Lattice Boltzmann Models Minimizing Kolmogorov–Smirnov Distance and Some Applications to Rarefied Flows

In this paper, classes of six and eight velocity (per spatial dimension) lattice Boltzmann models on Cartesian lattices are studied. We deduce several high-order lattices and monitor the difference between the first and the third semi-moments (wall half-moments) of the discrete velocity equilibrium and the Maxwell equilibrium. The lattice Boltzmann schemes with minimal errors reproduce the Maxwell boundary conditions with better accuracy. On the other hand, for the flow regimes close to free molecular ones the errors in the high-order half-moments affect accuracy. Then, we propose considering the models with the lattices also yielding the minimal Kolmogorov–Smirnov distance between the lattice equilibrium and the Maxwell equilibrium (Gaussian distribution). Two-dimensional lattice Boltzmann schemes are constructed as a tensor product of one-dimensional models. The numerical experiments for the rarefied Poiseuille and Couette flows support the idea that the models which minimize the Kolmogorov–Smirnov distance and errors in the first half-moments show better precision in describing non-equilibrium flow features.