[Users] ML_BBSN shift evolution equation

[Erik Schnetter]

Robyn

There are two choices to evolve the shift, either as first-order or as
second-order equation. In the second case B^a is also evolved. In the
first case B^a is just a dummy variable that remains unused. The other
parameters have slightly different meanings in both cases.

The "traditional" BSSN equation for the shift uses B^a.

The "traditional" BSSN shift evolution also omits the Lie derivative
terms for the shift. Some people find that these terms are useful, and
so there are parameters to add them.

Finally, the code is slightly more complicated because a certain term
is identical to (part of) the RHS of the Gamma-tilde terms, and this
term is reused.

In the end we have unfortunately reached a state where the current
meaning of the parameters cannot be changed any more because this
would break existing parameter files. If in doubt the code is the
ground truth. You might be able to choose some parameters (e.g. "B is
evolved", "Lie derivative terms are omitted") and then manually
simplify the code to see what happens.

The story for the lapse alpha and its time derivative A is similar.

-erik


On Fri, Nov 17, 2023 at 5:47 AM Robyn Munoz <robyn.munoz at yahoo.fr> wrote:
>
> Hello everyone,
>
> Could someone please clarify the shift evolution equation implemented in ML_BSSN?
>
> Just looking at the parameter file it seems like it is:
>
>      d/dt beta^i = shiftGammaCoeff  Xt^i - betaDriver alpha^shiftAlphaPower beta^i
>
> looking at McLachlan_BSSN.m (for GammaDriver and without advection and dissipation), I think it is:
>
>      d/dt beta^a =  shiftGammaCoeff alpha^shiftAlphaPower B^a
> with
>      d/dt B^a = d/dt Xt^a - betaDriver B^a
>
> Whereas looking at Alcubierre's book Eq 4.3.33 and 4.3.34 I would expect it to be
>
>      d/dt beta^i =  B^i
> with
>      d/dt B^i = shiftGammaCoeff alpha^shiftAlphaPower d/dt Xt^a - betaDriver B^a
>
> on the McLachlan webpage: https://www.cct.lsu.edu/~eschnett/McLachlan/
> the link to describe the Gamma driver shift condition http://grwiki.physics.ncsu.edu/wiki/Shift_Conditions doesn't work but looking at the 2019 version on the wayback machine the evolution described is
>
>      d/dt beta^a = (3/4) B^a
> with
>      d/dt B^a = d/dt Xt^a - betaDriver B^a
>
> where here (and in Alcubierre) d/dt = partial_t - beta^c partial_c. Does the McLachlan code have that second term beta^c partial_c ?
>
> This is confusing me, so what is actually implemented? Where are the parameters shiftGammaCoeff, betaDriver and shiftAlphaPower in the evolution equation? And these are all constant positive reals right? In Alcubierre's book xi (that I think is shiftGammaCoeff) can be function of space and the lapse.
>
> Thank you,
> Robyn
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-- 
Erik Schnetter <schnetter at gmail.com>
http://www.perimeterinstitute.ca/personal/eschnetter/