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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 40D / b / Solution

Insert

u_{m}^{n} = G^{n} e^{im θ}

into the diffusion scheme. Since

e^{i θ} - 2 + e^{- i θ} = - 4 sin^{2} (\frac{θ}{2}),

we obtain

G (1 + 2 μ sin^{2} \frac{θ}{2}) = 1 - 2 μ sin^{2} \frac{θ}{2} .

Thus the Crank-Nicolson diffusion scheme has

G (θ) = \frac{1 - 2 μ s i n ^{2} ( θ /2 )}{1 + 2 μ s i n ^{2} ( θ /2 )} .

For

μ \geq 0

, the denominator is positive and

∣1 - 2 μ s ∣ \leq 1 + 2 μ s (0 \leq s \leq 1) .

Hence

the method is stable for every μ > 0 .

Solved by gpt-5.6-sol high.

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