  A new numerical framework, based on the use of a simple first order strongly
hyperbolic evolution equations, is introduced and tested in case of
4-dimensional spherically symmetric gravitating systems. The analytic setup is
chosen such that our numerical method is capable to follow the time evolution
even after the appearance of trapped surfaces, more importantly, until the true
physical singularities are reached. Using this framework, the gravitational
collapse of various gravity-matter systems are investigated, with distinguished
attention to the evolution in trapped regions. It is justified that in advance
to the formation of these curvature singularities, trapped regions develop in
all cases, thereby supporting the validity of the weak cosmic censor hypothesis
of Penrose. Various upper bounds on the rate of blow-up of the Ricci and
Kretschmann scalars and the Misner-Sharp mass are provided. In spite of the
unboundedness of the Ricci scalar, the Einstein-Hilbert action was found to
remain finite in all the investigated cases. In addition, important conceptual
issues related to the phenomenon of topology changes are also discussed.
