal Fourier basis [F1]. The $m$ -step update sends it to $\sum_{ℓ} c_{ℓ} g_{ℓ}^{m} ψ_{ℓ}$ , so $∥ T^{m} v ∥_{2, h}^{2} = \sum_{ℓ} ∣ c_{ℓ} ∣^{2} ∣ g_{ℓ} ∣^{2 m} \leq (\max_{ℓ} ∣ g_{ℓ} ∣^{m})^{2} ∥ v ∥_{2, h}^{2}$ . Since the finite set of modes has an index attaining this maximum and its basis vector is nonzero, equality is attained there; [F4] therefore gives $∥ T^{m} ∥ = \max_{ℓ} ∣ g_{ℓ} ∣^{m}$ . The bounds and growing modes in steps 1.1 and 1.2 now give the claimed contraction and instability statements, including the $N = 2$ aliasing cases.

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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)