Extension to a variable coefficient; Crank-Nicolson

Eevaluting $a(t_{n+\half})$ and using an average for $u$: $$\begin{equation} \frac{u^{n+1} - u^{n}}{\Delta t} = -a(t_{n+\half})\half(u^n + u^{n+1}) \end{equation}$$

Using an average for $a$ and $u$: $$\begin{equation} \frac{u^{n+1} - u^{n}}{\Delta t} = -\half(a(t_n)u^n + a(t_{n+1})u^{n+1}) \end{equation}$$