  Motivated by the goal for high accuracy modeling of gravitational radiation
emitted by isolated systems, recently, there has been renewed interest in the
numerical solution of the hyperboloidal initial value problem for Einstein's
field equations in which the outer boundary of the numerical grid is placed at
null infinity. In this article, we numerically implement the tetrad-based
approach presented in [J.M. Bardeen, O. Sarbach, and L.T. Buchman, Phys. Rev. D
83, 104045 (2011)] for a spherically symmetric, minimally coupled,
self-gravitating scalar field. When this field is massless, the evolution
system reduces to a regular, first-order symmetric hyperbolic system of
equations for the conformally rescaled scalar field which is coupled to a set
of singular elliptic constraints for the metric coefficients. We show how to
solve this system based on a numerical finite-difference approximation,
obtaining stable numerical evolutions for initial black hole configurations
which are surrounded by a spherical shell of scalar field, part of which
disperses to infinity and part of which is accreted by the black hole. As a
non-trivial test, we study the tail decay of the scalar field along different
curves, including one along the marginally trapped tube, one describing the
world line of a timelike observer at a finite radius outside the horizon, and
one corresponding to a generator of null infinity. Our results are in perfect
agreement with the usual power-law decay discussed in previous work. This
article also contains a detailed analysis for the asymptotic behavior and
regularity of the lapse, conformal factor, extrinsic curvature and the
Misner-Sharp mass function along constant mean curvature slices.
