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