In PhD thesis of Jonas (Sec 2.1.1) and to ensure lattice isotropy, he uses four of the six equation given in Eq. 2.1 (six equations) to derive the weight factors for different directions in symmetric LB schemes and the Cs value. In a standard symmetric D2Q9 scheme these are determined as t0=4/9 (rest state), ts=1/9 (short links) and tl=1/36 (long links).
But if we are dealing with an asymmetric D2Q9 scheme we will end up having ten (nine factors +Cs) unknowns which are more than the number of equations given in Eq.2.1 (six). Can someone please tell me what other isotropy equations should we use?
Hi,
I believe that these equations should be sufficient. Remember that each of these six equations contains several “subequations”, since the indices \alpha, \beta, etc. are arbitrary. For instance, using Lätt’s D2Q9 numbering, (b) becomes two different equations by choosing \alpha = x or y:
\alpha = x: t_1 - t_3 + t_5 - t_6 - t_7 + t_8 = 0,
\alpha = y: t_2 - t_4 + t_5 + t_6 - t_7 - t_8 = 0.
Thus, the system of equations is larger than it looks. Some of these equations will turn out to be linearly dependent, though. It might help to remember that if |c_\alpha| is 0 or 1, then c_\alpha^3 = c_\alpha^1 .
Good luck!
Much appreciated for your help. I will have a look at it and see how it goes.