Update closure model documentation

[jhp-lanl]

PR Summary

Updates the PTE solver documentation. Here is a summary of the additions/changes

• Various minor notation adjustments to make things more clear
• Reorganized things into a general discussion of the governing equations (separate from PTE) and the $\bar{\rho}$ formulation and the implication that volume fractions and densities are both inputs and outputs.
• Reformulated the constraint equations to better reflect what's in the code. Reorganized the density-energy/density-temperature discussion to reflect the changes.
• Added descriptions for the pressure/temperature known versions

PR Checklist

• N/A Adds a test for any bugs fixed. Adds tests for new features.
• N/A Format your changes by using the make format command after configuring with cmake.
• Document any new features, update documentation for changes made.
• Make sure the copyright notice on any files you modified is up to date.
• After creating a pull request, note it in the CHANGELOG.md file.
• N/A LANL employees: make sure tests pass both on the github CI and on the Darwin CI

If preparing for a new release, in addition please check the following:

• Update the version in cmake.
• Move the changes in the CHANGELOG.md file under a new header for the new release, and reset the categories.

[jhp-lanl]

See https://lanl.github.io/singularity-eos/jhp/docs/index.html to view my changes

[jhp-lanl]

@Yurlungur and @aematts I think I have everything updated. Please review

[Yurlungur]

Thanks for adding this more detailed documentation, @jhp-lanl !

[Yurlungur]

@aematts how does this look to you? If you're happy I'll merge.

[aematts]

I am looking at it now. Sorry about holding you up but I was on my Frday off and took my husband to the neurologist Monday.

[Yurlungur]

I am looking at it now. Sorry about holding you up but I was on my Frday off and took my husband to the neurologist Monday.

Not at all--there's no rush!

[aematts]

OK @jhp-lanl and @Yurlungur. I have gone through it once now. Your turn.

[aematts]

One more thing: We might want to point out that if you use sesame tables they should NOT be inverted if you use the pressure-temperature formalism, but they SHOULD BE inverted if you use the density-energy formalism.

[aematts]

It is all about what gives you the best derivatives.

[aematts]

Fixed pressure should also be done on original tables (NOT inverted).
Fixed temperature doesn't matter but using the original tables would probably be better, avoiding inversion when it is not needed.

[Yurlungur]

One more thing: We might want to point out that if you use sesame tables they should NOT be inverted if you use the pressure-temperature formalism, but they SHOULD BE inverted if you use the density-energy formalism.

These are good rules of thumb to recommend, though I'm sure there are exceptions that prove the rule.

[jhp-lanl]

One more thing: We might want to point out that if you use sesame tables they should NOT be inverted if you use the pressure-temperature formalism, but they SHOULD BE inverted if you use the density-energy formalism.

I have a note in there about the need to invert from density-temperature space to density-energy space for the density-energy formulation. I mentioned that there are both performance and accuracy implications (I'm not sure how much I trust the inverted derivatives for example).

[jhp-lanl]

I'll merge once the docs build and I see they look okay. Thanks for the review @aematts and @Yurlungur !

[aematts]

Looks good now.