(a) The advection equation

$$u_{t}=u_{x}, \quad 0 \leqslant x \leqslant 1, t \geqslant 0$$

is discretised using an equidistant grid with stepsizes $\Delta x=h$ and $\Delta t=k$. The spatial derivatives are approximated with central differences and the resulting ODEs are approximated with the trapezoidal rule. Write down the relevant difference equation for determining $\left(u_{m}^{n+1}\right)$ from $\left(u_{m}^{n}\right)$. What is the name of this scheme? What is the local truncation error?

The boundary condition is periodic, $u(0, t)=u(1, t)$. Explain briefly how to write the discretised scheme in the form $B \mathbf{u}^{n+1}=C \mathbf{u}^{n}$, where the matrices $B$ and $C$, to be identified, have a circulant form. Using matrix analysis, find the range of $\mu=\Delta t / \Delta x$ for which the scheme is stable. [Standard results may be used without proof if quoted carefully.]

[Hint: An $n \times n$ circulant matrix has the form

$$A=\left(\begin{array}{cccc} a_{0} & a_{1} & \cdots & a_{n-1} \\ a_{n-1} & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & a_{1} \\ a_{1} & \cdots & a_{n-1} & a_{0} \end{array}\right)$$

All such matrices have the same set of eigenvectors $\mathbf{v}_{\ell}=\left(\omega^{j \ell}\right)_{j=0}^{n-1}, \ell=0,1, \ldots, n-1$, where $\omega=e^{2 \pi i / n}$, and the corresponding eigenvalues are $\lambda_{\ell}=\sum_{k=0}^{n-1} a_{k} \omega^{k \ell}$.]

(b) Consider the advection equation on the unit square

$$u_{t}=a u_{x}+b u_{y}, \quad 0 \leqslant x, y \leqslant 1, t \geqslant 0$$

where $u$ satisfies doubly periodic boundary conditions, $u(0, y)=u(1, y), u(x, 0)=u(x, 1)$, and $a(x, y)$ and $b(x, y)$ are given doubly periodic functions. The system is discretised with the Crank-Nicolson scheme, with central differences for the space derivatives, using an equidistant grid with stepsizes $\Delta x=\Delta y=h$ and $\Delta t=k$. Write down the relevant difference equation, and show how to write the scheme in the form

$$\mathbf{u}^{n+1}=\left(I-\frac{1}{4} \mu A\right)^{-1}\left(I+\frac{1}{4} \mu A\right) \mathbf{u}^{n}$$

where the matrix $A$ should be identified. Describe how (*) can be approximated by Strang splitting, and explain the advantages of doing so.

[Hint: Inversion of the matrix $B$ in part (a) has a similar computational cost to that of a tridiagonal matrix.]