  We present a Galerkin-Collocation domain decomposition algorithm applied to
the evolution of cylindrical unpolarized gravitational waves. We show the
effectiveness of the algorithm in reproducing initial data with high localized
gradients and in providing highly accurate dynamics. We characterize the
gravitational radiation with the standard Newman-Penrose Weyl scalar $\Psi_4$.
We generate wave templates for both polarization modes, $\times$ and $+$,
outgoing and ingoing, to show how they exchange energy nonlinearly. In
particular, considering an initially ingoing $\times$ wave, we were able to
trace a possible imprint of the gravitational analog of the Faraday effect in
the generated templates.
