A Comparative Study of an Asymptotic Preserving Scheme and Unified Gas-kinetic Scheme in Continuum Flow Limit

TL;DR

This paper compares an implicit-explicit asymptotic preserving scheme and the unified gas kinetic scheme for simulating continuum flows, demonstrating UGKS's superior accuracy in capturing viscous solutions in the lid-driven cavity benchmark.

Contribution

It provides a detailed comparison of two AP schemes in continuum flow regimes, highlighting the importance of coupling collision and transport in numerical treatment.

Findings

UGKS accurately captures viscous flow solutions

IMEX AP scheme struggles to match benchmark solutions

Coupling collision and transport improves scheme accuracy

Abstract

Asymptotic preserving (AP) schemes are targeting to simulate both continuum and rarefied flows. Many AP schemes have been developed and are capable of capturing the Euler limit in the continuum regime. However, to get accurate Navier-Stokes solutions is still challenging for many AP schemes. In order to distinguish the numerical effects of different AP schemes on the simulation results in the continuum flow limit, an implicit-explicit (IMEX) AP scheme and the unified gas kinetic scheme (UGKS) based on Bhatnagar-Gross-Krook (BGk) kinetic equation will be applied in the flow simulation in both transition and continuum flow regimes. As a benchmark test case, the lid-driven cavity flow is used for the comparison of these two AP schemes. The numerical results show that the UGKS captures the viscous solution accurately. The velocity profiles are very close to the classical benchmark solutions. However, the IMEX AP scheme seems have difficulty to get these solutions. Based on the analysis and the numerical experiments, it is realized that the dissipation of AP schemes in continuum limit is closely related to the numerical treatment of collision and transport of the kinetic equation. Numerically it becomes necessary to couple the convection and collision terms in both flux evaluation at a cell interface and the collision source term treatment inside each control volume.