(i) The difference equation

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

where $\mu=\Delta t /(\Delta x)^{2}$, is the basic equation used in the second-order AdamsBashforth method and can be employed to approximate a solution of the diffusion equation $u_{t}=u_{x x}$. Prove that, as $\Delta t \rightarrow 0$ with constant $\mu$, the local error of the method is $O(\Delta t)^{2}$.

(ii) By applying the Fourier stability test, show that the above method is stable if and only if $\mu \leqslant 1 / 4$.

(iii) Define the leapfrog scheme to approximate the diffusion equation and prove that it is unstable for every choice of $\mu>0$.