$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
The Crank-Nicolson scheme
$$
[D_t u = \overline{v}^t]^{n+\half},$$
$$
[D_t v = -\omega \overline{u}^t]^{n+\half}$$
The result is also a coupled system:
$$
\begin{align}
u^{n+1} - \half\Delta t v^{n+1} &= u^{n} + \half\Delta t v^{n},\\
v^{n+1} + \half\Delta t \omega^2 u^{n+1} &= v^{n}
- \half\Delta t \omega^2 u^{n}
\tp
\end{align}
$$
