$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Step 2: Fulfilling the equation at discrete time points
- The ODE holds for all \( t\in (0,T] \) (infinite no of points)
- Idea: let the ODE be valid at the mesh points only (finite no of points)
$$
\begin{equation}
u'(t_n) = -au(t_n),\quad n=1,\ldots,N_t
\tag{3}
\end{equation}
$$
