(i) The diffusion equation

$$\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}$$

is discretized by the finite-difference method

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

where $u_{m}^{n} \approx u(m \Delta x, n \Delta t), \mu=\Delta t /(\Delta x)^{2}$ and $\alpha$ is a constant. Derive the order of magnitude (as a power of $\Delta x$ ) of the local error for different choices of $\alpha$.

(ii) Investigate the stability of the above finite-difference method for different values of $\alpha$ by the Fourier technique.