Simple question… how to define in Thermal Lattice Boltzmann Method this part of a heat equation:
LaTeX:
\nabla ^2T
You can use an equilibrium function defined in terms of T rather than rho. If you don’t have any advection then u=0. You can also eliminate the quadratic terms in the equilibrium function. Note that the Chapman-Enskog expansion will show that you have errors at higher orders.
OK, so maybe I’ll try to explain my quastion in details 
I need to know with part of Thermal Lattice Boltzman Discreet Equation (in BGK approximation) is responsible for recovering this part of Heat Equation:
\nabla ^2T
For example in TLBM heat flux is defined as:
q=\sum_{i}^{8}g_{i}e_{i}
So how to define this: “\nabla ^2T”? Of course it can be easy computed from temperature field by 2nd Order Central difference, but I think that it can be obtainet strait from TLBM.
pleb01 what do You exacly mean. Can You explain me Your idea in details ?
It comes from the divergence of the non-equilibrium part of the flux, in a similar way to how the viscous stress comes from the equivalent expression in fluid mechanical LBEs. Perhaps have a look at this paper http://www.sciencedirect.com/science/article/pii/S004579301000352X It’s not for thermal LBE, but it explains how to get an advection-reaction-diffusion equation, and it is second order in time. Of course, you can ignore the reaction term in this article! I think there are quite a few articles out there that explicitly deal with TLBE or diffusion equations - Ginzburg, for example, has a very interesting articles for TRT collisions.