Lattice Boltzmann methods beyond BGK-style collision operators

Question

I am trying to educate myself on the lattice Boltzmann method to see if it may be useful for simulating some problems in plasma physics that I am interested in. One thing that strikes me is the (seemingly) universal adoption of BGK-style collision operators. Is the method intractable with more sophisticated collision operators like full Boltzmann (for neutral gases) or Fokker-Planck (for charged particles)? I understand that BGK-style operators are useful pedagogically for learning about the method, but does this also represent the state of the art?

Answer 1

An excellent starting point for a more in-depth look at different collision operators is Part 2, Chapter 10 of the book by Krüger et. al which introduces multiple-relaxation-time (MRT) and two-relaxation-time (TRT) collision operators.

These have been further refined by Karlin, Bösch and Chikatamarla (KBC) in an entropy maximizing manner, see their original publications:

Gibbs' principle for the lattice-kinetic theory of fluid dynamics I. V. Karlin, F. Bösch, and S. S. Chikatamarla Phys. Rev. E 90, 031302(R)

Entropic multi-relaxation lattice Boltzmann scheme for turbulent flows

This is of course not the end of the story, depending on the application you are interested in you might use specialized methods/relaxation times, see for instance

W. Li, Y. Chen, M. Desbrun, C. Zheng, and X. Liu, “Fast and scalable turbulent flow simulation with two-way coupling,” ACM Trans. Graph. 39, 20 (2020).

or

Lattice-Boltzmann Simulations of Particle-Fluid Suspensions A. J. C. Ladd & R. Verberg Journal of Statistical Physics volume 104, pages 1191–1251 (2001)

just to name two publications tackling special use-cases.