Higher order LBM

Dear all,

With sending best greetings to you.
I am Mohammad, M.Sc. student of mechanical engineering
my final thesis is about micro flow simulation by
lattice Boltzmann method, i have done a micro flow simulation by LBM
based on D2Q9 and effective mean free path to capture Knudsen layer.
I also have used diffuse scattering boundary condition(DSBC) and I got
good results! For higher accuracy, i am going to approach toward higher
order lattice Boltzmann methods like D2Q13 and D2Q16. When i was
searching, i saw your website! (LBMethod.org).
I search in your website which was very perfect.

I start to write a code for micro flow based on D2Q13 LBM.
But there are many questions in my mind such as, “what kind of boundary condition (BCs)
can we use in D2Q13 model?”, “if we can use diffuse scattering
boundary condition(DSBC), how can i implement it for D2Q13
numerically?”. I don’t know if the BCs for D2Q13 would be like D2Q9
or not!. There are several papers for D2Q9 that perfectly explained
how we can implement DSBC for D2Q9, but i have not seen any paper or thesis that explains
about boundary conditions for D2Q13 or D2Q16 and implementing boundary
conditions numerically for higher order!. I also read some
papers about higher order LBM and about boundary conditions in LBM but I could not find
how we can implement boundary conditions numerically in discrete lattice! in higher-order LB .

Another question:

What is the magnitude of sound speed “Cs” in D2Q13?
I have seen in a paper, for D2Q13, that sound speed is “Cs^2=(C^2)/2”,
where “C=(2RT)^0.5”. I set C=1.0 and Cs=1/(2^0.5) in my code, is it true?
If your answer is yes! Why?!
I know that R is the gas constant and T is the temprature that is
assumed constant(for isothermal simulation), but i don not know what is
the magnitude of R and T in the LBM! I need the values of R and T because in thermal LBM i guesse we should
calculate the C with avrage temprature! so we should know the magnitude of R!

“how can i find the magnitude of R and T in the LBM?” and “if the magnitude of Cs in D2Q13 is
Cs^2=(C^2)/2 what is the magnitude of Cs and Wi( weight factors) in
D2Q16 or in higher models?”

I have used periodic boundary condition in inlet and outlet for micro
Couette flow!.
I write my inlet boundary condition as you can see as follows (it would be high
appreciate if you check it for me).

Schematic of discrete velocities in D2Q13

        C6 C2 C5
             \ | /

/ |
C7 C4 C8

Schematic of discrete lattice

NJ * ----------–… …–-----------------*
| | | | | |
| | | | | |
----------–… …–-----*------------ *
| | | | | |
: : : : : :

:     :     :           :      :               :
|     |     |           |       |               |

2 ---------–… …–----------------- -
| | | | | |
| | | | | |
----------–… …–-------------------
1 2 NI-1 NI

first indice in the parenthesis indicates discrete directions, second
and third indices show the position of nodes in x direction and y
direction respectively!

(for inlet boundary)
F(1,1,J) = F(1,NI,J)
F(5,1,J) = F(5,NI,J)
F(8,1,J) = F(8,NI,J)
F(9,1,J) = F(9,NI-1,J)

Next question:
“In D2Q13, some particles move two lattices at each time step. For
these particles, if they are leaving a wall boundary toward the fluid
domain, what is the distribution functions for lattices between the
wall boundary lattice and the lattice where the particles arrive after dt?”.

By the way, I am studying the micro flow simulation by higher accuracy
models in details. So, I appreciate you if you send me information in
this field (such as: answer to the above questions, any codes, articles or
and any useful help to solve my problems).

I am looking forward to hear from you, soon.
sorry for bad ENGLISH!
Best regards.


I think there has been a similar question: How to treat boundary conditions in higher order LBM where populations may propagate farther than 1 grid node. I do not know the answer.
Are you sure that you have considered all the literature about D2Q13 and D2Q16?
The same goes for the speed of sound. In principle, one has to compute it for each lattice structure. In higher order LBM it may be different. This should have been done by the people introducing the lattice.
I am very sorry that I cannot help you with this, but I have no experience in this sub-topic.


Dear Timm,
thank you for your replay

could you please mention the literature about higher LBM like D2Q13 and D2Q16?
I searched but i could not find any useful one for solving my problem!

I do not know references. If you have not tried yet, you could use search engines like google scholar, or better:

Dear Timm
I have tried it but could not find anything!
maybe i am not a professional !
by the way
Thank you for your attention.

Dear Mohammad,

You can obtain sound of speed and other characteristics for D2Q12 model or D2Q16 from this paper:
2006 Cambridge University Press
J. Fluid Mech. (2006), vol. 550, pp. 413–441.
Kinetic theory representation of hydrodynamics:
a way beyond the Navier–Stokes equation
By Xiaowen Shan, Xue-Feng Yuan and Hudong Chen
However, it’s off-lattice schemes, and authors do not provide you with exact view of equilibrium function and weights (they just mention it). However, if you really want to understand you can go and dig into that paper to figure out the parameters needed.

Also, I recommend to look to papers of Karlin and his group. They did some advance multi-models, matching off-lattice models to lattice (previous paper give you exact derivation) and boundary conditions for them. But please note that it would be an entropic LBM (another construction of the equilibrium distribution function) if it’s OK with you.

The question you posed is not easy and will take time,
Good luck,

Hello, I also have the same questions. Have you solved these problems? Could you give me some help?

Thanks. best regards.