Initial condition for quadratic damping

Simply use that $u'=V$ in the scheme when $t=0$ ($n=0$): $$[mD_tD_t u + bV|V| + s(u) = F]^0$$

which gives $$u^1 = u^0 + \Delta t V + \frac{\Delta t^2}{2m}\left(-bV|V| - s(u^0) + F^0\right)$$