### 11.3 Adaptive mesh refinement and curvilinear grids

In [331, 333] the authors introduced an approach, which combines the advantages of adaptive mesh
refinement near the “sources” (say, black holes) with curvilinear coordinates adapted to the wave zone; see
Figure 12. The patches are communicated using polynomial interpolation in its Lagrange form, as explained
in Section 8.1, and centered stencils are used, both for finite differencing and interpolation. Up to
eighth-order finite differencing is used, with an observed convergence rate between six and eight in the
modes of the computed gravitational waves (parts of the scheme have a lower-order
convergence rate, but they do not appear to dominate). Presently, the BSSN formulation of
Einstein’s equations as described in Section 4.3 is used directly in its second-order-in-space form,
with outgoing boundary conditions for all the fields. The implementation is generic and flexible
enough to allow for other systems of equations, though. As in most approaches using curvilinear
coordinates in numerical relativity, the field variables are expressed in a global coordinate frame. This
might sound unnatural and against the idea of using local patches and coordinates. However, it
simplifies dramatically any implementation. It is also particularly important when taking into
account that most formulations of Einstein’s equations and coordinate conditions used are not
covariant.
This hybrid approach has been used in several applications, including the validation of extrapolation
procedures of gravitational waves extracted from numerical simulations at finite radii to large distances
from the “sources” [331]. Since the outermost grid structure is well adapted to the wave zone, the outer
boundary can be located at large distances with only linear cost on its location. Other applications include
Cauchy-Characteristic extraction (CCE) of gravitational waves [350, 55], a waveform hybrid
development [371], and studies of memory effect in gravitational waves [330]. The accuracy necessary to
study small memory effects is enabled both by the grid structure – being able to locate the outer boundary
far away – and CCE.