1. Dynamic control of discrete constraint violations in 3D black hole evolutions
Manuel Tiglio
Hearne Institute for Theoretical Physics
Louisiana State University
Mexico, Dec 2003

2. The technique (M. Tiglio, gr-qc/0304062)
Given a set of evolution equations, add the constraints Ci = Ci (u, Du), to the right hand side
These evolution equations determine how these constraints propagate
Define an energy for these constraints Ec = (C,C). You can show that It satisfies the evolution equation
Choose m to get the desired behavior, e.g.
Do it on the fly, during evolution. In order to achieve symmetric hyperbolicity, do it for one resolution, interpolate, and keep that m(t) from there on. It is then a priori given.
If you control the constraints (=they do not grow) for one resolution, and you are in the convergence regime, you are controlling them for higher resolutions as well
The discretized constraints will not grow in time

3. Some general remarks
Instead of asking the constraints to be exactly satisfied, fix a tolerance value T, and ask the constraints to decay to T after na=1 timesteps:
That is, ask N(t+Dt)=T and solve for na=1 :
A good tolerance value is the initial discretization value for the constraints
Ideally, one would like na=1, in that way controlling the constraints at each timestep.
Although a fully discrete approach could be followed, here all these calculations are semidiscrete: space is assumed to be discrete (with gridspacing not necessariliy small) and time continuous. Will come back to this later.

4. DBF! DBF! A proof of concept: a spherically symmetric black hole.
Without constraint minimization, crashes at ~10M
DBF!
With constraint minimization, runs for at least 10,000M. Here na=1, and m is defined by the coarsest resolution run.

5. The 3D equations (O. Sarbach + M
The 3D equations (O. Sarbach + M. Tiglio, PRD 66, (2002), gr-qc/ )
Symmetric hyperbolic formulation with (a slight generalization of) the Bona-Masso slicing conditions.
Two completely free functions of time (and space, if one wants) to use to minimize the constraints growth, here called g and h.
It really trivial to add more 21 more free functions.

6. 3D black hole simulations (M. Tiglio, L. Lehner and D
3D black hole simulations (M. Tiglio, L. Lehner and D. Neilsen, gr-qc/ ). Very generous help from P. Diener and O. Sarbach.
Without constraint minimization (g=0=h).
The run with coarsest resolution crashes around 20M

7. Constraint minimization: how good is the semidiscrete picture?
Although fully discrete calculations could be
done, for simplicity we use semidiscrete ones. How accurately do they represent the fully discrete simulation?
Black hole simulation of the previous slide
Gauge wave
Time derivative of the constraints energy, using: a)the semidiscrete energy calculations.b) a posteriori computing the time derivative of the actual energy through finite differencing.
The agreement is remarkable. Actually not so surprising…

8. Dynamic minimization: preliminaries
Let’s warm up by keeping one of the free constraint-functions fixed, say h=0,
and dynamically defining g by setting a tolerance value, as described in the second slide. Here the boundaries are at 5M and he tolerance value for the energy here is equal to the initial discretization one.
One notices that large (but not too large) values of na perform better. Why? na=1 would be the ideal choice. However, small values of na make the free parameters vary too much in order to keep the constraints under control. Not possible with a fixed Courant factor.
On the other hand, large values of
na allow the constraints to grow toomuch between two iterations.

9. Resulting functions for two runs of the previous slide.
The titles are wrong. The minimization is done at every timestep, in one case with na =1, and in the with other na =1000.
Small values of na unfortunately cause the free constraint-function h to vary too much. What can we do about it?
One could fine-tune na, but this is against the whole spirit of minimizing the needed numerical experimentation.

10. Bi-parametric minimization
Having found the limitation of adjusting the equations to suppress instabilies, it is clear how to proceed:
Namely, use more than one free constraint-function, in order to avoid large or quick
variations in them. Two unkowns for one equation, can satisfy one more equation. Ask the code to choose the pair (h and g) that not only achieves the desired constraint behaviour, but also but minimizes the change in this pair between two timesteps.

11. Not only quite stable, but also very accurate.
Apparent horizon mass, using Jonathan Thornburg’s new fast apparent horizon finder, for the runs of the previous slide. The run with higher resolution ran out of computing time.
With the coarsest (very coarse) resolution the errors are initially less than one percent, and later less than one part in For the higher (not that high at all) resolution the errors after a while are of the order of one part in 105 !

12. Moving out the boundaries: 10M and 15M
The dependence of the liftetime on the position of the outer boundaries is not monotonic:
Boundaries at 5M runs for ~800M.
Boundaries at 10M runs for ~700M.
Boundaries at 15M runs for ~1000M.
No fine tuning here on the tolerance value or na (here na typically is 100, still not 1 !!!).
All these runs are fully 3D ones, no octant, bitant or any other symmetry is imposed. The run with boundaries at 15M is single resolution only because of lack of computing time.

13. Understanding the mechanism in more detail
The resulting, dynamically defined constraint-functions h and g
After some time they settle down to some values, and at late times the change very quickly. These late time variations are consequence and not cause of the code crashing.
Comparing a run with parametrs from this figure to another run where those parameters are kept fixed after t=750M. The code still crashes, therefore the late time oscillations
in the constraint-functions are consequence and not cause of the code crash.

14. Fixed, fine-tuned constraint-functions
After some time the constraint-functions remain fairly constant until the code crashes. What happens if one runs with fixed constraint-functions, with their values given the those asymptotic, quite constant values?.
Not big difference. However, presumably this would change in the case of a dynamical solution

15. How much do the constraint-functions depend on the position of the outer boundaries?
a) Boundaries at 5M: they settle down to h=-1.9, g=-1.0 b) Boundaries at 10M: they settle down to h= , g=-2.5 c) Boundaries at 15M: they settle down to h= , g=-3.4
Not yet converging to anything when the domain is enlarged.
Constraint functions obtained with certain boundary and used for other computational domain is not optimal, but much better than doing nothing. This plot shows a run with boundaries at 10M and fixed constraint-functions given by (a) and (b).

16. Don’t Be Fooled (TM)!, ask for the convergence tests!
DBF!
Definition 16 (IDIOT) Anyone who publishes a calculation without checking it against an identical computation with smaller N OR without evaluating the residual of the … approximations via finite differences is an IDIOT
John P. Boyd, in "Chebyshev and Fourier Spectral Methods", Second Edition.