Add volume integrands for XCTS asymptotic quantities.

[iago-mendes]

Proposed changes

This PR adds volume integrands for the calculation of ADM mass, ADM linear momentum, and center of mass in the XCTS system.

The primary goal of using infinite volume integrals instead of infinite surface integrals is to improve numerical accuracy. We haven't seen significant accuracy improvements yet, and we have found some analytic test cases in which the volume integrals do not give the expected results (which could be a bug in the method or an assumption being broken somewhere in these cases). Then, we'll keep using the only surface integrals in the ID parameter control for now.

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Further comments

[nilsvu]

I haven't reviewed the tests yet. Please push changes as fixup commits.

[nilsvu]

I think this is missing 2 terms: $+\gamma^{jk} \Gamma_l \Gamma^l_{jk} - \gamma^{lj} \Gamma_l \Gamma_j$ where $\Gamma_l = \Gamma^k_{kl}$.
I don't think these cancel since $\gamma^{jk}\Gamma^l_{jk} \neq \gamma^{lj}\Gamma^k_{jk}$ (or do they somehow?)

[iago-mendes]

If I understand it right, these extra terms would only show up if we take the covariant derivative $\bar D_i$ when applying Gauss' law instead of a partial derivative $\partial_i$ (this is what I'm currently doing). I remember we briefly talked about these two options (referencing Saul's recent DG paper), and we decided that $\partial_i$ was the way to go.

That said, after redoing the calculations, I found an extra term involving the conformal factor first derivative:
$\partial_i ( - 8 \bar\gamma^{ij} \partial_j \psi ) = - 8 \bar\gamma^{ij} \partial_i \partial_j \psi - 8 \partial_i \bar\gamma^{ij} \partial_j \psi = - 8 \bar D^2 \psi - 8 \partial_i \bar\gamma^{ij} \partial_j \psi $.
Again, this is only if we use $\partial_i$ in Gauss' Law. Looking at Baumgarte-Shapiro's equation (3.146), they seem to use $\bar D_i$ in Gauss' Law.

I'm not sure what approach we should take here, but both of them require updates to these integrands.

[nilsvu]

You need the covariant Laplacian of $\psi$ to replace it with the Hamiltonian constraint, so I would go with the covariant derivatives and the Christoffel terms I wrote above. As you say, either way you have to account for the covariant derivatives with Christoffel terms.

[iago-mendes]

Ok, that makes sense. This raises another point. If we're using $\bar D_i$ when applying Gauss' law, then can we still assume a flat space in this transformation?

[nilsvu]

For Gauss' law you transform $\oint n_i \sqrt{\gamma} F^i = \int \partial_i (\sqrt{\gamma} F^i) = \int \sqrt{\gamma} \nabla_i F^i$ (see Eqs. (7-10) in https://arxiv.org/abs/physics/9802027v1). So you get the covariant derivative from moving the metric determinant through the partial derivative.
So, you integrate with the curved area/volume element, get a covariant derivative from Gauss' law, and expand it into Christoffel terms where needed.

[iago-mendes]

This is helpful, thanks! The test that I had in Test_AdmMass was using flat area/volume elements and started failing after these changes. Once I updated this test to use curved area/volume elements, I started getting the correct values again. So, things seem consistent.

[nilsvu]

Ok great! Both approaches work, we just have to be consistent. Please note in the docs for the integrand function if it needs to be integrated with the flat or conformal area/volume element.

[nilsvu]

You can squash these two commits (including this change)

[nilsvu]

Looks good, you can squash the fixup.

[nilsvu]

New fixups look good too, you can squash

[nilsvu]

Looks good. You can squash everything, including this.

[nilsvu]

At the moment you don't use this yet, right? Same for Ricci scalar/tensor and longitudinal shift below. Maybe hold off with adding these until you use them in this test.

[nilsvu]

Can you keep this at O(1/distance)?

[iago-mendes]

It doesn't work for this test because the CoM integral "diverges" as we increase R. I believe this happens due to round-off errors when canceling larger and larger terms. I describe a bit of this at the top of this file.

[nilsvu]

This is very high refinement. My understanding is that the accuracy of the integral is currently not limited by resolution, so can you lower this?

[iago-mendes]

This was the lowest resolution that I was able to find in order to keep the tolerance at $10^{-3}$. Figure 3.2.1 in these convergence plots shows that the volume integral residual gets slightly better for higher resolution.

I think there must be some change that we can make in the calculation of the CoM integrals to get around these round-off errors, but this is probably something for a future PR.

[nilsvu]

ADM mass test times out, probably just relax the timeout

[iago-mendes]

ADM mass test times out, probably just relax the timeout

How do I relax the timeout?

[nilsvu]

Add // [[TimeOut, 10]] above the SPECTRE_TEST_CASE

[iago-mendes]

It seems like it's still timing out for some compilers. Should I relax it even further?

[nilsvu]

Yes just set a reasonable timeout, and/or speed up the test

[iago-mendes]

Yes just set a reasonable timeout, and/or speed up the test

Done. The ADM mass test now succeeds after <20s for the compilers that were failing before (gcc-9, gcc-11). Now the fails are from PartialDerivs, so it's unrelated to my changes.