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 Incoming links: Semidiscrete Fourier transform

Past exam of the numerical solutions of partial differential equations course of the University of Oxford / 2023 / 3 / b / Solution 2026-10-25

For a grid sequence define its semidiscrete Fourier transform in the dimensionless frequency variable

θ \in [- π, π]

by

W (θ) = \sum_{j \in Z} W_{j} e^{- ij θ}, W_{j} = \frac{1}{2 π} \int_{- π}^{π} W (θ) e^{ij θ} d θ .

For general square-summable data the transform is interpreted as an limit in the L2 norm, rather than an absolutely convergent pointwise sum. The Parseval identity in this normalization is

∥ W ∥_{ℓ_{2}}^{2} = Δ x \sum_{j \in Z} ∣ W_{j} ∣^{2} = \frac{Δ x}{2 π} \int_{- π}^{π} ∣ W (θ) ∣^{2} d θ .

The Fourier symbols of

D_{x}^{0}

and

D_{x}^{+} D_{x}^{-}

are

i sin θ /Δ x

and

- 4 sin^{2} (θ /2) / (Δ x)^{2}

. Transforming the implicit equation gives

(1 + 4 a μ sin^{2} (θ /2) + i ν sin θ) U^{m + 1} = U^{m} .

Its amplification factor is consequently

G_{I} (θ) = \frac{1}{1 + 4 a μ s i n ^{2} ( θ /2 ) + i ν s i n θ} .

The denominator has real part at least one because

a \geq 0

. It never vanishes, so multiplication by

G_{I}

defines a unique next value in the l2 sequence space, and

∣ G_{I} (θ) ∣^{2} = \frac{1}{( 1 + 4 a μ s i n ^{2} ( θ /2 ) ) ^{2} + ν ^{2} s i n ^{2} θ} \leq 1.

The Parseval identity now gives

∥ U^{m + 1} ∥_{ℓ_{2}} \leq ∥ U^{m} ∥_{ℓ_{2}} \leq ∥ U^{0} ∥_{ℓ_{2}} for every m \geq 0, Δ x, Δ t > 0.

This is unconditional practical stability of a finite difference scheme: in the course's terminology the numerical evolution does not increase this norm. Linearity gives the same estimate for perturbations of the initial data. This contraction convention is stronger than a finite-time mesh-uniform bound allowing physical exponential growth.^[ref]

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Past exam of the numerical solutions of partial differential equations course of the University of Oxford / 2023 / 3 / d / Solution 2026-10-25

Using the semidiscrete Fourier transform from (b), the explicit scheme has amplification factor

G_{E} (θ) = 1 - 4 a μ sin^{2} (θ /2) - i ν sin θ .

r = a μ

and

s = sin^{2} (θ /2) \in [0, 1]

, so

sin^{2} θ = 4 s (1 - s)

. Direct expansion gives

∣ G_{E} ∣^{2} = (1 - 4 rs)^{2} + 4 ν^{2} s (1 - s), ∣ G_{E} ∣^{2} - 1 = 4 s [(ν^{2} - 2 r) + (4 r^{2} - ν^{2}) s] .

The expression in square brackets is an affine function of

s

. At the two endpoints its values are

ν^{2} - 2 r \leq 0, 4 r^{2} - 2 r = 2 r (2 r - 1) \leq 0,

using precisely

ν^{2} \leq 2 r \leq 1

. Every intermediate value is a convex combination of these endpoints, so

∣ G_{E} ∣^{2} \leq 1

for every frequency. The Parseval identity proves

∥ U^{m} ∥_{ℓ_{2}} \leq ∥ U^{0} ∥_{ℓ_{2}} (m \geq 0), ν^{2} \leq 2 a μ \leq 1.

Thus the scheme is practically stable under the given condition. For

a > 0

it is equivalent to

Δ t \leq 2 a

and

Δ t \leq (Δ x)^{2} / (2 a)

. This is the Forward Euler stability for centered advection-diffusion calculation with the diffusion coefficient retained. The proof does not require all three stencil weights to be nonnegative.

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