The Crank-Nicolson scheme; ideas

Idea 1: let the ODE hold at $t_{n+\half}$ $$u'(t_{n+\half}) = -au(t_{n+\half})$$

Idea 2: approximate $u'(t_{n+\half}$ by a centered difference $$\begin{equation} u'(t_{n+\half}) \approx \frac{u^{n+1}-u^n}{t_{n+1}-t_n} \tag{10} \end{equation}$$

Problem: $u(t_{n+\half})$ is not defined, only $u^n=u(t_n)$ and $u^{n+1}=u(t_{n+1})$

Solution: $$u(t_{n+\half}) \approx \half(u^n + u^{n+1})$$