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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2025

Paper 3

Insert
$$
u_m^n=G^n e^{im\theta}
$$
into the diffusion scheme. Since
$$
e^{i\theta}-2+e^{-i\theta}
=-4\sin^2\left(\frac\theta2\right),
$$
we obtain
$$
G\left(1+2\mu\sin^2\frac\theta2\right)
=1-2\mu\sin^2\frac\theta2.
$$
Thus the <Crank-Nicolson diffusion scheme> has
$$
G(\theta)=
\frac{1-2\mu\sin^2(\theta/2)}
{1+2\mu\sin^2(\theta/2)}.
$$
For $\mu\geq0$, the denominator is positive and
$$
|1-2\mu s|\leq1+2\mu s
\qquad(0\leq s\leq1).
$$
Hence
$$
\boxed{\text{the method is stable for every }\mu>0}.
$$

Solution

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

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

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

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