The Canberra code employs a null quasi-spherical (NQS) gauge (not to be confused with the
quasi-spherical approximation in which quadratically-aspherical terms are ignored [56]). The NQS gauge
takes advantage of the possibility of mapping the angular part of the Bondi metric conformally onto a
unit-sphere metric, so that
. The required transformation
is
in general dependent upon
and
so that the NQS angular coordinates
are not
constant along the outgoing null rays, unlike the Bondi–Sachs angular coordinates. Instead
the coordinates
display the analogue of a shift on the null hypersurfaces
.
In addition, the NQS spheres
are not the same as the Bondi spheres. The
radiation content of the metric is contained in a shear vector describing this shift. This results in
the description of the radiation in terms of a spin-weight 1 field, rather than the spin-weight
2 field associated with
in the Bondi–Sachs formalism. In both the Bondi–Sachs and
NQS gauges, the independent gravitational data on a null hypersurface is the conformal part of
its degenerate 3-metric. The Bondi–Sachs null data consist of
, which determines the
intrinsic conformal metric of the null hypersurface. In the NQS case,
and the shear
vector comprises the only non-trivial part of the conformal 3-metric. Both the Bondi–Sachs and
NQS gauges can be arranged to coincide in the special case of shear-free Robinson–Trautman
metrics [95, 32
].
The formulation of Einstein’s equations in the NQS gauge is presented in [31], and the
associated gauge freedom arising from dependent rotation and boosts of the unit sphere is
discussed in [32]. As in the PITT code, the main equations involve integrating a hierarchy of
hypersurface equations along the radial null geodesics extending from the inner boundary to
null infinity. In the NQS gauge the source terms for these radial ODEs are rather simple when
the unknowns are chosen to be the connection coefficients. However, as a price to pay for this
simplicity, after the radial integrations are performed on each null hypersurface, a first-order
elliptic equation must be solved on each
cross-section to reconstruct the underlying
metric.
The components of Einstein’s equations independent of the hypersurface and evolution equations,
were called supplementary conditions by Bondi et al. [63] and Sachs [258]. They showed that the Bianchi identityAs a result, the supplementary conditions can be replaced by the condition that the Einstein tensor satisfy
on the worldtube, where These conservation laws (44) can also be expressed in terms of the intrinsic metric of the
worldtube.
The worldtube conservation laws can also be interpreted as a symmetric hyperbolic system governing the evolution of certain components of the extrinsic curvature [306]. This leads to the
Worldtube Theorem:
Given ,
and
, the worldtube constraints constitute a well-posed initial-value problem,
which determines the remaining components of the extrinsic curvature
.
These extrinsic curvature components are related to the integration constants for the Bondi–Sachs
system, which leads to possible applications of the worldtube theorem. One application is to waveform
extraction. In that case, the data necessary to apply the worldtube theorem are
supplied by the numerical results of a 3 + 1 Cauchy evolution. The remaining components of the
extrinsic curvature can then be determined by means of a well-posed initial-value problem on the
boundary. The integration constants
, for the Bondi–Sachs equations at
are then determined. This approach can be used to enforce the constraints in the
numerical computation of waveforms at
by means of Cauchy-characteristic extraction (see
Section 6).
Another possible application is to the characteristic initial-boundary value problem, for which boundary data consistent with the constraints must be prescribed a priori, i.e., independent of the evolution. The object is to obtain a well-posed version of the characteristic initial-boundary value problem. However, the complicated coupling between the Bondi–Sachs evolution system and the boundary constraint system prevents any definitive results.
For a {3 + 1} evolution algorithm based upon a system of wave equations, or any other symmetric
hyperbolic system, numerical dissipation can be added in the standard Kreiss–Oliger form [186].
Dissipation cannot be added to the {2 + 1 + 1} format of characteristic evolution in this standard way for
{3 + 1} Cauchy evolution. In the original version of the PITT code, which used square stereographic
patches with boundaries aligned with the grid, numerical dissipation was only introduced in the
radial direction [195]. This was sufficient to establish numerical stability. In the new version of
the code with circular stereographic patches, whose boundaries fit into the stereographic grid
in an irregular way, angular dissipation is necessary to suppress the resulting high-frequency
error.
Angular dissipation can be introduced in the following way [13]. In terms of the spin-weight 2 variable
Similarly, dissipation can be introduced in the radial integration of (48) through the substitution
The PITT code was originally formulated in the second differential form of Equations (36, 37
, 38
, 39
),
which in the spin-weighted version leads to an economical number of 2 real and 2 complex variables.
