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