(i)

Given the finite-difference method

$$\sum_{k=-r}^{s} \alpha_{k} u_{m+k}^{n+1}=\sum_{k=-r}^{s} \beta_{k} u_{m+k}^{n}, \quad m, n \in \mathbb{Z}, n \geqslant 0$$

define

$$H(z)=\frac{\sum_{k=-r}^{s} \beta_{k} z^{k}}{\sum_{k=-r}^{s} \alpha_{k} z^{k}}$$

Prove that this method is stable if and only if

$$\left|H\left(e^{i \theta}\right)\right| \leqslant 1, \quad-\pi \leqslant \theta \leqslant \pi .$$

[You may quote without proof known properties of the Fourier transform.]

(ii) Find the range of the parameter $\mu$ such that the method

$$(1-2 \mu) u_{m-1}^{n+1}+4 \mu u_{m}^{n+1}+(1-2 \mu) u_{m+1}^{n+1}=u_{m-1}^{n}+u_{m+1}^{n}$$

is stable. Supposing that this method is used to solve the diffusion equation for $u(x, t)$, determine the order of magnitude of the local error as a power of $\Delta x$.