$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
The ODE problem has a continuous and discrete version
Continuous problem
$$
\begin{equation}
u' = -au,\ t\in (0,T], \quad u(0)=I
\tag{1}
\end{equation}
$$
(varies with a continuous \( t \))
Discrete problem
Numerical methods applied to the continuous problem turns it into
a discrete problem
$$
\begin{equation}
u^{n+1} = \mbox{const} u^n, \quad n=0,1,\ldots N_t-1, \quad u^n=I
\tag{2}
\end{equation}
$$
(varies with discrete mesh points \( t_n \))
