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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}
