$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Computing truncation errors in nonlinear problems (1)
$$
\begin{equation}
u'=f(u,t)
\tag{34}
\end{equation}
$$
Crank-Nicolson scheme:
$$
\begin{equation}
[D_t u'=\overline{f}^{t}]^{n+\half}\tp
\tag{34}
\end{equation}
$$
Truncation error:
$$
\begin{equation}
[D_t \uex' - \overline{f}^{t}= R]^{n+\half}\tp
\tag{35}
\end{equation}
$$
Using (19)-(20) for the arithmetic mean:
$$
\begin{align*}
\lbrack\overline{f}^{t}\rbrack^{n+\half} &=
\half(f(\uex^n,t_n) + f(\uex^{n+1},t_{n+1}))\\
&= f(\uex^{n+\half},t_{n+\half}) +
\frac{1}{8}\uex''(t_{n+\half})\Delta t^2
+ \Oof{\Delta t^4}\tp
\end{align*}
$$
