The difference equation

$$u_{m}^{n+1}=u_{m}^{n}+\frac{3}{2} \mu\left(u_{m-1}^{n}-2 u_{m}^{n}+u_{m+1}^{n}\right)-\frac{1}{2} \mu\left(u_{m-1}^{n-1}-2 u_{m}^{n-1}+u_{m+1}^{n-1}\right),$$

where $\mu=\Delta t /(\Delta x)^{2}$, is used to approximate a solution of the diffusion equation $u_{t}=u_{x x}$.

(a) Prove that, as $\Delta t \rightarrow 0$ with constant $\mu$, the local error of the method is $\mathcal{O}(\Delta t)^{2}$.

(b) Applying the Fourier stability test, show that the method is stable if and only if $\mu \leqslant \frac{1}{4}$.