(a) The diffusion equation

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

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

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

where $\mu=k / h^{2}, k=\Delta t, h=1 /(M+1), u_{m}^{n} \approx u(m h, n k)$, and $a_{\alpha}=a(\alpha h)$.

Assuming that the function $a$ and the exact solution are sufficiently smooth, prove that the exact solution satisfies the numerical scheme with error $O\left(h^{3}\right)$ for constant $\mu$.

(b) For the problem in part (a), assume that there exist $0<a_{-}<a_{+}<\infty$ such that $a_{-} \leqslant a(x) \leqslant a_{+}$for all $0 \leqslant x \leqslant 1$. State (without proof) the Gershgorin theorem and prove that the method is stable for $0<\mu \leqslant 1 /\left(2 a_{+}\right)$.