The incompressible Navier–Stokes limit from the discrete-velocity BGK Boltzmann equation

In this paper, we extend the Bardos–Golse–Levermore program (Bardos et al 1993 Commun. Pure Appl. Math. 46 667–753) to prove that a local weak solution to the d-dimensional incompressible Navier–Stokes equations ( d⩾2) can be constructed by taking the hydrodynamic limit of a discrete-velocity Boltzmann equation with a simplified Bhatnagar–Gross–Krook collision operator. Moreover, in the case when the dimension is d=2,3, we characterise the combinations of finitely many particle velocities and probabilities that lead to the incompressible Navier–Stokes equations in the hydrodynamic limit. Numerical computations conducted in two-dimensional indicate that in the case of the simplest velocity lattice (D2Q9), the rate with which this hydrodynamic limit is achieved is of order O(ε2), where ε→0 is the Knudsen number. For the future investigations, it is worth considering if the hydrodynamic limit of the discrete-velocity Boltzmann equation can be also rigorously justified in the presence of non-trivial boundary conditions.