Let

$$\frac{\mu}{4} u_{m-1}^{n+1}+u_{m}^{n+1}-\frac{\mu}{4} u_{m+1}^{n+1}=-\frac{\mu}{4} u_{m-1}^{n}+u_{m}^{n}+\frac{\mu}{4} u_{m+1}^{n},$$

where $n$ is a positive integer and $m$ ranges over all integers, be a finite-difference method for the advection equation

$$\frac{\partial u}{\partial t}=\frac{\partial u}{\partial x}, \quad-\infty<x<\infty, \quad t \geqslant 0$$

Here $\mu=\frac{\Delta t}{\Delta x}$ is the Courant number.

(a) Show that the local error of the method is $O\left((\Delta x)^{3}\right)$.

(b) Determine the range of $\mu>0$ for which the method is stable.