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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
