5.2 The computational matching strategy
CCM evolves a mixed spacelike-null initial value problem in which Cauchy data is given in a spacelike
hypersurface bounded by a spherical boundary
and characteristic data is given on a null hypersurface
emanating from
. The general idea is not entirely new. An early mathematical investigation combining
spacelike and characteristic hypersurfaces appears in the work of Duff [88]. The three chief ingredients for
computational implementation are: (i) a Cauchy evolution module, (ii) a characteristic evolution module
and, (iii) a module for matching the Cauchy and characteristic regions across their interface. In the simplest
scenario, the interface is the timelike worldtube which is traced out by the flow of
along the
worldlines of the Cauchy evolution, as determined by the choice of lapse and shift. Matching
provides the exchange of data across the worldtube to allow evolution without any further
boundary conditions, as would be necessary in either a purely Cauchy or purely characteristic
evolution. Other versions of CCM involve a finite overlap between the characteristic and Cauchy
regions.
The most important application of CCM is anticipated to be the waveform and momentum recoil in the
binary black hole inspiral and merger. The 3D Cauchy codes being applied to simulate this problem employ
a single Cartesian coordinate patch. In principle, the application of CCM to this problem might seem
routine, tantamount to translating into finite difference form the textbook construction of an atlas
consisting of overlapping coordinate patches. In practice, it is a complicated project. The computational
strategy has been outlined in [45
]. The underlying algorithm consists of the following main submodules:
- The boundary module which sets the grid structures. This defines masks identifying which
points in the Cauchy grid are to be evolved by the Cauchy module and which points are
to be interpolated from the characteristic grid, and vice versa. The reference structures for
constructing the mask is the inner characteristic boundary, which in the Cartesian Cauchy
coordinates is the “ spherical” extraction worldtube
, and the outer Cauchy
boundary
, where the Cauchy boundary data is injected. The choice of
lapse and shift for the Cauchy evolution governs the dynamical and geometrical properties of
these worldtubes.
- The extraction module whose input is Cauchy grid data in the neighborhood of the extraction
worldtube at
and whose output is the inner boundary data for the exterior characteristic
evolution. This module numerically implements the transformation from Cartesian {3 + 1}
coordinates to spherical null coordinates. The algorithm makes no perturbative assumptions
and is based upon interpolations of the Cauchy data to a set of prescribed grid points near
. The metric information is then used to solve for the null geodesics normal to the slices of
the extraction worldtube. This provides the Jacobian for the transformation to null coordinates
in the neighborhood of the worldtube. The characteristic evolution module is then used to
propagate the data from the worldtube to null infinity, where the waveform is calculated.
- The injection module which completes the interface by using the exterior characteristic
evolution to inject the outer boundary data for the Cauchy evolution at
. This is the
inverse of the extraction procedure but must be implemented with
to allow for
overlap between the Cauchy and characteristic domains. The overlap region can be constructed
either to have a fixed physical size or to shrink to zero in the continuum limit. In the latter
case, the inverse Jacobian describing the transformation from null to Cauchy coordinates can
be obtained to prescribed accuracy in terms of an affine parameter expansion along the null
geodesics emanating from the worldtube. The numerical stability of this element of the scheme
is not guaranteed.
The above strategy provides a model of how Cauchy and characteristic codes can be pieced together as modules
to form a global evolution code.
The full advantage of CCM lies in the numerical treatment of nonlinear systems where its error
converges to zero in the continuum limit of infinite grid resolution [38
, 39, 76
]. For high accuracy, CCM is
also the most efficient method. For small target error
, it has been shown that the relative amount of
computation required for CCM (
) compared to that required for a pure Cauchy calculation (
)
goes to zero,
as
[49
, 45
]. An important factor here is the use of a
compactified characteristic evolution, so that the whole spacetime is represented on a finite grid.
From a numerical point of view this means that the only error made in a calculation of the
radiation waveform at infinity is the controlled error due to the finite discretization. Accuracy of a
Cauchy algorithm which uses an ABC requires a large grid domain in order to avoid error from
nonlinear effects in its exterior. The computational demands of CCM are small because the
interface problem involves one less dimension than the evolution problem. Because characteristic
evolution algorithms are more efficient than Cauchy algorithms, the efficiency can be further
enhanced by making the matching radius as small as possible consistent with the avoidance of
caustics.
At present, the computational strategy of CCM is mainly the tool of numerical relativists, who are used
to dealing with dynamical coordinate systems. The first discussion of its potential was given in [38] and its
feasibility has been more fully explored in [76
, 77
, 87
, 42
, 236
]. Recent work has been stimulated
by the requirements of the binary black hole problem, where CCM is one of the strategies to
provide boundary conditions and determine the radiation waveform. However, it also has inherent
advantages in dealing with other hyperbolic systems in computational physics, particularly nonlinear
3-dimensional problems. A detailed study of the stability and accuracy of CCM for linear and
nonlinear wave equations has been presented in [43
], illustrating its potential for a wide range of
problems.