### 6.4 Alternative approaches

Although the formulation of Einstein’s equations on a finite space domain with an artificial time-like
boundary is currently the most used approach in numerical simulations, there are a number of difficulties
associated with it. First, as discussed above, spurious reflections from the boundary surface may
contaminate the solution unless the boundary conditions are chosen with great care. Second, in principle
there is a problem with wave extraction, since gravitational waves can only be defined in an
unambiguous (gauge-invariant) way at future null infinity. Third, there is an efficiency problem, since in
the far zone the waves propagate along outgoing null geodesics so that hyperboloidal surfaces,
which are asymptotically null, should be better adapted to the problem. These issues have
become more apparent as numerical simulations have achieved higher accuracy to the point
that boundary and wave extraction artifacts are noticeable, and have driven a number of other
approaches.
One of them is that of compactification schemes, which include spacelike or null infinity
into the computational domain. For schemes compactifying spacelike infinity; see [335, 336].
Conformal compactifications are reviewed in [172, 183], and a partial list of references to date
includes [328, 176, 177, 180, 179, 170, 245, 172, 247, 100, 446, 447, 316, 87, 451, 452, 448, 449, 450, 305, 364, 42].

Another approach is Cauchy-characteristic matching (CCM) [99, 392, 401, 143, 148, 53], which
combines a Cauchy approach in the strong field regime (thereby avoiding the problems that the presence of
caustics would cause on characteristic evolutions) with a characteristic one in the wave zone. Data from the
Cauchy evolution is used as inner boundary conditions for the characteristic one and, viceversa,
the latter provides outer boundary conditions for the Cauchy IBVP. An understanding of the
Cauchy IBVP is still a requisite. CCM is reviewed in [432]. A related idea is Cauchy-perturbative
matching [455, 356, 4, 370], where the Cauchy code is instead coupled to one solving gauge-invariant
perturbations of Schwarzschild black holes or flat spacetime. The multipole decomposition in the
Regge–Wheeler–Zerilli equations [347, 453, 376, 294, 307] implies that the resulting equations are
1+1 dimensional and can therefore extend the region of integration to very large distances
from the source. As in CCM, an understanding of the IBVP for the Cauchy sector is still a
requisite.

One way of dealing with the ambiguity of extracting gravitational waves from Cauchy evolutions at
finite radii is by extrapolating procedures; see, for example, [72, 331] for some approaches
and quantification of their accuracies. Another approach is Cauchy characteristic extraction
(CCE) [350, 37, 349, 32, 34, 54]. In CCE a Cauchy IBVP is solved, and the numerical data on a world
tube is used to provide inner boundary conditions for a characteristic evolution that “transports” the
data to null infinity. The difference with CCM is that in CCE there is no “feedback” from the
characteristic evolution to the Cauchy one, and the extraction is done as a post-processing
step.