Is there not a common root of Lattice Boltzmann Methods (LBM) and Smoothed-particle Hydrodynamics (SPH) modelling incompressible fluid flows? I see common grounds by means of a mesocopic description based on the Boltzmann equation, where LBM is an Euler and SPH is a Lagrange approach discretizing a siplification of the Boltzmann equation, e.g. the BGK-Boltzmann equation in the hydrodynamic (incompressible) limit. If you look at e.g. the kind of pressure scaling in both methods you will see similarieties. Are there any other thoughts or even papers on that topic?
Hi there, I do not share your viewpoint, as I believe I do not agree with your definition of a « discretization ».
LBM recovers the solution of the NSE in the limit of low Mach and Knudsen numbers. Because when you perform a multiscale Chapman-Enskog expansion you can show that the LBM gets back to NSE + some terms in Mach and Knudsen. So, you are not per se discretizing the NSE, but recovering its solution/behavior at the macroscopic scale (velocity and pressure).
As a matter of fact, to come back to your initial question, you cannot with SPH easily recover, at least at first glance, the probability distribution functions (which are the quantities that are truly evolved in LBM). This shows to me that the two methods are completely different.
For your information, we compared the two methods in a recent article:
https://link.springer.com/article/10.1007%2Fs40571-019-00250-3
This being said, I (and others!) do use LBM as a solver for NSE and not for solving a simplified formulation (I agree) of the Boltzmann equation ! I would just not call LBM a discretization of the NSE... It is not to me. I believe our disagreement is just about the definition of what is a discretization ! :-)
One can see it like that, but in LBM trajectories are fixed while in SPH methods they are not. Interestingly in SPH methods trajectories become more and more ordered the higher one chose the background pessure. The question remains open, is there a common root?
Well, LBM discretizes the Boltzmann Equation (a statistical mechanics representation of a fluid) and SPH discretizes the Navier-Stokes equations (a continuum mechanics representation of a fluid). Two different approaches. However, (quasi-)incompressibility is enforced similarly by means of a stiff equation of state relating pressure to density. This is the only similarity that I can see at first glance...
David Vidal: LBM does discretise the BGK-Boltzmann equation, a simlification of the Boltzmann equation, in a certain limit which leads to the incompressible Navier-Stokes equation (NSE). And so, LBM does also discretise the NSE.
Hi there, I do not share your viewpoint, as I believe I do not agree with your definition of a « discretization ».
LBM recovers the solution of the NSE in the limit of low Mach and Knudsen numbers. Because when you perform a multiscale Chapman-Enskog expansion you can show that the LBM gets back to NSE + some terms in Mach and Knudsen. So, you are not per se discretizing the NSE, but recovering its solution/behavior at the macroscopic scale (velocity and pressure).
As a matter of fact, to come back to your initial question, you cannot with SPH easily recover, at least at first glance, the probability distribution functions (which are the quantities that are truly evolved in LBM). This shows to me that the two methods are completely different.
For your information, we compared the two methods in a recent article:
https://link.springer.com/article/10.1007%2Fs40571-019-00250-3
This being said, I (and others!) do use LBM as a solver for NSE and not for solving a simplified formulation (I agree) of the Boltzmann equation ! I would just not call LBM a discretization of the NSE... It is not to me. I believe our disagreement is just about the definition of what is a discretization ! :-)
Intresting paper! Do you know a paper only comparing singe phase flows in the NSE regime?
There are papers of Junk http://www.math.uni-konstanz.de/numerik/personen/junk/publications/index.php which relate LBM to NSE (e.g. http://www.math.uni-konstanz.de/numerik/personen/junk/publications/paper/JY03.pdf). They see LBM as finite difference discretisation and provide even a finite difference stencil.
Mathias J. Krause Hi Mathias, we also compared in single phase, but the article was too long so we cut this part. Otherwise, the following article compared the two methods in terms of the parallelization:
https://dl.acm.org/doi/10.5555/2048577.2048599
Any other ? Not that I remember and if there are Thomas (first author) must have included them in the article.
Yes, you are right, it has been shown that LBM scheme comes back to a finite difference discretization scheme of the NSE (after multiscale Chapman-Enskog expansion). However, I believe it is in the limit of low Ma and Kn numbers, that is when the spurious terms are negligible. This is why I prefer to not see the LBM as a discretization of the NSE.
Greetings.
David.
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