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Hi Ian, Erik,</div>
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Thanks for your feedback. I've finally tracked the problem to what I <u>thought</u> the ET was doing at the boundary vs what was
<u>actually happening</u>. My par file has</div>
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Coordinates::outer_boundary_size = 1<br>
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and together with the none BC, the initial values of the system variables remain as such at these points. This is different to my Python code, which is evolving the outer boundary points (in preparation for the SAT method). Thus, I set the time derivative of
all fields to zero at the outer boundary points and switched back to RK4 (Euler is indeed unstable!) and everything works now. So, the issue can be summarised as my misunderstanding of what is happening in the ET.</div>
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Thanks again for your helpful comments!</div>
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Chris</div>
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<div id="divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" style="font-size:11pt" color="#000000"><b>From:</b> Ian Hinder <ian.hinder@manchester.ac.uk><br>
<b>Sent:</b> 17 May 2022 19:32<br>
<b>To:</b> Chris Stevens <chris.stevens@canterbury.ac.nz><br>
<b>Cc:</b> Erik Schnetter <schnetter@gmail.com>; Einstein Toolkit Users <users@einsteintoolkit.org><br>
<b>Subject:</b> Re: [Users] Implementing a simple 1+1 BSSN in Python</font>
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<div>Hi Chris,
<div class=""><br class="">
</div>
<div class="">- Note that the Euler method you are using is unstable, and will probably amplify any roundoff errors you have. </div>
<div class="">- Does everything work if you use periodic boundary conditions?</div>
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<div class="">Erik said:</div>
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<div class="">Given your description, I assume that all derivatives would be identically zero, that there would not even be a floating-point round-off error. Is that so? If not, why not?</div>
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<div class="">I think this would be true for something like (u_{j+1} - u_j) / dx, but it might not be true for ( u_{j+2} - 2u_{j+1} + u_j ) / dx^2 which would evaluate to ( u - 2u + u ) / dx^2, and depending on how the compiler implements it, (u - 2 u) for
example might not be exactly representable in floating point, and might not cancel exactly the (+u). If u were zero, I would agree.</div>
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<div class="">On 14 May 2022, at 04:22, Chris Stevens <<a href="mailto:chris.stevens@canterbury.ac.nz" class="">chris.stevens@canterbury.ac.nz</a>> wrote:</div>
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<div dir="auto" class="">Hi Erik,</div>
<div dir="auto" class=""><br class="">
</div>
<div dir="auto" class="">Thanks for your quick response!</div>
<div dir="auto" class=""><br class="">
</div>
<div dir="auto" class="">I use the none bc type for now, rather than the more complicated radiative bc. I use a stencil that leans over as you approach the boundary. Currently this is a 4th order interior 3rd order sbp fd operator of Strand.</div>
<div dir="auto" class=""><br class="">
</div>
<div dir="auto" class="">By looking at the output on the boundary in both codes, there are round off level numbers appearing after the first Euler step in multiple variables. There is no use comparing these exactly as they're so small but note that the same
variables have these in both codes. These come from taking spatial derivatives.</div>
<div dir="auto" class=""><br class="">
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<div dir="auto" class="">Sorry, the lapse is initially one.</div>
<div dir="auto" class=""><br class="">
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<div dir="auto" class="">I have compared output by saving the data in both codes. It seems like, since the first few time steps everything is round off errors, it is hard to see exactly where things start to go wrong.</div>
<div dir="auto" class=""><br class="">
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<div dir="auto" class="">I will try your small grid and see if that yields anything more obvious.</div>
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<div dir="auto" class="">Thanks,</div>
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<div dir="auto" class="">Chris </div>
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<div id="x_divRplyFwdMsg" dir="ltr" class=""><font face="Calibri, sans-serif" class="" style="font-size:11pt"><b class="">From:</b> Erik Schnetter <<a href="mailto:schnetter@gmail.com" class="">schnetter@gmail.com</a>><br class="">
<b class="">Sent:</b> Saturday, May 14, 2022 3:09:17 PM<br class="">
<b class="">To:</b> Chris Stevens <<a href="mailto:chris.stevens@canterbury.ac.nz" class="">chris.stevens@canterbury.ac.nz</a>><br class="">
<b class="">Cc:</b> Einstein Toolkit Users <<a href="mailto:users@einsteintoolkit.org" class="">users@einsteintoolkit.org</a>><br class="">
<b class="">Subject:</b> Re: [Users] Implementing a simple 1+1 BSSN in Python</font>
<div class=""> </div>
</div>
<div class="">
<div dir="ltr" class="">Chris
<div class=""><br class="">
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<div class="">What do you mean by "Boundary conditions are not applied"? What do you do at the boundaries? Are you using one-sided differencing there?</div>
<div class=""><br class="">
</div>
<div class="">Given your description, I assume that all derivatives would be identically zero, that there would not even be a floating-point round-off error. Is that so? If not, why not?</div>
<div class=""><br class="">
</div>
<div class="">Are you really setting the lapse to zero initially? It should be one for Minkowski, same as e.g. g_xx.</div>