Subsequently, the variable
In particular, in initial attempts to simulate a white-hole fission, Gómez [133] encountered an
oscillatory error pattern in the angular directions near the time of fission. The origin of the problem was
tracked to numerical error of an oscillatory nature introduced by
terms in the hypersurface and
evolution equations. Gómez’s solution was to remove the offending second angular derivatives by
introducing additional variables and reducing the system to first differential order in the angular directions.
This suppressed the oscillatory mode and subsequently improved performance in the simulation of the
white-hole fission problem [136
] (see Section 4.4.2).
This success opens the issue of whether a completely first differential order code might perform even
better, as has been proposed by Gómez and Frittelli [135]. By the use of as a fundamental
variable, they cast the Bondi system into Duff’s first-order quasilinear canonical form [103
]. At the analytic
level this provides standard uniqueness and existence theorems (extending previous work for
the linearized case [124]) and is a starting point for establishing the estimates required for
well-posedness.
At the numerical level, Gómez and Frittelli point out that this first-order formulation provides a bridge
between the characteristic and Cauchy approaches, which allows application of standard methods for
constructing numerical algorithms, e.g., to take advantage of shock-capturing schemes. Although true
shocks do not exist for vacuum gravitational fields, when coupled to hydro the resulting shocks couple back
to form steep gradients, which might not be captured by standard finite-difference approximations. In
particular, the second derivatives needed to compute gravitational radiation from stellar oscillations
have been noted to be a troublesome source of inaccuracy in the characteristic treatment of
hydrodynamics [270]. Application of standard versions of AMR is also facilitated by the first-order
form.
The benefits of this completely first-order approach are not simple to decide without code
comparison. The part of the code in which the operator introduced the oscillatory error mode
in [133] was not identified, i.e., whether it originated in the inner boundary treatment or in the
interpolations between stereographic patches where second derivatives might be troublesome.
There are other possible ways to remove the oscillatory angular modes, such as adding angular
dissipation (see Section 4.2.2). The finite-difference algorithm in the original PITT code only
introduced numerical dissipation in the radial direction [195
]. The economy of variables and other
advantages of a second-order scheme [187] should not be abandoned without further tests and
investigation.
The PITT code is an explicit finite-difference evolution algorithm based upon retarded time steps on a uniform three-dimensional null coordinate grid based upon the stereographic coordinates and a compactified radial coordinate. The straightforward numerical implementation of the finite-difference equations has facilitated code development. The Canberra code uses an assortment of novel and elegant numerical methods. Most of these involve smoothing or filtering and have obvious advantage for removing short wavelength noise but would be unsuitable for modeling shocks.
There have been two recent projects, to improve the performance of the PITT code by using the cubed-sphere method to coordinatize the sphere. They both include an adaptation of the eth-calculus to handle the transformation of spin-weighted variables between the six patches.
In one of these projects, Gómez, Barreto and Frittelli develop the cubed-sphere approach into an
efficient, highly parallelized 3D code, the LEO code, for the characteristic evolution of the coupled
Einstein–Klein–Gordon equations in the Bondi–Sachs formalism [134]. This code was demonstrated to be
convergent and its high accuracy in the linearized regime with a Schwarzschild background was
demonstrated by the simulation of the quasinormal ringdown of the scalar field and its energy-momentum
conservation.
Because the characteristic evolution scheme constitutes a radial integration carried out for each angle on
the sphere of null directions, the natural way to parallelize the code is to distribute the angular grid among
processors. Thus, given processors one can distribute the
points in each spherical
patch (cubed-sphere or stereographic), assigning to each processor equal square grids of extent
in each direction. To be effective this requires that the communication time between
processors scales effectively. This depends upon the ghost point location necessary to supply
nearest neighbor data and is facilitated in the cubed-sphere approach because the ghost points
are aligned on one-dimensional grid lines, whose pattern is invariant under grid size. In the
stereographic approach, the ghost points are arranged in an irregular pattern, which changes in
an essentially random way under rescaling and requires a more complicated parallelization
algorithm.
Their goal is to develop the LEO code for application to black-hole–neutron-star binaries in a close orbit regime, where the absence of caustics make a pure characteristic evolution possible. Their first anticipated application is the simulation of a boson star orbiting a black hole, whose dynamics is described by the Einstein–Klein–Gordon equations. They point out that characteristic evolution of such systems of astrophysical interest have been limited in the past by resolution due to the lack of necessary computational power, parallel infrastructure and mesh refinement. Most characteristic code development has been geared toward single processor machines, whereas the current computational platforms are designed toward performing high-resolution simulations in reasonable times by parallel processing.
