About the time step delta_t , contradiction ..?

On the one hand, we all know that the LBM main equation is f(x+1,t+1)=f(x,t)-(1/Tau)*(f(x,t)-feq(x,t)). There is a more general equation in which the (1/Tau) is written (delta_t / Tau). So the first equation assumes that delta_t=1 (NB: most of the time we assume that delta_t= delta_x=1).

On the other hand, when one reads the article of Dr.Jonas Latt http://wiki.palabos.org/_media/howtos:lbunits.pdf , it is found that delta_t and delta_x are defined by : delta_x=1/N and delta_t=1/ N_iter , and that they are closely related to stability by the relation (delta_t< delta_x/sqrt(3)). This is in contradiction with the first two lines i wrote above.

Who can explain this? Are the delta_t in the two cases different? Did i confuse some notions?
Please help .
Thanks in advance.


I think (please correct me if I am wrong), that what is referred to as delta_t is delta_t in the simulation which for convenience is always taken as 1. On the other hand delta_t_physical is delta_t * C_t, where C_t is the conversion factor for time. Check this excellent set of slides that I found most helpful

Dear student,

the doubts you have from this “contradiction” are provoked because of the different set of units used.

the delta_x = 1/N is the non-dimensional delta_x.
the delta_x = 1 is the delta_x typically used for LBM simulations.

therefore, to pass from non-dimensional to lattice units, you need to divide the non-dimensional delta_x with the same delta_x. It is just the same procedure as you have been doing with all other macroscopic variables. delta_x can be actually transformed into a real distance with physical units of longitude(meters)

non-dimensional delta_x * characteristic length = physical delta_x

that delta_t - delta_x relation is obviously in non-dimensional units. because on the LBM they are usually 1 as I previously said.

Hope this clarifies something.