An Implicit Unified Gas-kinetic Scheme for Unsteady Flow in All Knudsen Regimes
The unified gas-kinetic scheme (UGKS) is a direct modeling method for multiple scale transports. The modeling scale of the scheme is the mesh size and time step, and the ratios of the mesh size over particle mean free path or the time step over particle collision time determine the local evolution regime of flow physics. For unsteady flow computation, due to the CFL condition the time step in the explicit UGKS is limited by the smallest cell size in the computational domain. As a result, for a largely stretched non-uniform mesh the global time step becomes very small and the ratio of the time step over local particle collision time may get a very small value. Under such a circumstance, the physics in explicit UGKS may be constrained by the kinetic scale physics only. The multiscale UGKS reduces into a single scale discrete velocity method where the free transport is used for the flux evaluation. In order to keep the advantages of UGKS and improve the efficiency for unsteady flow computation, the restriction from global CFL condition on the time step has to be removed. In this paper, we develop an implicit UGKS (IUGKS) for unsteady flow by alternatively solving the macroscopic and microscopic governing equations within a step iteratively. In order to preserve the coherent flow evolution and multiscale nature, the numerical flux is still evaluated by the explicit UGKS with the local CFL condition. Therefore, the multiscale property of the UGKS modeling is preserved over non-uniform meshes. Many numerical examples are included to validate the scheme for both continuum and rarefied flows. The IUGKS presents reasonably good results with second order accuracy for unsteady flow computation and the efficiency has been improved by dozen of times in comparison with the explicit UGKS.