$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Step 3: Replacing derivatives by finite differences
Now it is time for the finite difference approximations of
derivatives:
$$
\begin{equation}
u'(t_n) \approx \frac{u^{n+1}-u^{n}}{t_{n+1}-t_n}
\tag{4}
\end{equation}
$$
