  In this paper we present in detail the numerical solution of the conformally
invariant wave equation on top of a fixed background space-time corresponding
to two different cases: i) 1+1 Minkowski space-time in Cartesian coordinates
and ii) Schwarzschild space-time. In both cases we use hyperboloidal constant
mean curvature slices and scri-fixing conformal compactification, and solve the
wave equation on the conformal space-time. In the case of the Schwarzschild
space-time we study the quasinormal mode oscillations and the late-time
polynomial tail decay exponents corresponding to a mass-less scalar field. We
also present general formulas to construct hyperboloidal constant mean
curvature slicings of spherically symmetric, static, space-times in spherical
coordinates.
