$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Computing \( u^{\prime} \)
\( u \) is often displacement/position, \( u^{\prime} \) is velocity and can be computed by
$$
u^{\prime}(t_n) \approx \frac{u^{n+1}-u^{n-1}}{2\Delta t} = [D_{2t}u]^n
$$
