$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Operator notation; initial condition
$$
[u = I]^0,\quad [D_{2t} u = 0]^0
$$
where \( [D_{2t} u]^n \) is defined as
$$
[D_{2t} u]^n = \frac{u^{n+1} - u^{n-1}}{2\Delta t}
\tp
$$
