Consider discretisation of the diffusion equation

$$\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}, \quad 0 \leqslant t \leqslant 1$$

by the Crank-Nicholson method:

$$u_{m}^{n+1}-\frac{1}{2} \mu\left(u_{m-1}^{n+1}-2 u_{m}^{n+1}+u_{m+1}^{n+1}\right)=u_{m}^{n}+\frac{1}{2} \mu\left(u_{m-1}^{n}-2 u_{m}^{n}+u_{m+1}^{n}\right), \quad n=0, \ldots, N,$$

where $\mu=\frac{k}{h^{2}}$ is the Courant number, $h$ is the step size in the space discretisation, $k=\frac{1}{N+1}$ is the step size in the time discretisation, and $u_{m}^{n} \approx u(m h, n k)$, where $u(x, t)$ is the solution of $(*)$. The initial condition $u(x, 0)=u_{0}(x)$ is given.

(a) Consider the Cauchy problem for $(*)$ on the whole line, $x \in \mathbb{R}$ (thus $m \in \mathbb{Z}$ ), and derive the formula for the amplification factor of the Crank-Nicholson method ( $\dagger$ ). Use the amplification factor to show that the Crank-Nicholson method is stable for the Cauchy problem for all $\mu>0$.

[You may quote basic properties of the Fourier transform mentioned in lectures, but not the theorem on sufficient and necessary conditions on the amplification factor to have stability.]

(b) Consider $(*)$ on the interval $0 \leqslant x \leqslant 1$ (thus $m=1, \ldots, M$ and $h=\frac{1}{M+1}$ ) with Dirichlet boundary conditions $u(0, t)=\phi_{0}(t)$ and $u(1, t)=\phi_{1}(t)$, for some sufficiently smooth functions $\phi_{0}$ and $\phi_{1}$. Show directly (without using the Lax equivalence theorem) that, given sufficient smoothness of $u$, the Crank-Nicholson method is convergent, for any $\mu>0$, in the norm defined by $\|\boldsymbol{\eta}\|_{2, h}=\left(h \sum_{m=1}^{M}\left|\eta_{m}\right|^{2}\right)^{1 / 2}$ for $\boldsymbol{\eta} \in \mathbb{R}^{M}$.

[You may assume that the Trapezoidal method has local order 3 , and that the standard three-point centred discretisation of the second derivative (as used in the CrankNicholson method) has local order 2.]