  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.
