$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
The Forward Euler scheme with operator notation
$$
\begin{equation}
[D_t^+ u = -au]^n
\end{equation}
$$
