D1Q3 as a projection


I recently got back a review on an article where I examine purely one-dimensional plane wave behaviour using a D1Q3 lattice. The reviewer disagrees with my choice of lattice, claiming that using D2Q9 or D3Q27 would make more sense here.

I don’t agree with that. Since the D1Q3 lattice and its lattice weights is a one-dimensional projection of D2Q9, D3Q15, D3Q19, D3Q27 and their weights, I would think that one-dimensional behaviour along a major axis of any of these higher-dimensional lattices is identical with corresponding behaviour in a D1Q3 lattice. (I am not concerned with isotropy in my article.) I found a similar view in an article by A.J. Wagner (DOI: 10.1103/PhysRevE.74.056703).

Am I wrong here? I don’t think I am, but since I believe the reviewer to be a certain well-known LB researcher, I don’t want to dismiss his opinion out of hand.

Erlend M

Dear Erlend,

I am nearly perfectly sure that you are right. As you say, one can show that the D1Q3 lattice is the 1D projection of the lattices you have mentioned. In fact, I have recently read an article where this is explicitly shown. It would be a major surprise if the results differed (always assuming that everything is perfectly 1D). Possible deviations should be a machine accuracy artifact. You should defend your point. Maybe, it is necessary to perform a comparative 2D simulation.

Good luck,

Thanks for the support, Timm! :slight_smile:

I unfortunately don’t have space in my article to show this explicitly myself, but I’d like to refer to an article to support my approach. The article you mention sounds perfect, do you remember which one it was?

Erlend M

I do. But it is not published yet. :frowning:
Are there more references in Alexander Wagner’s article?
Maybe this article also helps: Chikatamarla, Karlin. Complete Galilean invariant lattice Boltzmann models. Comput. Phys. Commun. 179 (2008) 140-143


Ah, too bad.

I read the article you referred to, and it was quite interesting, but went in the opposite direction: It constructed a two-dimensional lattice from a one-dimensional one. Still very relevant, but I’d like a clearer statement if I’m going to cite an article for support.

I’ve managed so far to find only three articles which state that D1Q3 is a projection of the higher-dimensional lattices. The Wagner article I referred to above, and two others by Wagner and Li (DOIs: 10.1103/PhysRevE.76.036701 and 10.1016/j.physa.2005.09.030). All of these simply state that D1Q3 is a projection, without referring to other articles. I guess they consider it obvious enough that it doesn’t need a citation.

I find it strange that this doesn’t seem to be more widely stated, since this is quite fundamental. Ah well, I guess I’ll cite one of Wagner’s articles to support my approach. They’re very clear on stating that D1Q3 is a projection.

Erlend M

Dear Erlend,

It is the same from the point of simulation view. However, they are quite different in terms of stability, though multidimensional models can be reduced to the one-dimensional case in terms of stability - I suppose it’s the same case as how to choose weights parameters to reduce multidimensional models to one dimensional. You can refer to this article but too hard to read:

The role of the kinetic parameter in the stability of two-relaxation-time advection–diffusion lattice Boltzmann schemes
I. Ginzburg , and A.A. Mohamad

By the way, if you draw a D2Q9 lattice and assume that it’s uniform in y direction - that right away implies that the information is only transported in x direction by D1Q3 population which automatically implies the same behaviour of D2Q9 and D1Q3.

Hopefully it will help,