So far, the discretization of the equations has
been rather straightforward. One of the schemes that has been used
is the higher dimensional Lax-Wendroff scheme, also called the
rotated Richtmyer scheme, a discretization scheme with second order
accuracy. It has been employed alone [46] or together with
Strang splitting [95
] to treat the
principal part of the equations differently from the source part.
Since a second order scheme requires much more computing resources
compared to higher order methods to achieve the same accuracy,
Hübner started to use the method of lines [86
] with a fourth order
scheme to compute the spatial derivatives and fourth order
Runge-Kutta for the evolution in time. He reports [95
] that the fourth
order method is very much superior to the second order scheme in
terms of efficiency. (The feasibility of the method of lines in
relativity has been studied by Hungerbühler [97],
using pseudo-spectral methods for the spatial derivatives and a
combination of Adams-Bashforth and Adams-Moulton schemes for the
time evolution.) A similar method has been employed by Frauendiener
and Hein [52
] in a code that
evolves the conformal field equations under the assumption of
axi-symmetry. The code makes use of the “cartoon method” [2] (see
also [50]),
which can be used to treat axi-symmetric systems using Cartesian
coordinates. The code has been used to reproduce exact solutions
such as the boost-rotation symmetric solutions of Bičák and
Schmidt [21, 22
]. It allows one to
compute the entire future of an initial hyperboloidal data set
within that class up to time-like infinity
.
The conformal field equations and the propagation
equations derived from them are quasi-linear. This implies that the
characteristics of the system depend on the current solution, and
this, in turn, means that one has to be able to change the
time-step between successive time-slices in order
to keep a stable evolution scheme. This is necessary because the
schemes are explicit schemes and, therefore, subject to the
Courant-Friedrichs-Lewy condition [37],
which states that the numerical domain of dependence of a point
should always include the analytical domain of dependence. This
requirement already excludes the popular leap-frog scheme, which is
nevertheless used sometimes also for evolving the Einstein
equations. A general criterion for computing the maximal time-step
allowed in each iteration in arbitrary dimension has been derived
in [46
].
Another important point in the development of evolution codes is the numerical treatment of the boundaries. As explained already above, it is one of the advantages in the conformal approach that the outer boundary is not as influential as it is in the conventional approach using the standard Cauchy problem. It was also pointed out that in this case it is enough to impose a boundary condition that results in a numerically stable code because the outer boundary is located in the unphysical region and, therefore, cannot influence the physical space-time.
The proper way to treat the boundary is to
prescribe conditions that are compatible with the full conformal
field equations, in particular with their restriction to the
boundary manifold. This has not been done so far. Since the outer
boundary is not important for the physical effects, other ways of
dealing with the boundary have been devised. One way is to forget
about the restrictions of the conformal field equations to the
boundary and to analyze the possible boundary conditions for the
propagation equations. To first order, one can define ingoing and
outgoing fields on the boundary. Then a sufficiently general
boundary condition will be obtained by specifying the ingoing
fields in terms of the outgoing ones. Although this boundary
treatment is not necessarily compatible with the restriction of the
conformal field equations to the boundary, it is compatible with
the evolution equations. This means that the evolution can remain
stable, although the solution will not satisfy the constraints in
the domain of influence of the boundary, which, however, is always
in the unphysical part of space-time. This method has been used
in [46] with satisfactory
results. In particular, the boundary did not give rise to
non-physical modes. These findings are in agreement with the
analysis of numerical boundary conditions by Trefethen [158].
Another method for dealing with the boundary has been found by Hübner [94]. He realized that it is sufficient to solve the conformal field equations in the physical space-time only, and not necessary to solve them in the unphysical region as long as the characteristics remain such that the information created in the unphysical part of the computational domain cannot reach the physical part. Consequently, in his treatment the grid is divided into three zones: the inner zone, the outer zone, and a transition zone. The inner zone covers the physical space-time (flagged by a positive conformal factor) and some part of the adjacent unphysical region. On this part of the grid the conformal field equations are solved. In the outer zone, which is located in a neighbourhood of the grid boundary, one solves an advection equation that propagates outwards, off the grid. In the transition zone, a sufficiently smooth interpolation between these two systems of equations is solved. The effect is that the boundary condition that has to be imposed on the grid boundaries is very simple, and that the noise generated in the transition region is propagated away from the physical region outward towards the grid boundary.
