Hello Seyed. That's a very interesting discussion indeed. May you indicate some of these papers based on this Eulerian discretization of the DVM?

Dear Ernesto,

There are quite a lot of them; Here are a few:

1 - Article Two-Dimensional Lattice Boltzmann Model For Compressible Flo...

2 - Article Finite-Difference Lattice Boltzmann Methods for binary fluids

3 - Article Two-dimensional finite-difference lattice Boltzmann method f...

etc.

Dear Seyed, thank you very much for the references!

Going even farther, it might also be interesting to answer this question: Assuming NS level dynamics are the targeted equations, what would the advantage of solving the truncated discrete Boltzmann (with an Eulerian treatment of time and space) over classical solvers for the NS equations be?

Dear Mohamed Hssikou ,

Thank you for your interest. Two points on your comments:

- Although it is clear that the NSF system of equations does not perform well in such situations, it is not a surprise. However, one can readily perform such simulations based on macroscopic balance equations (although granted they are derived from the kinetic theory), e.g. Grad's system of equations, Burnett and super Burnett equations, or the NSF equations in combination with proper slip velocity boundaries etc.

- The lattice Boltzmann method does not (at least in its classical form) recover any additional kinetic level information? Unless I have misunderstood it, it only correctly recovers moments of the EDF up to order two. It is a discretised form of the Boltzmann equation intended for the hydrodynamic regime. To be able to go beyond the limits of the NSF (in terms of Knudsen numbers) you'd have to use an extended set of discrete velocities, assuming the scheme stays stable and the BGK operator is valid in the regime you're targeting...

But apart from those issues, my question was more on the physical space and time discretisation strategy for the Discrete Velocity Boltzmann equation...

For me LBM is a consitent discretization method for kinetic equations coupling one or several model and discretisation parameters such that a particular target equation is reached in the limit.

Thank you for your reply. Based on your definition how would you distinguish "LBM" from other discrete velocity kinetic models that preceded the late 80s and early 90s? Because if we put them in the same category then what appeared in those years had nothing new?

Would an Eulerian space/time discretisation of the phase-space discretised Boltzmann equation with a linear relaxation collision operator qualify as LBM?

Historically (ties to a digital Lagrangian scheme, i.e. the LGA), and theoretically it seems like the on-lattice Lagrangian (thus the word lattice in the name) ingredient should play a central and essential role in the definition of this category of solvers?

Often during presentations, the LBM is presented as a "fully local" and "explicit" scheme, making it suitable for parallel processing. An Eulerian solver of the discrete Boltzmann system of hyperbolic PDEs relying on an implicit time-stepping scheme would not really have these advantages. Meaning that maybe the explicit nature of the solver is also an essential component?

Dear Mohamed Hssikou ,

Thank you for the comment, however this does not answer my question, namely "what is the lattice boltzmann?".

Is an Eulerian Discretisation of the phase-space-discretised Boltzmann equation considered LB (lattice Boltzmann)?

and if I may, I disagree with "LBM is a second-order accurate solver for the weakly compressible Navier-Stokes equation". It is a second order solver for the corresponding space-continuous discrete Boltzmann equation. It can be said to be a second-order approximation (and not a direct second-order solver) to the Navier-Stokes equation only under specific scaling.

Dear all,

Thank you for your contributions to this very interesting discussion. I agree with Mohamed Hssikou that a main feature of LBM is that it separates the transport and the non-linear part of the physics which should be modeled. I think we can also agree that the fact that the non-linearity is computed locally is also one of the main plus points in terms of ease of implementation and efficiency.

Regarding the main question, if schemes that perform advection in a Lagrangian manner by streaming from one lattice point to another should be the only ones that can be called LBMs I am not sure but it would perhaps make the taxonomy easier. If we should apply the same rigor regarding locality which could be applied in space also in time (as mentioned above by Seyed Ali Hosseini if LBMs need to be explicit to be called LBMs), I am even less sure. On the other hand, the community found for instance, legitimate ways for stretching the nomenclature of Lagrangian schemes by adding a word as was done for the semi-Lagrangian schemes. Recently, we needed a scheme for the high Knudsen number regime. We extended LBM to avoid the ray effect coming from the normally rather low number of streaming directions of the normally used LBM stencils. We did that by approximating arbitrary streaming directions by alternating basis LBM grid directions in time. So the advection is not approximated exactly anymore but can have a maximum of one grid spacing deviation from the desired trajectory for any time. On the other hand, we land always on a grid point when streaming, so it can be called fully Lagrangian. Anyhow, we dare to call it a LBM scheme and for obvious reasons we denote it the worm-LBM.

Preprint The worm-LBM, an algorithm for a high number of propagation ...

BR,

René