Consider the advection equation $u_{t}=u_{x}$ on the unit interval $x \in[0,1]$ and $t \geqslant 0$, where $u=u(x, t)$, subject to the initial condition $u(x, 0)=\varphi(x)$ and the boundary condition $u(1, t)=0$, where $\varphi$ is a given smooth function on $[0,1]$.

(i) We commence by discretising the advection equation above with finite differences on the equidistant space-time grid $\{(m \Delta x, n \Delta t), m=0, \ldots, M+1, n=0, \ldots, T\}$ with $\Delta x=1 /(M+1)$ and $\Delta t>0$. We obtain an equation for $u_{m}^{n} \approx u(m \Delta x, n \Delta t)$ that reads

$$u_{m}^{n+1}=u_{m}^{n}+\frac{1}{2} \mu\left(u_{m+1}^{n}-u_{m-1}^{n}\right), \quad m=1, \ldots, M, n \in \mathbb{Z}^{+}$$

with the condition $u_{0}^{n}=0$ for all $n \in \mathbb{Z}^{+}$and $\mu=\Delta t / \Delta x$.

What is the order of approximation (that is, the order of the local error) in space and time of the above discrete solution to the exact solution of the advection equation? Write the scheme in matrix form and deduce for which choices of $\mu$ this approximation converges to the exact solution. State (without proof) any theorems you use. [You may use the fact that for a tridiagonal $M \times M$ matrix

$$\left(\begin{array}{cccc} \alpha & \beta & 0 & 0 \\ -\beta & \ddots & \ddots & 0 \\ 0 & \ddots & \ddots & \beta \\ 0 & 0 & -\beta & \alpha \end{array}\right)$$

the eigenvalues are given by $\lambda_{\ell}=\alpha+2 i \beta \cos \frac{\ell \pi}{M+1}$.]

(ii) How does the order change when we replace the central difference approximation of the first derivative in space by forward differences, that is $u_{m+1}^{n}-u_{m}^{n}$ instead of $\left(u_{m+1}^{n}-u_{m-1}^{n}\right) / 2 ?$ For which choices of $\mu$ is this new scheme convergent?

(iii) Instead of the approximation in (i) we consider the following method for numerically solving the advection equation,

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

where we additionally assume that $u_{m}^{1}$ is given. What is the order of this method for a fixed $\mu$ ?