These instabilities came as a rude shock and led to a retreat to the simpler problem of axisymmetric scalar waves propagating in Minkowski space, with the metric
in outgoing null cone coordinates. A null cone code for this
problem was constructed using an algorithm based upon Eq. (8), with the angular part of the flat space Laplacian replacing
the curvature terms in the integrand on the right hand side. This
simple setting allowed the instability to be traced to a subtle
violation of the CFL condition near the vertices of the cones. In
terms of grid spacings
, the CFL condition in this coordinate system takes the explicit
form
where the coefficient
K, which is of order 1, depends on the particular startup
procedure adopted for the outward integration. Far from the
vertex, the condition (10) on the time step
is quantitatively similar to the CFL condition for a standard
Cauchy evolution algorithm in spherical coordinates. But
condition (10
) is strongest near the vertex of the cone where (at the equator
) it implies that
This is in contrast to the analogous requirement
for stable Cauchy evolution near the origin of a spherical
coordinate system. The extra power of
is the price that must be paid near the vertex for the
simplicity of a characteristic code. Nevertheless, the
enforcement of this condition allowed efficient global simulation
of axisymmetric scalar waves. Global studies of backscattering,
radiative tail decay and solitons were carried out for nonlinear
axisymmetric waves [92
], but three-dimensional simulations extending to the vertices of
the cones were impractical on existing machines.
Aware now of the subtleties of the CFL condition near the
vertices, the Pittsburgh group returned to the Bondi problem,
i.e. to evolve the Bondi metric [35]
by means of the three hypersurface equations
and the evolution equation
The beauty of the Bondi equations is that they form a clean
hierarchy. Given
on an initial null hypersurface, the equations can be integrated
radially to determine
,
U,
V
and
on the hypersurface (in that order) in terms of integration
constants determined by boundary conditions, or smoothness if
extended to the vertex of a null cone. The initial data
is unconstrained except by smoothness conditions. Because
represents a spin-2 field, it must be
near the poles of the spherical coordinates and must consist of
spin-2 multipoles.
In the computational implementation of this system by the
Pittsburgh group [74], the null hypersurfaces were chosen to be complete null cones
with nonsingular vertices, which (for simplicity) trace out a
geodesic worldline
r
=0. The smoothness conditions at the vertices were formulated in
local Minkowski coordinates.
The vertices of the cones were not the chief source of
difficulty. A null parallelogram marching algorithm, similar to
that used in the scalar case, gave rise to an instability that
sprang up throughout the grid. In order to reveal the source of
the instability, physical considerations suggested looking at the
linearized version of the Bondi equations, since they must be
related to the wave equation. If this relationship were
sufficiently simple, then the scalar wave algorithm could be used
as a guide in stabilizing the evolution of
. A scheme for relating
to solutions
of the wave equation had been formulated in the original paper
by Bondi, Metzner and van den Burgh [35]. However, in that scheme, the relationship of the scalar wave
to
was nonlocal in the angular directions and was not useful for
the stability analysis.
A local relationship between
and solutions of the wave equation was found [74
]. This provided a test bed for the null evolution algorithm
similar to the Cauchy test bed provided by Teukolsky waves [140]. More critically, it allowed a simple von Neumann linear
stability analysis of the finite difference equations, which
revealed that the evolution would be unstable if the metric
quantity
U
was evaluated on the grid. For a stable algorithm, the grid
points for
U
must be staggered between the grid points for
,
and
V
. This unexpected feature emphasizes the value of linear
stability analysis in formulating stable finite difference
approximations.
It led to an axisymmetric code for the global Bondi problem
which ran stably, subject to a CFL condition, throughout the
regime in which caustics and horizons did not form. Stability in
this regime was verified experimentally by running arbitrary
initial data until it radiated away to
. Also, new exact solutions as well as the linearized null
solutions were used to perform extensive convergence tests that
established second order accuracy. The code generated a large
complement of highly accurate numerical solutions for the class
of asymptotically flat, axisymmetric vacuum spacetimes, a class
for which no analytic solutions are known. All results of
numerical evolutions in this regime were consistent with the
theorem of Christodoulou and Klainerman [44] that weak initial data evolve asymptotically to Minkowski space
at late time.
An additional global check on accuracy was performed using Bondi's formula relating mass loss to the time integral of the square of the news function. The Bondi mass loss formula is not one of the equations used in the evolution algorithm but follows from those equations as a consequence of a global integration of the Bianchi identities. Thus it not only furnishes a valuable tool for physical interpretation but it also provides a very important calibration of numerical accuracy and consistency.
An interesting feature of the evolution arises in regard to
compactification. By construction, the
u
-direction is timelike at the origin where it coincides with the
worldline traced out by the vertices of the outgoing null cones.
But even for weak fields, the
u
-direction generically becomes spacelike at large distances along
an outgoing ray. Geometrically, this reflects the property that
is itself a null hypersurface so that all internal directions
are spacelike, except for the null generator. For a flat space
time, the
u
-direction picked out at the origin leads to a null evolution
direction at
, but this direction becomes spacelike under a slight deviation
from spherical symmetry. Thus the evolution generically becomes
``superluminal'' near
. Remarkably, there were no adverse numerical effects. This
fortuitous property apparently arises from the natural way that
causality is built into the marching algorithm so that no
additional resort to numerical techniques, such as ``causal
differencing'' [141
], was necessary.
The code was intended to study gravitational waves from an
axisymmetric star. Since only the vacuum equations are evolved,
the outgoing radiation from the star is represented by data (
in Newman-Penrose notation) on an ingoing null cone forming the
inner boundary of the evolved domain. The inner boundary data is
supplemented by Schwarzschild data on the initial outgoing null
cone, which models an initially quiescent state of the star. This
provides the necessary data for a double-null initial value
problem. The evolution would normally break down where the
ingoing null hypersurface develops caustics. But by choosing a
scenario in which a black hole is formed, it is possible to
evolve the entire region exterior to the horizon. An obvious test
bed is the Schwarzschild spacetime for which a numerically
satisfactory evolution was achieved (convergence tests were not
reported).
Physically interesting results were obtained by choosing data
corresponding to an outgoing quadrupole pulse of radiation. By
increasing the initial amplitude of the data
, it was possible to evolve into a regime where the energy loss
due to radiation was large enough to drive the total Bondi mass
negative. Although such data is too grossly exaggerated to be
consistent with an astrophysically realistic source, the
formation of a negative mass is an impressive test of the
robustness of the code.
The code is designed to insure Bondi coordinate conditions at
infinity, so that the metric has the asymptotically Minkowskian
form corresponding to null-spherical coordinates. In order to
achieve this, the hypersurface equation for the Bondi metric
variable
must be integrated radially inward from infinity, where the
integration constant is specified. The evolution of the dynamical
variables proceeds radially outward as dictated by
causality [115
]. This differs from the Pittsburgh code in which all the
equations are integrated radially outward, so that the coordinate
conditions are determined at the inner boundary and the metric is
asymptotically flat but not asymptotically Minkowskian. The
Southampton scheme simplifies the formulae for the Bondi news
function and mass in terms of the metric. It is anticipated that
the inward integration of
causes no numerical problems because this is a gauge choice
which does not propagate physical information. However, the code
has not yet been subject to convergence and long term stability
tests so that these issues cannot be properly assessed at the
present time.
The matching of the Southampton axisymmetric code to a Cauchy interior is discussed in Sec. 4.5 .
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Characteristic Evolution and Matching
Jeffrey Winicour http://www.livingreviews.org/lrr-2001-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |