Past exam of the numerical solutions of partial differential equations course of the University of Oxford / 2023 / 3 / b

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- Solution b

Solution

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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