$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
A centered scheme for linear damping
$$
[mD_tD_t u + f(D_{2t}u) + s(u) = F]^n
$$
Written out
$$
m\frac{u^{n+1}-2u^n + u^{n-1}}{\Delta t^2}
+ f(\frac{u^{n+1}-u^{n-1}}{2\Delta t}) + s(u^n) = F^n
$$
Assume \( f(u') \) is linear in \( u'=v \):
$$
u^{n+1} = \left(2mu^n + (\frac{b}{2}\Delta t - m)u^{n-1} +
\Delta t^2(F^n - s(u^n))
\right)(m + \frac{b}{2}\Delta t)^{-1}
$$
