Comparative study of discrete velocity method and high-order lattice Boltzmann method for simulation of rarefied flows

A comparative study of discrete velocity method (DVM) and high-order lattice Boltzmann method (HLBM) is made in this work. The key difference between the DVM and HLBM lies in the use of equilibrium distribution function and discrete velocity. In the DVM, the complete Maxwellian distribution function or the Shakhov distribution function is used and the discrete velocity is to represent the continuous velocity domain in the particle velocity space. As a result, the integral forms for the moments of distribution function are approximated by numerical integration. In order to make the quadrature error to be small enough, a large number of discrete velocity points are required for the DVM in general case. In contrast, in the HLBM, the lattice Boltzmann model is utilized as the equilibrium distribution function and discrete velocity. The construction of lattice Boltzmann model is intended to seek a minimal set of velocities in the particle velocity space to represent the objective hydrodynamics. With lattice properties of the lattice Boltzmann model, the weighted sums of distribution functions at those velocity points can exactly satisfy the physical conservation laws. However, the choice of discrete velocity points may have considerable effect on the accuracy of HLBM for fluid flows with large Knudsen number as reported in literatures [26, 29-30, 34-36]. From this study, it was found that HLBM provides reasonable results for fluid flows at moderate Knudsen number and low temperature difference with less computational cost than that of DVM. For fluid flows with large Knudsen number and/or high temperature difference, the DVM is preferred although more computational cost is required.