3.4 The Bondi Mass3 Characteristic Evolution Codes3.2 {2 + 1} Codes

3.3 The Bondi Problem

The first characteristic code based upon the original Bondi equations for a twist-free axisymmetric spacetime was constructed by Isaacson, Welling and Winicour [92Jump To The Next Citation Point In The Article] at Pittsburgh. The spacetime was foliated by a family of null cones, complete with point vertices at which regularity conditions were imposed. The code accurately integrated the hypersurface and evolution equations out to compactified null infinity. This allowed studies of the Bondi mass and radiation flux on the initial null cone, but it could not be used as a practical evolution code because of an instability at the vertices of the null cones.

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. (8Popup Equation), 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 tex2html_wrap_inline1941, 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 (10Popup Equation) on the time step tex2html_wrap_inline1945 is quantitatively similar to the CFL condition for a standard Cauchy evolution algorithm in spherical coordinates. But condition (10Popup Equation) is strongest near the vertex of the cone where (at the equator tex2html_wrap_inline1947) 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 tex2html_wrap_inline1949 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 [92Jump To The Next Citation Point In The Article], 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 [35Jump To The Next Citation Point In The Article]


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 tex2html_wrap_inline1951 on an initial null hypersurface, the equations can be integrated radially to determine tex2html_wrap_inline1863, U, V and tex2html_wrap_inline1959 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 tex2html_wrap_inline1951 is unconstrained except by smoothness conditions. Because tex2html_wrap_inline1951 represents a spin-2 field, it must be tex2html_wrap_inline1965 near the poles of the spherical coordinates and must consist of tex2html_wrap_inline1967 spin-2 multipoles.

In the computational implementation of this system by the Pittsburgh group [74Jump To The Next Citation Point In The Article], 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 tex2html_wrap_inline1951 . A scheme for relating tex2html_wrap_inline1951 to solutions tex2html_wrap_inline1975 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 tex2html_wrap_inline1951 was nonlocal in the angular directions and was not useful for the stability analysis.

A local relationship between tex2html_wrap_inline1951 and solutions of the wave equation was found [74Jump To The Next Citation Point In The Article]. 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 tex2html_wrap_inline1951, tex2html_wrap_inline1863 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 tex2html_wrap_inline1655 . 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 tex2html_wrap_inline1691 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 tex2html_wrap_inline1691, but this direction becomes spacelike under a slight deviation from spherical symmetry. Thus the evolution generically becomes ``superluminal'' near tex2html_wrap_inline1691 . 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'' [141Jump To The Next Citation Point In The Article], was necessary.

3.3.1 The Conformal-Null Tetrad Approach

J. Stewart has implemented a characteristic evolution code which handles the Bondi problem by a null tetrad, as opposed to metric, formalism [135Jump To The Next Citation Point In The Article]. The geometrical algorithm underlying the evolution scheme, as outlined in [136, 62], is H. Friedrich's [59Jump To The Next Citation Point In The Article] conformal-null description of a compactified spacetime in terms of a first order system of partial differential equations. The variables include the metric, the connection, and the curvature, as in a Newman-Penrose formalism, but in addition the conformal factor (necessary for compactification of tex2html_wrap_inline1691) and its gradient. Without assuming any symmetry, there are more than 7 times as many variables as in a metric based null scheme, and the corresponding equations do not decompose into as clean a hierarchy. This disadvantage, compared to the metric approach, is balanced by several advantages:
  1. The equations form a symmetric hyperbolic system so that standard theorems can be used to establish that the system is well-posed.
  2. Standard evolution algorithms can be invoked to ensure numerical stability.
  3. The extra variables associated with the curvature tensor are not completely excess baggage, since they supply essential physical information.
  4. The regularization necessary to treat tex2html_wrap_inline1691 is built in as part of the formalism so that no special numerical regularization techniques are necessary as in the metric case. (This last advantage is somewhat offset by the necessity of having to locate tex2html_wrap_inline1691 by tracking the zeroes of the conformal factor.)

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 (tex2html_wrap_inline2011 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 tex2html_wrap_inline2011, 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.

3.3.2 Twisting Axisymmetry 

The Southampton group, as part of its long range goal of combining Cauchy and characteristic evolution has developed a code [52Jump To The Next Citation Point In The Article, 53Jump To The Next Citation Point In The Article, 115Jump To The Next Citation Point In The Article] which extends the Bondi problem to full axisymmetry, as described by the general characteristic formalism of Sachs [122]. By dropping the requirement that the rotational Killing vector be twist-free, they are able to include rotational effects, including radiation in the ``cross'' polarization mode (only the ``plus'' mode is allowed by twist-free axisymmetry). The null equations and variables are recast into a suitably regularized form to allow compactification of null infinity. Regularization at the vertices or caustics of the null hypersurfaces is not necessary, since they anticipate matching to an interior Cauchy evolution across a finite worldtube.

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 tex2html_wrap_inline1863 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 [115Jump To The Next Citation Point In The Article]. 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 tex2html_wrap_inline1863 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 .

3.4 The Bondi Mass3 Characteristic Evolution Codes3.2 {2 + 1} Codes

image Characteristic Evolution and Matching
Jeffrey Winicour
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de