  A new numerical method is introduced to study the problem of time evolution
of generic non-linear dynamical systems in four-dimensional spacetimes. It is
assumed that the time level surfaces are foliated by a one-parameter family of
codimension two compact surfaces with no boundary and which are conformal to a
Riemannian manifold C. The method is based on the use of a multipole expansion
determined uniquely by the induced metric structure on C. The approach is fully
spectral in the angular directions. The dynamics in the complementary 1+1
Lorentzian spacetime is followed by making use of a fourth order finite
differencing scheme with adaptive mesh refinement.
  In checking the reliability of the introduced new method the evolution of a
massless scalar field on a fixed Kerr spacetime is investigated. In particular,
the angular distribution of the evolving field in to be superradiant scattering
is studied. The primary aim was to check the validity of some of the recent
arguments claiming that the Penrose process, or its field theoretical
correspondence---superradiance---does play crucial role in jet formation in
black hole spacetimes while matter accretes onto the central object. Our
findings appear to be on contrary to these claims as the angular dependence of
a to be superradiant scattering of a massless scalar field does not show any
preference of the axis of rotation. In addition, the process of superradiance,
in case of a massless scalar field, was also investigated. On contrary to the
general expectations no energy extraction from black hole was found even though
the incident wave packets was fine tuned to be maximally superradiant. Instead
of energy extraction the to be superradiant part of the incident wave packet
fails to reach the ergoregion rather it suffers a total reflection which
appears to be a new phenomenon.
