Improve Par_EquilibriumIC

[hsinhaoHHuang]

Goal

• To resolve Remove Cloud_Par_Num_Ratio #188
• To resolve Potential danger for using convertToString().c_str() in Par_EquilibriumIC.cpp #201
• To resolve Make tables in Par_EquilibriumIC.cpp more flexible #202
• To resolve Improve Par_EquilibriumIC.cpp #295

Changes

1. Adjust the user interface

   1. Move the part with reading the parameters from the input files into TestProblem/Hydro/ParticleEquilibriumIC/Init_TestProb_Hydro_ParEqmIC.cpp like other test problems.
   2. Use setXXX() as the interface to set the parameters for the cloud constructors from test problems, and only the necessary parameters are needed to be set.
   3. Support adding analytical external potential. Users can define their analytical function. Also, the potentials of some analytical models are provided.
   4. Output the relative error of the total enclosed mass by comparing it to the analytical solution instead.
   5. Output the radial distribution and energy distribution for debugging.
   6. Separate the number of bins of the input density profile, input external potential, inner radial profiles, and inner energy distribution.

2. Adjust the calculation

   1. Unify the processes for density profiles from the table and from the analytical models.
   2. Construct the radial profiles more consistently, from the density profile to the enclosed mass profile, the potential profile, and finally the drho_dPsi profile.
   3. Use the central difference for the differentiation instead of linear regression.
   4. Add the external potential before the usage of potential to be more consistent.
   5. Extend the radial profile to a radius larger than the given MaxR for scattered particles for calculation of the distribution function to reduce the truncation error.
   6. Include the low-binding-energy (high-velocity) distribution function. When the E is smaller than the Psi_min ( at large r ), the dRho/dPsi is approximated using power-law extrapolation.
   7. Use linear extrapolation for the density profile at r = 0.
   8. Integrate the enclosed mass profile with a higher-order shell mass for the density table.
   9. Use interpolation to sample E from the probability distribution.
   10. Use interpolation to find the drho/dPsi for a given Psi.

3. Adjust the format

   1. Rename some variables and functions.
   2. Remove some no longer used variables and functions.
   3. Add more description.
   4. Adjust the coding styles.

Verification Tests

1. Compare the distribution to the old branch with the default double-cloud setup. ->The results can roughly match the previous results.
   1. Type = NFW

2. Type = Table

2. Compare the distribution to the analytical solutions. -> The results can match analytical solutions.

   1. Hernquist
      
   2. Plummer
      

3. Check the stability of a single NFW profile with the default setup. -> The results are stable.

New	Old

4. Check the stability of a single Einasto density profile (scale density=1.0, scale radius=0.1)with the Plummer external potential (scale density=80.0, scale radius=0.05) (constructed with external potential and run with the same external potential). -> The results are stable.

Case	Projections	Profiles	Error
Construct with ExtPot, Run with ExtPot
Construct without ExtPot, Run with ExtPot
Construct with ExtPot, Run without ExtPot

[hsinhaoHHuang]

I have updated it to the latest main branch and resolved the conflicts.

[koarakawaii]

Temporarily review break point.

[hsinhaoHHuang]

Updated to the latest main branch.

[hyschive]

@hsinhaoHHuang This PR is neat. Thanks for the excellent contributions! I only have minor comments and will proceed with merge once they are addressed.

In addition to the inline comments, please also take care of the following tasks:

• Update the wiki to provide instructions on how to use this initial condition constructor
• Add Record__ParEquilibriumIC_Model_* to example/test_problem/Hydro/ParticleEquilibriumIC/clean.sh

[hyschive]

@hsinhaoHHuang This PR is neat. Thanks for the excellent contributions! I only have minor comments and will proceed with merge once they are addressed.

In addition to the inline comments, please also take care of the following tasks:

• Update the wiki to provide instructions on how to use this initial condition constructor
• Add Record__ParEquilibriumIC_Model_* to example/test_problem/Hydro/ParticleEquilibriumIC/clean.sh

[hsinhaoHHuang]

@hyschive I have updated this PR according to the comments, including adding sanity checks for the order of function calls and adding instructions to the wiki.
Besides,

• I changed the default Cloud_Type to Hernquist because I only have analytical distribution functions for the Plummer and Hernquist models, and Plummer has its own test problem already.
• I fixed more grammar errors and style issues.
• I updated the script to plot the phase-space distribution.

Please have another look when you have time and let me know whether there is anything that needs further modifications.
Thank you so much again for your thorough review and all of your helpful suggestions to improve this PR.

[hyschive]

@hsinhaoHHuang Looks good. I only have a few minor comments before merging. Please also check this comment.

[hsinhaoHHuang]

I have updated the PR according to the comments.