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}