$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
The central difference for \( u'(t) \) (1)
$$
u(t_{n+\half}) - u(t_{n-1/2}) = u'(t_n)\Delta t + \frac{1}{24}u'''(t_n) \Delta t^3 + \Oof{\Delta t^5}
\tp
$$
By collecting terms in \( [D_t u]^n - u(t_n) \) we find \( R^n \) to be
$$
\begin{equation}
R^n = \frac{1}{24}u'''(t_n)\Delta t^2 + \Oof{\Delta t^4},
\end{equation}
$$
Note:
- Second-order accuracy since the leading term is \( \Delta t^2 \)
- Only even powers of \( \Delta t \)
