A comparative study of an asymptotic preserving scheme and unified gas-kinetic scheme in continuum flow limit

Asymptotic preserving (AP) schemes target to simulate both continuum and rarefied flows. Many existing AP schemes are capable of recovering the Euler limit in the continuum regime. However, to get accurate Navier-Stokes solutions is still challenging. In order to distinguish physical mechanism underlining different AP schemes, an implicit-explicit (IMEX) AP method and the unified gas kinetic scheme (UGKS) are employed to solve the Bhatnagar-Gross-Krook (BGK) kinetic equation in both transition and continuum flow regimes. As a benchmark test case, the lid-driven cavity flow is used for evaluating numerical performance of these two AP schemes. The numerical results show that the UGKS captures the viscous solution accurately. The velocity profiles converge to the classical benchmark solutions in continuum regime with the mesh size being much larger than the local particle mean free path. However, the IMEX AP scheme seems to have difficulty to get these solutions in the corresponding limit. The analysis demonstrates that the dissipation of AP schemes has to be properly controlled in the continuum flow regime through a delicate numerical treatment of collision and convection of the kinetic equation. Physically, it becomes necessary to couple both the convection and collision terms in the flux evaluation in order to recover correct Navier-Stokes limit.