Quadratic damping

With \( f(v)=b|v|v \), we can use a geometric mean $$ b|v^n|v^n\approx b|v^{n-\half}|v^{n+\half}, $$ resulting in $$ \begin{align*} u^n & = u^{n-1} + {\Delta t}v^{n-\half},\\ v^{n+\half} &= (1 + \frac{b}{m}|v^{n-\half}|\Delta t)^{-1}\left( v^{n-\half} + {\Delta t} m^{-1}\left(F^n - s(u^n)\right)\right)\tp \end{align*} $$