5.5 Cauchy-characteristic matching for 1D gravitational systems

The first numerical implementations of CCM were 1D feasibility studies. These model problems provided a controlled environment for the development of CCM, in which either exact solutions or independent numerical solutions were known. In the following studies CCM worked like a charm in a variety of 1D applications, i.e., the matched evolutions were essentially transparent to the presence of the interface.

5.5.1 Cylindrical matching

The Southampton group chose cylindrically-symmetric systems as their model problem for developing matching techniques. In preliminary work, they showed how CCM could be consistently carried out for a scalar wave evolving in Minkowski spacetime but expressed in a nontrivial cylindrical coordinate system [89].

They then tackled the gravitational problem. First they set up the analytic machinery necessary for investigating cylindrically-symmetric vacuum spacetimes [90]. Although the problem involves only one spatial dimension, there are two independent modes of polarization. The Cauchy metric was treated in the Jordan–Ehlers–Kompaneets canonical form, using coordinates (t,r,ϕ, z) adapted to the (ϕ,z) cylindrical symmetry. The advantage here is that u = t − r is then a null coordinate, which can be used for the characteristic evolution. They successfully recast the equations in a suitably regularized form for the compactification of + ℐ in terms of the coordinate ∘ ---- y = 1∕r. The simple analytic relationship between Cauchy coordinates (t,r) and characteristic coordinates (u, y) facilitated the translation between Cauchy and characteristic variables on the matching worldtube, given by r = const.

Next they implemented the scheme as a numerical code. The interior Cauchy evolution was carried out using an unconstrained leapfrog scheme. It is notable that they report no problems with instability, which have arisen in other attempts at unconstrained leapfrog evolution in general relativity. The characteristic evolution also used a leapfrog scheme for the evolution between retarded time levels u, while numerically integrating the hypersurface equations outward along the characteristics.

The matching interface was located at points common to both the Cauchy and characteristic grids. In order to update these points by Cauchy evolution, it was necessary to obtain field values at the Cauchy “ghost” points, which lie outside the worldtube in the characteristic region. These values were obtained by interpolation from characteristic grid points (lying on three levels of null hypersurfaces in order to ensure second-order accuracy). Similarly, the boundary data for starting up the characteristic integration was obtained by interpolation from Cauchy grid values inside the worldtube.

The matching code was first tested [102] using exact Weber–Wheeler cylindrical waves [301], which come in from ℐ−, pass through the symmetry axis and expand out to ℐ+. The numerical errors were oscillatory with low growth rate, and second-order convergence was confirmed. Of special importance, little numerical noise was introduced by the interface. Comparisons of CCM were made with Cauchy evolutions using a standard outgoing radiation boundary condition [230]. At high amplitudes the standard condition developed a large error very quickly and was competitive only for weak waves with a large outer boundary. In contrast, the matching code performed well even with a small matching radius. Some interesting simulations were presented in which an outgoing wave in one polarization mode collided with an incoming wave in the other mode, a problem studied earlier by pure Cauchy evolution [232]. The simulations of the collision were qualitatively similar in these two studies.

The Weber–Wheeler waves contain only one gravitational degree of freedom. The code was next tested [98] using exact cylindrically-symmetric solutions, due to Piran, Safier and Katz [231], which contain both degrees of freedom. These solutions are singular at + ℐ so that the code had to be suitably modified. Relative errors of the various metric quantities were in the range 10–4 to 10–2. The convergence rate of the numerical solution starts off as second order but diminishes to first order after long time evolution. This performance could perhaps be improved by incorporating subsequent improvements in the characteristic code made by Sperhake, Sjödin, and Vickers (see Section 3.1).

5.5.2 Spherical matching

A joint collaboration between groups at Pennsylvania State University and the University of Pittsburgh applied CCM to the EKG system with spherical symmetry [138]. This model problem allowed simulation of black-hole formation as well as wave propagation.

The geometrical setup is analogous to the cylindrically-symmetric problem. Initial data were specified on the union of a spacelike hypersurface and a null hypersurface. The evolution used a 3-level Cauchy scheme in the interior and a 2-level characteristic evolution in the compactified exterior. A constrained Cauchy evolution was adopted because of its earlier success in accurately simulating scalar wave collapse [80]. Characteristic evolution was based upon the null parallelogram algorithm (19View Equation). The matching between the Cauchy and characteristic foliations was achieved by imposing continuity conditions on the metric, extrinsic curvature and scalar field variables, ensuring smoothness of fields and their derivatives across the matching interface. The extensive analytical and numerical studies of this system in recent years aided the development of CCM in this non-trivial geometrical setting by providing basic knowledge of the expected physical and geometrical behavior, in the absence of exact solutions.

The CCM code accurately handled wave propagation and black-hole formation for all values of M ∕R at the matching radius, with no symptoms of instability or back-reflection. Second-order accuracy was established by checking energy conservation.

