Column 1: ENZO

Column 2: Phase diagram for All Gas ( top ) and Neutral Hydrogen ( bottom ). Temperature is internally computed by Grackle on run time.

Column 3: Phase diagram for the fraction of the gas that uses the Total Internal Energy to compute pressure.

Column 4: Phase diagram for the fraction of the gas that uses the Adavected Internal Energy to compute pressure.

Column 5: Phase diagram showing the temperature computed from the Total Internal Energy for ALL the gas.

Column 6: Phase diagram showing the temperature computed from the Advected Internal Energy for ALL the gas.

Here is a comparison of the phase diagram using PLMP reconstruction changing the way the Advected Internal Energy is evolved.

Row 1: The $p \nabla \cdot \mathbf{v}$ term is applied at the half-step and at the complete time step.

Row 2: The $p \nabla \cdot \mathbf{v}$ term is not applied at the half-step, it’s only applied at the complete time step.

Row 3: The $p \nabla \cdot \mathbf{v}$ term is not applied at the half-step. At the end of the half update an internal energy is selected ( Advected or Total ) and that value is used to set the Total Energy ( kinetic + internal ), for the subsequent steps the pressure is always computed from the Total Energy avoiding mixing of Advected Internal Energy or Total Internal Energy for individual cells.

We discussed the possible change of using a consistent internal energy for the reconstruction, this means if a cell uses advected internal energy as its centered value for the internal energy then this cell should read the the advected internal energies of its neighbors to make the reconstruction, this will avoid spatial mixing of advected or total internal energy when computing the reconstructed values. One problem with this approach is that this requires the advected and total energy to be out of synchronization, having the internal energies unsynchronized caused many more problems, this can be seen in the animation below where I compare the phase diagram when the internal energies are unsynchronized.

Row 1: PCM Reconstruction

Row 2: PLMP Reconstruction

By looking at the column 5 where the Total Internal Energy in all the box is plotted, you can see that the temperature doesn’t drop when using the PCM reconstruction, I am unsure of what to conclude from this but it’s possible that the absence of low temperature gas when using PCM and syncronized Internal Energies could be because when using PCM the total internal energy is never lower than the adiabat.