$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Extension to quadratic damping
$$
\begin{equation}
mu'' + \beta |u'|u' + s(u) =F(t)\tp
\tag{41}
\end{equation}
$$
Centered scheme: \( |u'|u' \) gives rise to a nonlinearity.
Linearization trick: use a geometric mean,
$$ [|u'|u']^n \approx |[u']^{n-\half}|[u']^{n+\half}\tp$$
Scheme:
$$
\begin{equation}
[mD_t D_t u]^n + \beta |[D_{t} u]^{n-\half}|[D_t u]^{n+\half}
+ s(u^n)=F^n\tp
\end{equation}
$$
