General cylindrical waves are the simplest axisymmetrical gravitational waves that contain both $+$ and $times$ modes of polarization. In this paper, we have studied the evolution of general cylindrical gravitational waves in the realm of the characteristic scheme with a numerical code based on the Galerkin-Collocation method. The investigation consists of the numerical realization of concepts such as Bondi mass and the news functions adapted to cylindrical symmetry. The Bondi mass decays due to the presence of the news functions associated with both polarization modes. We have interpreted that each polarization mode as channels from which mass is extracted. Under this perspective, we have presented the enhancement effect of the polarization mode $+$ due to the nonlinear interaction with the mode $times$. After discussing the role of matter in cylindrical symmetry, we have extended the numerical code to include electromagnetic fields.
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