Consider the one-dimensional advection equation

$$u_{t}=u_{x}, \quad-\infty<x<\infty, \quad t \geqslant 0$$

subject to an initial condition $u(x, 0)=\varphi(x)$. Consider discretization of this equation with finite differences on an equidistant space-time $\left\{(m h, n k), m \in \mathbb{Z}, n \in \mathbb{Z}^{+}\right\}$with step size $h>0$ in space and step size $k>0$ in time. Define the Courant number $\mu$ and explain briefly how such a discretization can be used to derive numerical schemes in which solutions $u_{m}^{n} \approx u(m h, n k), m \in \mathbb{Z}$ and $n \in \mathbb{Z}^{+}$satisfy equations of the form

$$\sum_{i=r}^{s} a_{i} u_{m+i}^{n+1}=\sum_{i=r}^{s} b_{i} u_{m+i}^{n}$$

where the coefficients $a_{i}, b_{i}$ are independent of $m, n$.

(i) Define the order of a numerical scheme such as (1). Define what a convergent numerical scheme is. Explain the notion of stability and state the Lax equivalence theorem that connects convergence and stability of numerical schemes for linear partial differential equations.

(ii) Consider the following example of (1):

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

Determine conditions on $\mu$ such that the scheme $(2)$ is stable and convergent. What is the order of this scheme?