  Black holes represent extreme conditions of physical laws. Being predicted
about a century ago, they are now accepted as astrophysical reality by most of
the scientific community. Only recently more direct evidence of their existence
has been found - the detection of gravitational waves from black hole mergers
and of the shadow of a supermassive black hole in the center of a galaxy.
Astrophysical black holes are typically embedded in an active environment which
is affected by the strong gravity. When the environmental material emits
radiation, this radiation may carry imprints of the black hole that is hosting
the radiation source. In order to understand the physical processes that take
place in the close neighbourhood of astrophysical black holes, numerical
methods and simulations play an essential role. This is simply because the
dynamical evolution and the radiative interaction are far too complex in order
to allow for an analytic solution of the physical equations. A huge progress
has been made over the last decade(s) in the numerical code development, as
well as in the computer power that is needed to run these codes. This review
tries to summarize the basic questions and methods that are involved in the
undertaking of investigating the astrophysics of black holes by numerical
means. It is intended for a non-expert audience interested in an overview over
this broad field. The review comes along without equations and thus without a
detailed expert discussion of the underlying physical processes or numerical
specifics. Instead, it intends to illustrate the richness of the field and to
motivate for further reading. The review puts some emphasis on
magnetohydrodynamic simulations, but also touches radiation transfer and merger
simulations, in particular pointing out differences in these approaches.
