[Users] Einstein Toolkit and geodesic flow

[Charalampos Markakis]

Dear Cosima,

Regarding solving for thousands of geodesics, a faster alternative would be
to solve them as a PDE, i.e. as a dust fluid. This is analogous to solving
the Burgers equation (the pressureless Euler equation) for nonrelativistic
flows. The Einstein Toolkit is naturally designed to handle such scenarios.
But the details depend on what you are trying to simulate.

For example, for timelike geodesics, this could be done by evolving eq.
(10) in
https://arxiv.org/abs/1410.7777
with p_a given by eq. (8) and H given by eq. (16), setting the specific
enthalpy to h=1.

Some caveats:
1) For null geodesics, you can do something similar, but need a different
foliation.
2) If you try to use the Valencia formulation to evolve dust, you must
evolve the continuity equation for the density \rho, along with the Euler
equation for the velocity. This system is ill-posed and unstable for dust
(pressure = 0), because you are evolving an equation that is no-longer
needed: One needs to drop the continuity equation, eliminate \rho
completely from the equations, and evolve only 3 equations for the spatial
components of the 4-velocity. Then the system is well-posed (strongly
hyperbolic), just like the Burgers equation (whose relativistic
generalization you are solving). Eq. (10) in the above paper does this.
3) As Roland Haas pointed out, because you are evolving dust as a
pressureless fluid, even though the flow is geodesic, caustics or shocks
may form. If that happens, the way to handle it depends on what you are
trying to simulate with geodesics. If you need to simulate the actual
geodesics, and not dust, you can use the method of characteristics and
disregard shock solutions. (For null geodesics you won't have this problem.)
4) If you need to know where each geodesic particle is at a given time,
along with eq. (10) above, you can solve the other Hamilton equation for
the position x^a conjugate to p_a,
$ dx^a/dt=\partial H / \partial p_a $,
written in Eulerian PDE form. This evolves a 'label' of the initial
position of each particle.

For any clarification, please feel free to contact me or Roland Haas.

Best regards,

Haris Markakis

________________________________________________

Charalampos Markakis
Postdoctoral Research Associate

National Center for Supercomputing Applications
University of Illinois at Urbana-Champaign
1205 West Clark St, Room 4022, MC-257, Urbana, IL 61801
markakis at illinois.edu
________________________________________________



On 21 November 2016 at 10:56, Frank Loeffler <knarf at cct.lsu.edu> wrote:

> On Mon, Nov 21, 2016 at 12:26:59AM -0600, Frank Loeffler wrote:
>
>> Please consider joining the weekly Einstein Toolkit phone call at
>>
>
> The following group called in:
>
> Cosima (and a few others from the Frankfurt group;
>        I don't remember your names, sorry)
> Erik
> Frank
> Harry
> Ian
> Peter
> Roberto
> Vassili
> Yosef
> Zach
>
> One topic was how to optimized a geodesics iteration thorn written by Ian
> Hinder et.al., and now used by the Frankfurt group. The main problem is
> that it was written with a few geodesics in mind, but should now be used
> for millions. Erik strongly suggests to profile what exactly is slow before
> starting to heavily optimized what you might think is slow. Assuming it is
> communication, he gave a few hints which functions might help
> (locatePosition?).
>
> Connected to this it was mentioned that integrating GAs is currently
> broken if you have more than one component per process, as they are
> currently integrated (by MoL) fore each component. Current workaround: use
> more processes (to get 1 component per process).
>
> Release related issues:
>
> - Llama was added to the thornlist, including testsuites
> - initial test results look quite good
>  (a few failures from usual suspects)
> - Intel 17 compiler crashes on bbox.cc in Carpet.
>  - current workaround: lower optimization of that file to -O1
>  - maybe better: use bbox1-class instead of newer bbox2 class.
>                  however: you will have worse scaling for large runs
>  - current machines showing this problem: marconi, stampede-knl
> - Only a few machines saw testing so far. We need more!
>
> A first shot at a new design of the ET pages is online:
>
>  http://einsteintoolkit.org/new/
>
> These aren't polished yet, but should give a first impression. They are
> responsive, i.e., they should look presentable also on small screens
> (phones). When you test them, please use different browsers, different font
> sizes (zoom levels), different window sized, and different devices.
>
> Please send feedback either to the list, or directly to Frank <
> knarf at cct.lsu.edu>.
>
> Frank
>
>
> _______________________________________________
> Users mailing list
> Users at einsteintoolkit.org
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>
>
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