Comparative Study on Flux Solution Methods of Discrete Unified Gas Kinetic Scheme.

In this work, the Simpson method is proposed to calculate the interface flux of a discrete unified gas kinetic scheme (DUGKS) according to the distribution function at the node and the midpoint of the interface, which is noted by Simpson-DUGKS. Moreover, the optimized DUGKS and Simpson-DUGKS considering the force term are derived. Then, the original DUGKS, optimized DUGKS, and Simpson-DUGKS are compared and analyzed in theory. Finally, the numerical tests are performed under different grid numbers (N). In the steady unidirectional flow (Couette flow and Poiseuille flow), the three methods are stable under different Courant-Friedrichs-Lewy (CFL) numbers, and the calculated L₂ errors are the same. In the Taylor-Green vortex flow, the L₂ error of the optimized DUGKS is the smallest with respect to the analytical solution of velocity, but the L₂ error of the optimized DUGKS is the largest with respect to the analytical solution of density. In the lid-driven cavity flow, the results of the optimized DUGKS deviate more from the reference results in terms of accuracy, especially in the case of a small grid number. In terms of computational efficiency, it should be noted that the computational time of optimized DUGKS increases by about 40% compared with the original DUGKS when CFL = 0.1 and N = 16, and the calculation time of Simpson-DUGKS is reduced by about 59% compared with the original DUGKS when CFL = 0.95 and N = 16.