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Introduce an averaging operator: $$ \begin{equation} [\overline{u}^{t}]^n = \half (u^{n-\half} + u^{n+\half} ) \approx u(t_n) \tag{22} \end{equation} $$
The Crank-Nicolson scheme can then be written as $$ \begin{equation} [D_t u = -a\overline{u}^t]^{n+\half} \tag{23} \end{equation} $$
Test: use the definitions and write out the above formula to see that it really is the Crank-Nicolson scheme!
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