Initial conditions

$u(0)=I$, $u'(0)=V$: $$\begin{align*} \lbrack u &=I\rbrack^0\quad\Rightarrow\quad u^0=I\\ \lbrack D_{2t}u &=V\rbrack^0\quad\Rightarrow\quad u^{-1} = u^{1} - 2\Delta t V \end{align*}$$ End result: $$u^1 = u^0 + \Delta t\, V + \frac{\Delta t^2}{2m}(-bV - s(u^0) + F^0)$$ Same formula for $u^1$ as when using a centered scheme for $u''+\omega u=0$.