<div class=""><br class="">
</div>
<div class="">What happens if you choose a very small grid size (e.g. just 10 points) and compare the initial conditions between the Einstein Toolkit and the Python code? Are they identical? You would need to compare ASCII output and compare all the digits.</div>
<div class=""><br class="">
</div>
<div class="">What happens if you take a single Euler time step? Are the values still all identical? Try to find out and which iteration the first grid point differs between the two codes.</div>
<div class=""><br class="">
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<div class="">-erik</div>
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<div dir="ltr" class="x_x_gmail_attr">On Fri, May 13, 2022 at 9:59 PM Chris Stevens <<a href="mailto:chris.stevens@canterbury.ac.nz" class="">chris.stevens@canterbury.ac.nz</a>> wrote:<br class="">
</div>
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Hi all,</div>
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<div class="" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; background-color:rgb(255,255,255)">
I've been having trouble implementing the BSSN equations, as given by CTGamma, in Python using the package COFFEE, which I am very familiar with:</div>
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</div>
<div class="" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; background-color:rgb(255,255,255)">
<a href="https://gitlab.com/thebarista/coffee" id="x_x_gmail-m_5984039251753474632LPNoLPOWALinkPreview" target="_blank" class="">https://gitlab.com/thebarista/coffee</a><br class="">
</div>
<div class="" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; background-color:rgb(255,255,255)">
<a href="https://doi.org/10.1016/j.softx.2019.100283" rel="noreferrer noopener" title="Persistent link using digital object identifier" id="x_x_gmail-m_5984039251753474632LPNoLPOWALinkPreview_1" target="_blank" class="">https://doi.org/10.1016/j.softx.2019.100283</a><br class="">
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<div class="" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; background-color:rgb(255,255,255)">
The main comments are:</div>
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-- For simplicity, I am starting with a 1+1 code (set 2/3 spatial derivatives to be exactly zero).</div>
<div class="" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; background-color:rgb(255,255,255)">
-- The 1+log slicing for the lapse and a zero shift.</div>
<div class="" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; background-color:rgb(255,255,255)">
-- The "phi" conformal factor type </div>
<div class="" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; background-color:rgb(255,255,255)">
-- <span class="" style="background-color:rgb(255,255,255); display:inline">The evolution equations are written to Python by a very slight modification of the .m file in CTGEvolution. I've confirmed this .m file outputs the same file that is in the src directory.</span></div>
<div class="" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; background-color:rgb(255,255,255)">
<span class="" style="background-color:rgb(255,255,255); display:inline">-- The vanishing of the trace of A is enforced after each full timestep in the same way as CTGamma.</span></div>
<div class="" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; background-color:rgb(255,255,255)">
<span class="" style="background-color:rgb(255,255,255); display:inline">-- Boundary conditions are not applied.</span></div>
<div class="" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; background-color:rgb(255,255,255)">
<span class="" style="background-color:rgb(255,255,255); display:inline">-- Fourth order SBP operators that lean over toward the boundaries are used.</span></div>
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<span class="" style="background-color:rgb(255,255,255); display:inline">-- Minkowski initial data with all fields zero except \gamma11/22/33.</span></div>
<div class="" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; background-color:rgb(255,255,255)">
<span class="" style="background-color:rgb(255,255,255); display:inline">-- Euler step is used for time integration to easier compare between Python and the Einstein toolkit.</span></div>
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<span class="" style="background-color:rgb(255,255,255); display:inline"><br class="">
</span></div>
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<span class="" style="background-color:rgb(255,255,255); display:inline">This leads to a stable evolution in the Einstein toolkit with the attached .par file using a simple one patch Cartesian grid. However, in Python, I get that fields diverge, starting at
the boundaries and then propagating inward (including with RK4). This behaviour does not go away with higher resolution or reduced CFL. It is however, fixed completely by setting the second spatial derivative of \alpha to be exactly zero. Trying many different
SBP FD operators that have proved successful for other projects has not changed the behaviour. </span></div>
<div class="" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; background-color:rgb(255,255,255)">
<span class="" style="background-color:rgb(255,255,255); display:inline"><br class="">
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The COFFEE code has been used and tested in a variety of projects and so there should not be a problem here. Thus, the only thing I can think of, is that the Einstein toolkit is doing something that I am not realizing, that is helping it remain stable. Hopefully,
somebody on this mailing list might have an idea?<br class="">
<br class="">
Thanks in advance,</div>
<div class="" style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; background-color:rgb(255,255,255)">
<br class="">
Chris</div>
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