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.
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