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Centered advection is consistent and unstable under the CFL bound

Example

Fix $a, P, T > 0$ . For $q \geq 1$ , use periodic grids $N_{q} = 4 q$ , $h_{q} = P / N_{q}$ , and $k_{q} = h_{q} / (2 a)$ , so $ν = a k_{q} / h_{q} = 1 / 2$ . Every fixed smooth $P$ -periodic solution in the hypotheses of Smooth consistency orders for the five periodic advection stencils has centered residual $O (h_{q})$ . Nevertheless the mode $ℓ = q$ has $z_{ℓ} = i$ and multiplier $1 - i / 2$ , and the centered family has no uniform finite-time power bound, as proved in The nearest-neighbor CFL condition is necessary but not sufficient.

Facts & Assumptions

Given: Fixed $a, P, T > 0$ and periodic grids $N_{q} = 4 q$ , $h_{q} = P / (4 q)$ , $k_{q} = h_{q} / (2 a)$ , with Fourier index $ℓ = q$ .

[F1]

The centered residual is $O (h)$ under the two-sided mesh-ratio and $C^{6}$ hypotheses of Smooth consistency orders for the five periodic advection stencils.

[F2]

The centered amplification factor is $1 - i ν S$ (Amplification symbols for periodic scalar advection stencils).

[F3]

For this centered periodic family with $a k_{q} / h_{q} = 1 / 2$ , the finite-time power norms are unbounded as $q \to \infty$ (The nearest-neighbor CFL condition is necessary but not sufficient).

Verification

1.1givenF2algebra

Since $z_{ℓ} = i$ , $S_{ℓ} = 1$ , and the centered factor is $1 - i / 2$ . Its squared modulus is $1 + 1 / 4 = 5 / 4 > 1$ , while $k_{q} / h_{q} = 1 / (2 a)$ is constant and $h_{q}, k_{q} \to 0$ .

2.1F1F3step 1.1∎

By [F1], each fixed smooth solution satisfying the theorem's regularity and periodicity hypotheses has residual $O (h_{q})$ . By [F3], for every fixed $T > 0$ the centered update powers are not uniformly bounded over time levels $r k_{q} \leq T$ . Thus this family is consistent on fixed smooth data but fails finite-time power stability because of mesh-dependent high-frequency modes.

Depends on

Used by

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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)