High-order Unified Gas-kinetic Scheme

Abstract

In this paper, we present a high-order unified gas-kinetic scheme (UGKS) using the weighted essentially non-oscillatory with adaptive-order (WENO-AO) method for spatial reconstruction and the two-stage fourth-order scheme for time evolution. Since the UGKS updates both the macroscopic flow variables and microscopic distribution function, and provides an adaptive flux function by combining the equilibrium and non-equilibrium parts, it is possible to take separate treatment of the equilibrium and non-equilibrium calculation in the UGKS for the development of high-order scheme. Considering the fact that high-order techniques are commonly required for continuum flow with complex structures, and the rarefied flow structure are relatively simple and smooth in the physical space, we apply the highorder techniques in the equilibrium part of the UGKS for the capturing of macroscopic flow evolution, and retain the calculation of distribution function as a second-order method, so that a balance of computational cost and numerical accuracy could be well achieved. The high-order UGKS has been validated by several numerical test cases, including sine-wave accuracy test, sod-shock tube, Couette, oscillating Couette, lid-driven cavity and oscillating cavity flow. It is shown that the current method preserves the multiscale property of the original UGKS and obtains more accurate solutions in several cases.