Inner Boundary and Coordinate Conditions

Excerpt from BBH GC Newsletter Volume 2, Number 9.

• the state of development of the 3D and 1D (hyperbolic) Empire codes
• the state of development of the 3D ADM code
• the state of incorporation of DAGH in the Empire and ADM codes
• summary of implementation of shift vector capability in Empire
• successful and failed tests with shift vectors in 1D Empire code
• summary of recent tests of advecting a black hole with the ADM code
• discussion of need for dynamic & more general shift and lapse conditions
• the issue of coordinate shocks in simulations & methods of avoidance
• discussion of causal differencing versus characteristic differencing
• problem of `upwind' direction change in characteristic differencing
• use of buffer points inside black hole surface or not
• inclusion of apparent horizon finders in evolution codes
• new estimates of computational demands of test problems & full inspiral
• discussion of presently available machines: processor numbers & memory

The shift vector terms have been added to the Empire code. Experiments with various shift vectors have been tried with the 1D Empire code with mixed success, but it seems clear that effort in 1D experiment should be kept because of the long term evolutions that one has access in this case.

Experiments with analytically known coordinates, e.g. Eddington-Finkelstein, are highly useful but insufficient. One of the most important outstanding theoretical issues for this year is how to specify dynamic lapse and shift conditions that, when coupled at the apparent horizon, lead to regular coordinates that maintain adequately resolved holes and exterior regions.

Elliptic shift conditions may be crucial to maintaining smooth spatial coordinates (avoiding coordinate shocks). Elliptic conditions, if solved exactly, will be very expensive computationally. However, it may be feasible to use inexpensive approximate solutions to the elliptic conditions and obtain adequate coordinate smoothness. Approximate solutions to any specific elliptic condition will not introduce inconsistencies as long as the conditions are not also incorporated elsewhere in the form of the evolution equations (as many of us used to do). Experiments in 1D can be important demonstrations of feasibility.

Lapse conditions or evolutions of the lapse (in the hyperbolic scheme) may also lead to time slice irregularities. These were discussed and a suggested solution is to `restart' the lapse if irregularities begin to develop (more difficult may be knowing when time slice irregularities are developing). Again, experiments in 1D can be very useful in testing this idea. Experiments in 1D seem also to indicate that because of diffusion alternatives to causal differencing need consideration. The Empire team are considering characteristic variable differencing schemes which have different dissipative behaviors. The issue of whether to use buffer zone points inside the apparent horizon or not requires also careful attention. There is not clear resolution of the issue. Restricting the calculations to only those points in the black hole exterior seems to be philosophically the `royal road' but the alternative may allow less complicated differencing and interpolation. For locating the apparent horizon these buffer points might be useful but their use in updating advection terms may lead to instabilities or acausal behavior depending on resolution and coordinate conditions.

The following is a list of a series of numerical experiments and code development goals that appear to be important in gaining understanding of how to treat black hole inner boundaries and in assessing which approach to the Einstein equations is numerically preferable.

• Numerical experiments in shift choices using the 1D Empire code
• Numerical experiments with lapse restarts in the 1D Empire code
• Rapid development of a general 1D ADM code
• Lapse and shift choice experiments with 1D ADM code
• Continued experiments translating BH with EF coordinates with ADM code
• Experiments translating BH with dynamic lapse and shift (ADM)
• Experiments translating BH with dynamic lapse and shift (Empire)
• Evolution of Kerr hole with analytic lapse and shift with 3D ADM code
• Generalize evolution of Kerr hole using dynamical lapse and shift (ADM)
• Generalize evolution of Kerr hole using dynamical lapse and shift (Empire)
• Incorporate DAGH in 3D ADM code
• Experiments with characteristic variables & differencing in 1D & 3D Empire
• Stabilization of the outer boundary

During that meeting the following recommendations were made:

• There is insufficient basis for judging the numerical superiority of either the Empire code or ADM code at present. Development of both should be pressed, with the earliest possible demonstration of results on key 1D and 3D experiments.
• Experiments in 1D are particularly important at this stage since much needs to be learned about viable coordinate choices and differencing schemes and the turn-around time on 3D experiments is very long at present. Experiments in 1D can rapidly determine which shift choices lead to irregular coordinates (shocks) and which do not. Experiments in 1D can test lapse restarts as a means of avoiding the development of irregular time slices. A general 1D ADM code should be quickly written for these purposes and for side-by-side tests (where possible) with the 1D Empire code.
• Using analytic coordinates (lapse and shift) in experiments in 3D with known solutions for a translating black hole or a Kerr hole is a very useful means of breaking the problem down and testing black hole excision and whether causality is maintained numerically at the inner boundary. The black hole translation experiment needs to be pressed further and it is important to begin a parallel set of experiments trying to evolve a Kerr hole for many crossing times.
• While experiments in 3D with translating a black hole or evolving a Kerr hole with analytic coordinates are useful in many ways, such as testing black hole excision and numerical causality, these coordinates are of no use in the general case of a translating, rotating, and accelerating hole. Careful thought needs to be given to what coupled choice of the lapse and shift at the apparent horizon will capture the key properties of, for example, Eddington-Finkelstein coordinates for a non-rotating hole. This is a major outstanding theoretical issue that now must be tackled.
• We need to focus on the core elements of the problem and assemble teams to tackle some or all of the numerical experiments and theoretical issues listed above. Test (beta) versions of 1D and 3D codes are available and can be used for experimentation while further development proceeds simultaneously. The Syracuse meeting is an opportunity to try to assemble teams interested in this area.