On Accuracy of the Lattice Boltzmann Equations of Low and High Orders as Applied to Slow Isothermal Microflows

Application of two-dimensional lattice Boltzmann equations of various orders on standard lattices to the description of slow isothermal rarefied flows is considered. A two-dimensional Poiseuille flow at different Knudsen numbers is used as a test problem. This problem is solved numerically using several lattice Boltzmann equations of low and high orders that have from nine to 53 discrete velocities. The results are compared with the solutions to the linearized Boltzmann and Bhatnagar–Gross–Crook equations, which are used as test ones. Numerical experiments showed that increasing the order of the lattice Boltzmann equation (i.e., the number of first moments of the local Maxwell distribution reproduced by the discrete local equilibrium of the lattice Boltzmann equation) does not always lead to improved accuracy. In particular, a new low-order LB model with 16 velocities which reproduces diffuse reflection boundary conditions at a qualitative level is proposed. It is shown that for this model it is possible to obtain accurate values of the volumetric flow rate slip velocities for a wide range of Knudsen numbers in comparison with other models under consideration.