  With the example of the spherically symmetric scalar wave equation on
Minkowski space-time we demonstrate that a fully pseudospectral scheme (i.e.
spectral with respect to both spatial and time directions) can be applied for
solving hyperbolic equations. The calculations are carried out within the
framework of conformally compactified space-times. In our formulation, the
equation becomes singular at null infinity and yields regular boundary
conditions there. In this manner it becomes possible to avoid "artificial"
conditions at some numerical outer boundary at a finite distance. We obtain
highly accurate numerical solutions possessing exponential spectral
convergence, a feature known from solving elliptic PDEs with spectral methods.
Our investigations are meant as a first step towards the goal of treating time
evolution problems in General Relativity with spectral methods in space and
time.