Our next point is concerned with the extraction
of the radiative information from the numerically generated data.
This is the part of the entire numerical process where the
superiority of the conformal approach becomes apparent. How does
one determine the radiative field? First of all, one needs to find
on the current time-slice. Since
is the surface on which the conformal factor
vanishes and since
is explicitly
known during the evolution, the location of
is a simple task. The next problem is concerned with
the orientation of the tetrad on
. The asymptotic quantities
are defined with respect to a specific geometrically characterized
tetrad, a Bondi frame. But, in general, this tetrad is completely
unrelated to the “computational” tetrad used for the evolution.
Therefore, one needs to find the transformation from one to the
other at each point of
. Without going into too many
details (see [46
, 49
, 48
]), we remark that
most asymptotic quantities, in particular the radiation field, are
of a local character, so they can be read off without constructing
a Bondi frame. This is rather fortunate because there are global
issues involved in the transformation from the computational tetrad
to the Bondi frame. These have implications for the determination
of global quantities like the Bondi energy-momentum four-vector,
but they have no effect on the radiation field, which is defined as
that (complex) component of
that is entirely intrinsic
to
:
Here, is a null-vector tangent to
the generators of
, i.e.
, and
is any complex space-like null-vector that is
orthogonal to
. It is useful to require the space-like
vector to be tangent to the intersection of
with the current time-slice. Augmenting these two
vectors by a further real null-vector
yields a
null-tetrad which is fixed up to rotations in the (two-dimensional)
tangent space of that intersection and boosts in the plane
orthogonal to it. The behaviour of
under these
transformations is that of a GHP-weighted quantity [80, 134] with boost
weight
and spin weight
. This corresponds to the
quadrupole-like character of the gravitational radiation field.
However,
really depends only on the null-vector
. Suppose we perform a null-rotation around
, then
transforms into
for some complex valued function
on
. But
is invariant under this transformation.
So in order to find
it is only necessary to transform from
the given computational tetrad to the tetrad specified above, which
is rather straightforward. In fact, the computation of
involves only the combination of certain components
of the gravitational field with powers of
.
The final step in the correct determination of
the radiation is to find the correct time parameter. Suppose we
follow a specific null generator of crossing through
successive time-slices. On each slice we compute
on that generator. Then we obtain the radiation
emitted by the source into the direction specified by the generator
as a function of our computational time parameter. Since the time
coordinate is rather arbitrary, this means that the waveform
determined so far has no physical meaning. The problem is already
present in Maxwell’s theory: Suppose we have an emitter that sends
out a pure sine wave. A detector far away from the source cannot
determine the absolute frequency of the signal because the relative
velocity of emitter and receiver might be non-zero, but the
detector should also find a pure sine signal. However, this will be
true only if the detector records the signal as a function of
proper time. Any other time parameter along the detector’s
world-line will not produce a pure sine.
What one needs to do in the general case is to select among all parameters along the generator a specific, geometrically distinguished one, namely a Bondi parameter. A generator and such a parameter along it can be understood as a certain limit of freely falling observers with proper time clocks as they move towards infinity [49]. Bondi parameters are obtained as solutions of an ordinary, linear, second order differential equation, which is conformally invariant.
The computation of the Bondi energy-momentum is a
global procedure, i.e. it depends on properties of the entire
cut of with the current time-slice. There are two steps
involved in this procedure. First, one needs to obtain the
asymptotic translation group (see e.g. [135]) on each cut.
This provides four functions on the cut which are then, in a second
step, integrated against the “mass aspect” which is another
function obtained from the “Coulomb” part
of the gravitational field, and the “news function,”
which is a combination of components of the Ricci tensor and
connection coefficients. The first step, the determination of the
translation group, is the global step because it involves solving a
second order elliptic equation on the cut. These issues are
discussed in more detail in [48].
All these procedures for finding the relevant
data on have been worked out analytically and they have also
been tested (at least in part) numerically [46
]. The tests have
been performed under the assumption that null-infinity admits
toroidal cuts, which has the advantage that one can actually
compare the numerical results with analytical expressions because a
whole class of exact solutions [152
, 93
] is known to exist.
Admittedly, such space-times are rather unphysical, but since most
of the extraction procedures are local there is no doubt that they
will also work in more realistic cases.