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 (<i>N</i>). 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 <i>L</i>₂ errors are the same. In the Taylor-Green vortex flow, the <i>L</i>₂ error of the optimized DUGKS is the smallest with respect to the analytical solution of velocity, but the <i>L</i>₂ 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 <i>N</i> = 16, and the calculation time of Simpson-DUGKS is reduced by about 59% compared with the original DUGKS when CFL = 0.95 and <i>N</i> = 16.