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A centered scheme for quadratic damping
After some algebra:
\begin{align*}
u^{n+1} &= \left( m + b|u^n-u^{n-1}|\right)^{-1}\times \\
& \qquad \left(2m u^n - mu^{n-1} + bu^n|u^n-u^{n-1}| + \Delta t^2 (F^n - s(u^n))
\right)
\end{align*}
