Universe simulations in the past.

I think SPH stands for “smooth particle hydrodynamics”. Check – yes it does in this context. One of my friends at CSIRO does SPH. It’s a method that can work spectacularly well or spectacularly badly depending on how it mathematically treats the word “neighbour”.
The Illustris computer code uses moving unstructured meshes, which can also work well provided the code takes time to throw the mesh in the garbage bin fairly often. Otherwise the mesh tangles itself up in knots.
In my work for CSIRO I used the third method here called AMR which stands for adaptive mesh refinement on an Eulerian Grid. Where this needs care is in selecting the correct property (eg. Density) to control the adaption. If the choice is wrong then accuracy in areas where refinement is most needed can be exceptionally poor.
In other words, for all three methods there is an art to getting high accuracy.
I am actually tickled pink that all three methods give almost equally accurate results on a problem of this size. LOL. Proponents of all three methods have egg on face for falsely claiming that their method is much better.