The Southampton group chose cylindrically-symmetric systems as their model problem for developing matching techniques. In preliminary work, they showed how CCM could be consistently carried out for a scalar wave evolving in Minkowski spacetime but expressed in a nontrivial cylindrical coordinate system [89].
They then tackled the gravitational problem. First they set up the analytic machinery necessary for
investigating cylindrically-symmetric vacuum spacetimes [90]. Although the problem involves only one
spatial dimension, there are two independent modes of polarization. The Cauchy metric was treated in the
Jordan–Ehlers–Kompaneets canonical form, using coordinates adapted to the
cylindrical symmetry. The advantage here is that
is then a null coordinate, which can be used
for the characteristic evolution. They successfully recast the equations in a suitably regularized form
for the compactification of
in terms of the coordinate
. The simple analytic
relationship between Cauchy coordinates
and characteristic coordinates
facilitated the
translation between Cauchy and characteristic variables on the matching worldtube, given by
.
Next they implemented the scheme as a numerical code. The interior Cauchy evolution was carried out
using an unconstrained leapfrog scheme. It is notable that they report no problems with instability,
which have arisen in other attempts at unconstrained leapfrog evolution in general relativity.
The characteristic evolution also used a leapfrog scheme for the evolution between retarded
time levels , while numerically integrating the hypersurface equations outward along the
characteristics.
The matching interface was located at points common to both the Cauchy and characteristic grids. In order to update these points by Cauchy evolution, it was necessary to obtain field values at the Cauchy “ghost” points, which lie outside the worldtube in the characteristic region. These values were obtained by interpolation from characteristic grid points (lying on three levels of null hypersurfaces in order to ensure second-order accuracy). Similarly, the boundary data for starting up the characteristic integration was obtained by interpolation from Cauchy grid values inside the worldtube.
The matching code was first tested [102] using exact Weber–Wheeler cylindrical waves [301], which
come in from , pass through the symmetry axis and expand out to
. The numerical errors were
oscillatory with low growth rate, and second-order convergence was confirmed. Of special importance, little
numerical noise was introduced by the interface. Comparisons of CCM were made with Cauchy evolutions
using a standard outgoing radiation boundary condition [230]. At high amplitudes the standard condition
developed a large error very quickly and was competitive only for weak waves with a large outer
boundary. In contrast, the matching code performed well even with a small matching radius.
Some interesting simulations were presented in which an outgoing wave in one polarization
mode collided with an incoming wave in the other mode, a problem studied earlier by pure
Cauchy evolution [232]. The simulations of the collision were qualitatively similar in these two
studies.
The Weber–Wheeler waves contain only one gravitational degree of freedom. The code was next
tested [98] using exact cylindrically-symmetric solutions, due to Piran, Safier and Katz [231], which contain
both degrees of freedom. These solutions are singular at so that the code had to be suitably
modified. Relative errors of the various metric quantities were in the range 10–4 to 10–2. The
convergence rate of the numerical solution starts off as second order but diminishes to first
order after long time evolution. This performance could perhaps be improved by incorporating
subsequent improvements in the characteristic code made by Sperhake, Sjödin, and Vickers (see
Section 3.1).
A joint collaboration between groups at Pennsylvania State University and the University of Pittsburgh applied CCM to the EKG system with spherical symmetry [138]. This model problem allowed simulation of black-hole formation as well as wave propagation.
The geometrical setup is analogous to the cylindrically-symmetric problem. Initial data were specified on
the union of a spacelike hypersurface and a null hypersurface. The evolution used a 3-level Cauchy scheme
in the interior and a 2-level characteristic evolution in the compactified exterior. A constrained Cauchy
evolution was adopted because of its earlier success in accurately simulating scalar wave collapse [80].
Characteristic evolution was based upon the null parallelogram algorithm (19). The matching between the
Cauchy and characteristic foliations was achieved by imposing continuity conditions on the metric,
extrinsic curvature and scalar field variables, ensuring smoothness of fields and their derivatives
across the matching interface. The extensive analytical and numerical studies of this system in
recent years aided the development of CCM in this non-trivial geometrical setting by providing
basic knowledge of the expected physical and geometrical behavior, in the absence of exact
solutions.
The CCM code accurately handled wave propagation and black-hole formation for all values of
at the matching radius, with no symptoms of instability or back-reflection. Second-order accuracy was
established by checking energy conservation.
