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 , 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 
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 , 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 . 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 . This provided a test bed for the null evolution algorithm similar to the Cauchy test bed provided by Teukolsky waves . 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  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'' , 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 . 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 .
|Characteristic Evolution and Matching
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