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Crank–Nicolson is stable but does not strongly damp the highest mode

Example

Fix $Δ t > 0$ and consider even periodic one-dimensional grids with $h = L / N \to 0$ . Crank–Nicolson is weighted-L2 stable, but its Nyquist multiplier tends to $- 1$ and its magnitude tends to one, so the highest mode is not strongly damped.

Verification

Given: $κ > 0$ , a fixed $Δ t > 0$ , and an even periodic grid with spacing $h = L / N$ .

[F1] Fully discrete θ heat finite-difference scheme: Crank–Nicolson is the θ update with $θ = 1 / 2$ and zero source.

[F2] Periodic heat-stencil eigenvalues and θ amplification factor: the Nyquist eigenvalue is $λ = 4 κ / h^{2}$ , and its multiplier is $g_{θ} = (1 - (1 - θ) Δ t λ) / (1 + θ Δ t λ)$ .

1.1F1F2givenalgebra

Substituting $θ = 1 / 2$ and $λ = 4 κ / h^{2}$ into [F2] gives $g = (1 - 2 κ Δ t / h^{2}) / (1 + 2 κ Δ t / h^{2})$ .

2.1step 1.1algebra

As $h \to 0$ with $κ, Δ t > 0$ fixed, the numerator and denominator have leading terms $- 2 κ Δ t / h^{2}$ and $2 κ Δ t / h^{2}$ , so $g \to - 1$ and $∣ g ∣ \to 1$ .

3.1F2step 1.1algebra∎

For any $z = Δ t λ \geq 0$ , $∣ 1 - z / 2 ∣ \leq 1 + z / 2$ , so the factor in [F2] has magnitude at most one. Thus the mode is stable at each mesh while its high-frequency damping vanishes in the limit.

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