Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 341 / 3 / b

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- Solution b

Solution

Insert the Fourier mode

u_{m}^{n} = G^{n} e^{im θ}

into the scheme. Von Neumann stability analysis gives

G (θ) = \frac{1}{1 + 2 μ - 2 μ c o s θ} = \frac{1}{1 + 4 μ s i n ^{2} ( θ /2 )} .

For every physical Courant number

μ \geq 0

, one has

0 < G (θ) \leq 1

. Thus the method is unconditionally stable: the range is

μ \geq 0

.

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