  The time evolution of linear fields of spin $s = \pm 1$ and $s = \pm 2$ on
Kerr black hole spacetimes are investigated by solving the homogeneous
Teukolsky equation numerically. The applied numerical setup is based on a
combination of conformal compactification and the hyperbolic initial value
problem. The evolved basic variables are expanded in terms of spin-weighted
spherical harmonics which allows us to evaluate all the angular derivatives
analytically, whereas the evolution of the expansion coefficients, in the
time-radial section, is determined by applying the method of lines implemented
in a fourth order accurate finite differencing stencil. Concerning the
initialization, in all of our investigations single mode excitations---either
static or purely dynamical type initial data---are applied. Within this setup
the late time tail behavior is investigated. Due to the applied conformal
compactification the asymptotic decay rates are determined at three
characteristic locations---in the domain of outer communication, at the event
horizon and at future null infinity---simultaneously. Recently introduced new
type of `energy' and `angular momentum' balance relations are also applied in
order to demonstrate the feasibility and robustness of the developed numerical
schema, and also to verify the proper implementation of the underlying
mathematical model.
