$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
The Leapfrog scheme
Idea:
$$
\begin{equation}
u'(t_n)\approx \frac{u^{n+1}-u^{n-1}}{2\Delta t} = [D_{2t} u]^n
\end{equation}
$$
Scheme:
$$ [D_{2t} u = f(u,t)]^n$$
or written out,
$$
\begin{equation}
u^{n+1} = u^{n-1} + \Delta t f(u^n, t_n)
\tag{40}
\end{equation}
$$
- Some other scheme must be used as starter (\( u^1 \)).
- Explicit scheme - a nonlinear \( f \) (in \( u \)) is trivial to handle.
- Downside: Leapfrog is always unstable after some time.
