$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Linear model without damping
$$
\begin{equation}
u''(t) + \omega^2 u(t) = 0,\quad u(0)=I,\ u'(0)=0\tp
\tag{36}
\end{equation}
$$
Centered difference approximation:
$$
\begin{equation}
[D_tD_t u + \omega^2u=0]^n
\tag{37}
\tp
\end{equation}
$$
Truncation error:
$$
\begin{equation}
[D_tD_t \uex + \omega^2\uex =R]^n
\tp
\end{equation}
$$
Use (15)-(16) to expand \( [D_t D_t\uex]^n \):
$$ [D_tD_t \uex]^n = \uex''(t_n) + \frac{1}{12}\uex''''(t_n)\Delta t^2,$$
Collect terms: \( \uex''(t) + \omega^2\uex(t)=0 \). Then,
$$
\begin{equation}
R^n = \frac{1}{12}\uex''''(t_n)\Delta t^2 + \Oof{\Delta t^4}
\tp
\end{equation}
$$