At the same time the LEO code was being developed, Reisswig et al. [242] also constructed a
characteristic code for the Bondi–Sachs problem based upon the cubed-sphere infrastructure of
Thornburg [295, 294]. They retain the original second-order differential form of the angular
operators.
The Canberra code handles fields on the sphere by means of a 3-fold representation: (i) as discretized
functions on a spherical grid uniformly spaced in standard coordinates, (ii) as fast-Fourier
transforms with respect to
(based upon the smooth map of the torus onto the sphere), and
(iii) as a spectral decomposition of scalar, vector, and tensor fields in terms of spin-weighted
spherical harmonics. The grid values are used in carrying out nonlinear algebraic operations;
the Fourier representation is used to calculate
-derivatives; and the spherical harmonic
representation is used to solve global problems, such as the solution of the first-order elliptic
equation for the reconstruction of the metric, whose unique solution requires pinning down
the
gauge freedom. The sizes of the grid and of the Fourier and spherical-harmonic
representations are coordinated. In practice, the spherical-harmonic expansion is carried out
to 15th order in
, but the resulting coefficients must then be projected into the
subspace in order to avoid inconsistencies between the spherical harmonic, grid, and Fourier
representations.
The Canberra code solves the null hypersurface equations by combining an eighth-order Runge–Kutta
integration with a convolution spline to interpolate field values. The radial grid points are dynamically
positioned to approximate ingoing null geodesics, a technique originally due to Goldwirth and
Piran [132] to avoid the problems with a uniform -grid near a horizon, which arise from the
degeneracy of an areal coordinate on a stationary horizon. The time evolution uses the method of
lines with a fourth-order Runge–Kutta integrator, which introduces further high frequency
filtering.
In addition to these testbeds, a set of linearized solutions has recently been obtained in the
Bondi–Sachs gauge for either Schwarzschild or Minkowski backgrounds [47]. The solutions are
generated by the introduction of a thin shell of matter whose density varies with time and angle. This
gives rise to an exterior field containing gravitational waves. For a Minkowski background, the
solution is given in exact analytic form and, for a Schwarzschild background, in terms
of a power series. The solutions are parametrized by frequency and spherical harmonic
decomposition. They supply a new and very useful testbed for the calibration and further
development of characteristic evolution codes for Einstein’s equations, analogous to the role of
the Teukolsky waves in Cauchy evolution. The PITT code showed clean second-order
convergence in both the
and
error norms in tests based upon waves in a Minkowski
background. However, in applications involving very high resolution or nonlinearity, there
was excessive short wavelength noise, which degraded convergence. Recent improvements
in the code [16
] have now established clean second-order convergence in the nonlinear
regime.
It would be of great value to increase the accuracy of the code to higher order. However, the marching algorithm, which combines the radial integration of the hypersurface and evolution equations does not fall into the standard categories that have been studied in computational mathematics. In particular, there are no energy estimates for the analytic problem, which would could serve as a guide to design a higher-order stable algorithm. This is a an important area for future investigation.
The designed convergence rate of the operator used in the LEO code was verified
for second-, fourth- and eighth-order finite-difference approximations, using the spin-weight
2 spherical harmonic
as a test. Similarly, the convergence of the integral relations
governing the orthonormality of the spin-weighted harmonics was verified. The code includes
coupling to a Klein–Gordon scalar field. Although convergence of the evolution code was
not explicitly checked, high accuracy in the linearized regime with Schwarzschild background
was demonstrated in the simulation of quasinormal ringdown of the scalar field and in the
energy-momentum conservation of the scalar field.
In practical runs, the major source of inaccuracy is the spherical-harmonic resolution, which was
restricted to by hardware limitations. Truncation of the spherical-harmonic expansion
has the effect of modifying the equations to a system for which the constraints are no longer
satisfied. The relative error in the constraints is 10–3 for strong field simulations [33
].
A natural physical application of a characteristic evolution code is the nonlinear version of
the classic problem of scattering off a Schwarzschild black hole, first solved perturbatively by
Price [238]. Here the inner worldtube for the characteristic initial-value problem consists of the
ingoing branch of the
hypersurface (the past horizon), where Schwarzschild data are
prescribed. The nonlinear problem of a gravitational wave scattering off a Schwarzschild black hole
is then posed in terms of data on an outgoing null cone, which describe an incoming pulse
with compact support. Part of the energy of this pulse falls into the black hole and part is
backscattered to
. This problem has been investigated using both the PITT and Canberra
codes.
