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Crank-Nicolson: $$ \begin{equation} [D_t u = -au]^{n+\half}, \end{equation} $$ Truncation error: $$ \begin{equation} [D_t \uex + a\overline{\uex}^{t} = R]^{n+\half} \tp \tag{28} \end{equation} $$
From (3)-(4) and (19)-(20): $$ \begin{align*} \lbrack D_t\uex \rbrack^{n+\half} &= u'(t_{n+\half}) + \frac{1}{24}\uex'''(t_{n+\half})\Delta t^2 + \Oof{\Delta t^4},\\ [a\overline{\uex}^{t}]^{n+\half} &= u(t_{n+\half}) + \frac{1}{8}u''(t_{n})\Delta t^2 + + \Oof{\Delta t^4} \end{align*} $$ Inserted in the scheme we get $$ \begin{equation} R^{n+\half} = \left( \frac{1}{24}\uex'''(t_{n+\half}) + \frac{1}{8}u''(t_{n}) \right)\Delta t^2 + \Oof{\Delta t^4} \end{equation} $$ \( R^n = \Oof{\Delta t^2} \) (second-order scheme)
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