Bernd,

I know Ed Evans is going to respond to the first point of your message.

>   
>  2. 
>  You may also want to have a look at what the simplest way would be to  
>  compute the conformal derivatives. 
>   

I would suggest that in the first pass let us just make the conformal
factor equal to 1, so that we do not have to worry about it.  It is very
useful for our black hole studies.  But for the wave case, it will
not be useful.

>  3. 
>  We need simple test cases. Lot's of them. Since not very much has been 
>  done with real 3d elliptic shift conditions, trying various things is 
>  almost a paper in itself. Solving the momentum constraint is another 
>  topic about which little is known. 
>     

Right.  We are really going into new territories.  I suppose the
big unknown here is how the shift would function, and how to come
up with better conditions, ie., how to improve upon it.  This will
need a lot of investigation.

As for if we have solved the equations correctly, it would be
a simpler problem that we can use testbeds and convergence test.
Of course there will still be a lot of hard work!

>  How did Greg test it? Wai-Mo, any suggestions? 
>  Cactus has thorn_exact which is nice for comparisons. There are two or 
>  three interesting analytic metrics in there which have non-zero shift.   
>  Another good place to start is with the Suen etal papers on coordinate 
>  conditions. 

There are not that many anaytic shifts that we know are minimal
distortion.  We have to rely mostly on comparing with numerical 
constructions.  Comparing with 1D cases (e.g., Schwarzschild) are
easy, it is just integrating a second order ODE.  For comparing with 2D
cases, in the past we tested it with those results in
Bernstein's thesis.  Did you have a copy of that, Bernd (if not get
it from Ed)?  I can think of no good real 3D tests or comparison at
present, other than those old results we obtained with the G-code in the
our previous coordinate condition studies.  But I suppose if our 3D
implementation can obtain the correct 1D and 2D results, it should be
pretty safe.  I will think about this more.

Again this are for testing our ability in solving the MDS equations.
Whether it is good to use for a specific problem is a different matter.

At this point I am particularly eager to have the momentum
constraint solver up and running, one way or the other.  We need
it for doing the NS headon collision.  At present, we have already
a conformal flat momentum solver up and running by Hisaaki.  I am holding
my breath to see a full one working.

Wai-Mo

Created for the WUGRAV CoCoBoard Page Projects Page.

Created by The CoCoBoard.