  This article begins with a brief introduction to numerical relativity aimed
at readers who have a background in applied mathematics but not necessarily in
general relativity. I then introduce and summarise my work on the problem of
treating asymptotically flat spacetimes of infinite extent with finite
computational resources. Two different approaches are considered. The first
approach is the standard one and is based on evolution on Cauchy hypersurfaces
with artificial timelike boundary. The well posedness of a set of
constraint-preserving boundary conditions for the Einstein equations in
generalised harmonic gauge is analysed, their numerical performance is compared
with various alternate methods, and improved absorbing boundary conditions are
constructed and implemented. In the second approach, one solves the Einstein
equations on hyperboloidal (asymptotically characteristic) hypersurfaces. These
are conformally compactified towards future null infinity, where gravitational
radiation is defined in an unambiguous way. We show how the formally singular
terms arising in a $3+1$ reduction of the equations can be evaluated at future
null infinity, present stable numerical evolutions of vacuum axisymmetric black
hole spacetimes and study late-time power-law tails of matter fields in
spherical symmetry.
