Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 341 / 4 / c

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- Solution c

Solution

Forward Euler method gives

u_{m}^{n + 1} = u_{m}^{n} + r (u_{m - 1}^{n} - 2 u_{m}^{n} + u_{m + 1}^{n}) + s (u_{m + 1}^{n} - u_{m - 1}^{n}),

where

r = \frac{Δ t}{Δ x ^{2}}, s = \frac{α Δ t}{2Δ x} .

Its amplification factor is

G (ϑ) = 1 - 4 r sin^{2} \frac{ϑ}{2} + i \frac{α Δ t}{Δ x} sin ϑ .

X = sin^{2} (ϑ /2)

, the condition

∣ G ∣^{2} \leq 1

for every

0 \leq X \leq 1

is equivalent to

r \leq \frac{1}{2}, (\frac{α Δ t}{Δ x})^{2} \leq 2 r .

Therefore

0 \leq Δ t \leq min {\frac{Δ x ^{2}}{2}, \frac{2}{α ^{2}}}

with the second bound omitted when

α = 0

.

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