$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
The general model formulated as first-order ODEs
$$
\begin{equation}
mu'' + \beta |u'|u' + s(u) =F(t)\tp
\tag{41}
\end{equation}
$$
Rewritten as first-order system:
$$
\begin{align}
u' &= v,
\tag{42} \\
v' &= \frac{1}{m}\left( F(t) - \beta |v|v - s(u)\right)\tp
\tag{43}
\end{align}
$$
To solution methods:
- Forward-backward scheme
- Centered scheme on a staggered mesh