In further developmental work on the EKG model, the Pittsburgh group used CCM to formulate a
new treatment of the inner Cauchy boundary for a black-hole spacetime [141]. In the excision
strategy, the inner boundary of the Cauchy evolution is located at an apparent horizon, which
must lie inside (or on) the event horizon [300]. The physical rationale behind this apparent
horizon boundary condition is that the truncated region of spacetime cannot causally affect the
gravitational waves radiated to infinity. However, it should be noted that many Cauchy formalisms
contain superluminal gauge or constraint violating modes so that this strategy is not always fully
justified.
In the CCM excision strategy, illustrated in Figure 6, the interior black-hole region is evolved using an
ingoing null algorithm whose inner boundary is an MTS, and whose outer boundary lies outside
the black hole and forms the inner boundary of a region evolved by the Cauchy algorithm. In
turn, the outer boundary of the Cauchy region is handled by matching to an outgoing null
evolution extending to
. Data are passed between the inner characteristic and central
Cauchy regions using a CCM procedure similar to that already described for an outer Cauchy
boundary. The main difference is that, whereas the outer Cauchy boundary data is induced
from the Bondi metric on an outgoing null hypersurface, the inner Cauchy boundary is now
obtained from an ingoing null hypersurface, which enters the event horizon and terminates at an
MTS.
The translation from an outgoing to an incoming null evolution algorithm can be easily carried out. The
substitution in the 3D version of the Bondi metric (14
) provides a simple formal recipe for
switching from an outgoing to an ingoing null formalism [141
].
In order to ensure that trapped surfaces exist on the ingoing null hypersurfaces, initial data were chosen, which guarantee black-hole formation. Such data can be obtained from initial Cauchy data for a black hole. However, rather than extending the Cauchy hypersurface inward to an apparent horizon, it was truncated sufficiently far outside the apparent horizon to avoid computational problems with the Cauchy evolution. The initial Cauchy data were then extended into the black-hole interior as initial null data until an MTS was reached. Two ingredients were essential in order to arrange this. First, the inner matching surface must be chosen to be convex, in the sense that its outward null normals uniformly diverge and its inner null normals uniformly converge. (This is trivial to satisfy in the spherically-symmetric case.) Given any physically-reasonable matter source, the focusing theorem guarantees that the null rays emanating inward from the matching sphere continue to converge until reaching a caustic. Second, the initial null data must lead to a trapped surface before such a caustic is encountered. This is a relatively easy requirement to satisfy because the initial null data can be posed freely, without any elliptic or algebraic constraints other than continuity with the Cauchy data.
A code was developed, which implemented CCM at both the inner and outer boundaries [141]. Its
performance showed that CCM provides as good a solution to the black-hole excision problem in spherical
symmetry as any previous treatment [262, 263, 208
, 11]. CCM is computationally more efficient than
these pure Cauchy approaches (fewer variables) and much easier to implement. Depending
upon the Cauchy formalism adopted, achieving stability with a pure Cauchy scheme in the
region of an apparent horizon can be quite tricky, involving much trial and error in choosing
finite-difference schemes. There were no complications with stability of the null evolution at the
MTS.
The Cauchy evolution was carried out in ingoing Eddington–Finklestein (IEF) coordinates. The initial
Cauchy data consisted of a Schwarzschild black hole with an ingoing Gaussian pulse of scalar radiation.
Since IEF coordinates are based on ingoing null cones, it is possible to construct a simple transformation
between the IEF Cauchy metric and the ingoing null metric. Initially there was no scalar field present on
either the ingoing or outgoing null patches. The initial values for the Bondi variables and
were
determined by matching to the Cauchy data at the matching surfaces and integrating the hypersurface
equations (16
, 17
).
As the evolution proceeds, the scalar field passes into the black hole, and the MTS grows outward. The
MTS is easily located in the spherically-symmetric case by an algebraic equation. In order to excise the
singular region, the grid points inside the MTS were identified and masked out of the evolution. The
backscattered radiation propagated cleanly across the outer matching surface to . The strategy worked
smoothly, and second order accuracy of the approach was established by comparing it to an independent
numerical solution obtained using a second-order accurate, purely Cauchy code [208]. As discussed in
Section 5.9, this inside-outside application of CCM has potential application to the binary black-hole
problem.
In a variant of this double CCM matching scheme, Lehner [196] has eliminated the middle Cauchy
region between and
in Figure 6
. He constructed a 1D code matching the ingoing and outgoing
characteristic evolutions directly across a single timelike worldtube. In this way, he was able to
simulate the global problem of a scalar wave falling into a black hole by purely characteristic
methods.
http://www.livingreviews.org/lrr-2012-2 |
Living Rev. Relativity 15, (2012), 2
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