We apply the Forward Euler scheme to each component equation: $$ [D_t^+ u = v]^n,$$ $$ [D_t^+ v = -\omega^2 u]^n,$$ or written out, $$ \begin{align} u^{n+1} &= u^n + \Delta t v^n,\\ v^{n+1} &= v^n -\Delta t \omega^2 u^n \tp \end{align} $$