5.5.3 Excising 1D black holes

In further developmental work on the EKG model, the Pittsburgh group used CCM to formulate a new treatment of the inner Cauchy boundary for a black-hole spacetime [141Jump To The Next Citation Point]. In the excision strategy, the inner boundary of the Cauchy evolution is located at an apparent horizon, which must lie inside (or on) the event horizon [300]. The physical rationale behind this apparent horizon boundary condition is that the truncated region of spacetime cannot causally affect the gravitational waves radiated to infinity. However, it should be noted that many Cauchy formalisms contain superluminal gauge or constraint violating modes so that this strategy is not always fully justified.

In the CCM excision strategy, illustrated in Figure 6View Image, the interior black-hole region is evolved using an ingoing null algorithm whose inner boundary is an MTS, and whose outer boundary lies outside the black hole and forms the inner boundary of a region evolved by the Cauchy algorithm. In turn, the outer boundary of the Cauchy region is handled by matching to an outgoing null evolution extending to ℐ+. Data are passed between the inner characteristic and central Cauchy regions using a CCM procedure similar to that already described for an outer Cauchy boundary. The main difference is that, whereas the outer Cauchy boundary data is induced from the Bondi metric on an outgoing null hypersurface, the inner Cauchy boundary is now obtained from an ingoing null hypersurface, which enters the event horizon and terminates at an MTS.

View Image

Figure 6: Black hole excision by matching. A Cauchy evolution, with data at t0 is matched across worldtubes R0 and R1 to an ingoing null evolution, with data at v0, and an outgoing null evolution, with data at u0. The ingoing null evolution extends to an inner trapped boundary Q, and the outgoing null evolution extends to ℐ+.

The translation from an outgoing to an incoming null evolution algorithm can be easily carried out. The substitution β → β + iπ∕2 in the 3D version of the Bondi metric (14View Equation) provides a simple formal recipe for switching from an outgoing to an ingoing null formalism [141Jump To The Next Citation Point].

In order to ensure that trapped surfaces exist on the ingoing null hypersurfaces, initial data were chosen, which guarantee black-hole formation. Such data can be obtained from initial Cauchy data for a black hole. However, rather than extending the Cauchy hypersurface inward to an apparent horizon, it was truncated sufficiently far outside the apparent horizon to avoid computational problems with the Cauchy evolution. The initial Cauchy data were then extended into the black-hole interior as initial null data until an MTS was reached. Two ingredients were essential in order to arrange this. First, the inner matching surface must be chosen to be convex, in the sense that its outward null normals uniformly diverge and its inner null normals uniformly converge. (This is trivial to satisfy in the spherically-symmetric case.) Given any physically-reasonable matter source, the focusing theorem guarantees that the null rays emanating inward from the matching sphere continue to converge until reaching a caustic. Second, the initial null data must lead to a trapped surface before such a caustic is encountered. This is a relatively easy requirement to satisfy because the initial null data can be posed freely, without any elliptic or algebraic constraints other than continuity with the Cauchy data.

A code was developed, which implemented CCM at both the inner and outer boundaries [141Jump To The Next Citation Point]. Its performance showed that CCM provides as good a solution to the black-hole excision problem in spherical symmetry as any previous treatment [262, 263, 208Jump To The Next Citation Point, 11]. CCM is computationally more efficient than these pure Cauchy approaches (fewer variables) and much easier to implement. Depending upon the Cauchy formalism adopted, achieving stability with a pure Cauchy scheme in the region of an apparent horizon can be quite tricky, involving much trial and error in choosing finite-difference schemes. There were no complications with stability of the null evolution at the MTS.

The Cauchy evolution was carried out in ingoing Eddington–Finklestein (IEF) coordinates. The initial Cauchy data consisted of a Schwarzschild black hole with an ingoing Gaussian pulse of scalar radiation. Since IEF coordinates are based on ingoing null cones, it is possible to construct a simple transformation between the IEF Cauchy metric and the ingoing null metric. Initially there was no scalar field present on either the ingoing or outgoing null patches. The initial values for the Bondi variables β and V were determined by matching to the Cauchy data at the matching surfaces and integrating the hypersurface equations (16View Equation, 17View Equation).

As the evolution proceeds, the scalar field passes into the black hole, and the MTS grows outward. The MTS is easily located in the spherically-symmetric case by an algebraic equation. In order to excise the singular region, the grid points inside the MTS were identified and masked out of the evolution. The backscattered radiation propagated cleanly across the outer matching surface to ℐ+. The strategy worked smoothly, and second order accuracy of the approach was established by comparing it to an independent numerical solution obtained using a second-order accurate, purely Cauchy code [208]. As discussed in Section 5.9, this inside-outside application of CCM has potential application to the binary black-hole problem.

In a variant of this double CCM matching scheme, Lehner [196] has eliminated the middle Cauchy region between R0 and R1 in Figure 6View Image. He constructed a 1D code matching the ingoing and outgoing characteristic evolutions directly across a single timelike worldtube. In this way, he was able to simulate the global problem of a scalar wave falling into a black hole by purely characteristic methods.

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