General question about LBM

Hello there,

this will be my first post in this forum, so I will first make a small introduction of who I am.
Well, you don’t need to know my name… but at this moment I am still a Mathematics student who is doing an internship at some company that is concerned with flows inside greenhouses.

Currently, some CFD models with Navier-Stokes are used… but they want to see/try the advantages of the LBM. So, they “called” me to investigate it. After 1.5 week of research, reading into the problem/phsyics etc. I have keep stumbling on the following problems

  • In any lattice structure, take for example the D2Q9, how are the lattice weights determined? I can’t seem to figure it out and can’t find an explanation
  • I understand that the relaxation time is an important factor, it relates the meso/micro-domain to the macro-domain… if I am correct. But until now, I have not found a good way to connect some given pde (N-S, heat equation, adv-diff-eq) to the discrete LBE via the relaxation time.
  • After seeing some LBM code that models the diffusion equation and some other examples… I can’t seem to grasp the moment where you can “see” the original PDE that you are modeling. I assumed that this would be visible in the collision procedure… and all I see there is the equilibrium distribution and the relaxation time.
    So, then the whole “being” of the original PDE we are modeling, should be in the equilibrium dist… is this right?
  • Last question: in the Chapman-Enskog expansion, the distribution function is expanded in a powerserie with the Knudsen number… but this is only done up to the second order. Won’t this create a massive error in the calculations?? Why isn’t anyone concerned with this fact?

That is it for now… It would mean a lot to me if somebody could answer any of these questions. Im trying to figure it out on my own and that is taking some time…


One way to think about and derive the D2Q9 weight factors is summarized in Appendix B of my PhD thesis:

Hope you find it useful.


  • Abhijit

Oh wow thanks!

I just started reading your thesis and you’ve already helping me with some small stuff… like for example, why the power serie expansion of the e-power in the Maxwell-Boltzmann distribution can be truncated… so I am already glad for that.

Later this day I will also read the Apendix B… I’ve already skimmed through the text and it looked like an elaborate explanation, so I also thank you for that :slight_smile: