(i) The diffusion equation

$$\frac{\partial u}{\partial t}=\frac{\partial}{\partial x}\left(a(x) \frac{\partial u}{\partial x}\right), \quad 0 \leqslant x \leqslant 1, \quad t \geqslant 0$$

with the initial condition $u(x, 0)=\phi(x), 0 \leqslant x \leqslant 1$ and zero boundary conditions at $x=0$ and $x=1$, is solved by the finite-difference method

$$\begin{gathered} u_{m}^{n+1}=u_{m}^{n}+\mu\left[a_{m-\frac{1}{2}} u_{m-1}^{n}-\left(a_{m-\frac{1}{2}}+a_{m+\frac{1}{2}}\right) u_{m}^{n}+a_{m+\frac{1}{2}} u_{m+1}^{n}\right] \\ m=1,2, \ldots, N \end{gathered}$$

where $\mu=\Delta t /(\Delta x)^{2}, \quad \Delta x=\frac{1}{N+1}$ and $u_{m}^{n} \approx u(m \Delta x, n \Delta t), a_{\alpha}=a(\alpha \Delta x)$.

Assuming sufficient smoothness of the function $a$, and that $\mu$ remains constant as $\Delta x>0$ and $\Delta t>0$ become small, prove that the exact solution satisfies the numerical scheme with error $O\left((\Delta x)^{3}\right)$.

(ii) For the problem defined in Part (i), assume that there exist $0<a_{-}<a_{+}<\infty$ such that $a_{-} \leqslant a(x) \leqslant a_{+}, \quad 0 \leqslant x \leqslant 1$. Prove that the method is stable for $0<\mu \leqslant 1 /\left(2 a_{+}\right)$.

[Hint: You may use without proof the Gerschgorin theorem: All the eigenvalues of the matrix $A=\left(a_{k, l}\right)_{k, l=1, \ldots, M}$ are contained in $\bigcup_{k=1}^{m} \mathbb{S}_{k}$, where

$$\left.\mathbb{S}_{k}=\left\{z \in \mathbb{C}:\left|z-a_{k, k}\right| \leqslant \sum_{\substack{l=1 \\ l \neq k}}^{m}\left|a_{k, l}\right|\right\}, \quad k=1,2, \ldots, m . \quad\right]$$