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 is any vector field tangent to the worldtube and is the unit normal to the worldtube. Since , we can further replace (43) by the worldtube condition on the Ricci tensor The Ricci identity then gives rise to the strict Komar conservation law [181]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
the evolution equation (39) takes the form where represents the right-hand-side terms. We add angular dissipation to the -evolution through the modification where is the discretization size and is an adjustable parameter independent of . This leads to Integration over the unit sphere with solid angle element then gives Thus, the -term has the effect of damping high-frequency noise as measured by the norm of over the sphere.Similarly, dissipation can be introduced in the radial integration of (48) through the substitution
with . Angular dissipation can also be introduced in the hypersurface equations, e.g., in Equation (38) 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
was introduced to reduce Equation (37) to two first-order radial equations, which simplified the startup procedure at the boundary. Although the resulting code was verified to be stable and second-order accurate, its application to problems involving strong fields and gradients led to numerical errors, which made small-scale effects of astrophysical importance difficult to measure.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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