The Pittsburgh group studied the backscattered waveform (described by the Bondi news function) as a
function of incoming pulse amplitude. The computational eth-module smoothly handled the complicated
time-dependent transformation between the non-inertial computational frame at and the inertial
(Bondi) frame necessary to obtain the standard “plus” and “cross” polarization modes. In the
perturbative regime, the news corresponds to the backscattering of the incoming pulse off the
effective Schwarzschild potential. When the energy of the pulse is no larger than the central
Schwarzschild mass, the backscattered waveform still depends roughly linearly on the amplitude of the
incoming pulse. However, for very high amplitudes the waveform behaves quite differently. Its
amplitude is greater than that predicted by linear scaling and its shape drastically changes
and exhibits extra oscillations. In this very high amplitude case, the mass of the system is
completely dominated by the incoming pulse, which essentially backscatters off itself in a nonlinear
way.
The Canberra code was used to study the change in Bondi mass due to the radiation [33]. The
Hawking mass was calculated as a function of radius and retarded time, with the
Bondi mass
then obtained by taking the limit
. The limit had good numerical
behavior. For a strong initial pulse with
angular dependence, in a run from
to
(in units where the interior Schwarzschild mass is 1), the Bondi mass dropped from 1.8 to
1.00002, showing that almost half of the initial energy of the system was backscattered and that a
surprisingly negligible amount of energy fell into the black hole. A possible explanation is that the
truncation of the spherical harmonic expansion cuts off wavelengths small enough to effectively
penetrate the horizon. The Bondi mass decreased monotonically in time, as necessary theoretically,
but its rate of change exhibited an interesting pulsing behavior whose time scale could not be
obviously explained in terms of quasinormal oscillations. The Bondi mass loss formula was
confirmed with relative error of less than 10–3. This is impressive accuracy considering the potential
sources of numerical error introduced by taking the limit of the Hawking mass with limited
resolution. The code was also used to study the appearance of logarithmic terms in the asymptotic
expansion of the Weyl tensor [37]. In addition, the Canberra group studied the effect of the
initial pulse amplitude on the waveform of the backscattered radiation, but did not extend their
study to the very high amplitude regime in which qualitatively interesting nonlinear effects
occur.
The PITT code has also been implemented to evolve along an advanced time foliation by ingoing
null cones, with data given on a worldtube at their outer boundary and on the initial ingoing
null cone. The code was used to evolve a black hole in the region interior to the worldtube by
implementing a horizon finder to locate the MTS on the ingoing cones and excising its singular
interior [141]. The code tracks the motion of the MTS and measures its area during the evolution.
It was used to simulate a distorted “black hole in a box” [139
]. Data at the outer worldtube
was induced from a Schwarzschild or Kerr spacetime but the worldtube was allowed to move
relative to the stationary trajectories; i.e., with respect to the grid the worldtube is fixed but
the black hole moves inside it. The initial null data consisted of a pulse of radiation, which
subsequently travels outward to the worldtube, where it reflects back toward the black hole. The
approach of the system to equilibrium was monitored by the area of the MTS, which also equals its
Hawking mass. When the worldtube is stationary (static or rotating in place), the distorted
black hole inside evolved to equilibrium with the boundary. A boost or other motion of the
worldtube with respect to the black hole did not affect this result. The MTS always reached
equilibrium with the outer boundary, confirming that the motion of the boundary was “pure
gauge”.
This was the first code that ran “forever” in a dynamic black-hole simulation, even when the worldtube
wobbled with respect to the black hole to produce artificial periodic time dependence. An initially distorted,
wobbling black hole was evolved for a time of , longer by orders of magnitude than permitted by
the stability of other existing black hole codes at the time. This exceptional performance opens a promising
new approach to handle the inner boundary condition for Cauchy evolution of black holes by the matching
methods reviewed in Section 5.
Note that setting the pulse to zero is equivalent to prescribing shear free data on the initial null cone. Combined with Schwarzschild boundary data on the outer worldtube, this would be complete data for a Schwarzschild space time. However, the evolution of such shear free null data combined with Kerr boundary data would have an initial transient phase before settling down to a Kerr black hole. This is because the twist of the shear-free Kerr null congruence implies that Kerr data specified on a null hypersurface are not generally shear free. The event horizon is an exception but Kerr null data on other null hypersurfaces have not been cast in explicit analytic form. This makes the Kerr spacetime an awkward testbed for characteristic codes. (Curiously, Kerr data on a null hypersurface with a conical type singularity do take a simple analytic form, although unsuitable for numerical evolution [108].) Using some intermediate analytic results of Israel and Pretorius [236], Venter and Bishop [59] have recently constructed a numerical algorithm for transforming the Kerr solution into Bondi coordinates and in that way provide the necessary null data numerically.
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Living Rev. Relativity 15, (2012), 2
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