Numerical solution of the wave equation on particular space-times using CMC slices and scri-fixing conformal compactification

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.