The advection equation

$$u_{t}=u_{x}, \quad x \in \mathbb{R}, \quad t \geqslant 0,$$

is solved by the leapfrog scheme

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

where $n \geqslant 1$ and $\mu=\Delta t / \Delta x$ is the Courant number.

(a) Determine the local error of the method.

(b) Applying the Fourier technique, find the range of $\mu>0$ for which the method is stable.