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Result: $$ \begin{equation} \frac{u^{n+1}-u^n}{t_{n+1}-t_n} = -a\half (u^n + u^{n+1}) \tag{11} \end{equation} $$
Solve wrt to \( u^{n+1} \): $$ \begin{equation} u^{n+1} = \frac{1-\half a(t_{n+1}-t_n)}{1 + \half a(t_{n+1}-t_n)}u^n \tag{12} \end{equation} $$ This is a Crank-Nicolson (CN) scheme or a midpoint or centered scheme.
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