The unified gas-kinetic scheme (UGKS) has been developed for rarefied and continuum flow simulations. To further enhance the computational efficiency of the implicit UGKS (IUGKS) [Y. Zhu, C. Zhong, and K. Xu, J. Comput. Phys., 315 (2016), pp. 16--38] for the steady-state solution, a two-step IUGKS is proposed in this paper. The multiscale solution of the UGKS is determined by the integral solution of the kinetic model equation, which is composed of the Lagrangian integration of the equilibrium and the free particle transport of the nonequilibrium state. With the implicit evaluation of the macroscopic variables in the first iterative step, the integration of the equilibrium can be directly used in the flux calculation of macroscopic flow variables in the second iterative step. This is equivalent to including the viscous flux in the implicit scheme to accelerate the convergence of the solution instead of using the Euler flux in the iterative process of the original IUGKS. In the present IUGKS, the update of macroscopic flow variables is closely coupled with the implicit evolution of the gas distribution function. At the same time, to get a more accurate physical solution the full Boltzmann collision term has been incorporated into the current scheme through the penalty method. Different iterative techniques, such as lower-upper symmetric Gauss--Seidel and multigrid, are used for solving the linear algebraic system of coupled macroscopic and microscopic equations. The efficiency of the current IUGKS has reached an outstanding level among all implicit schemes for the kinetic equations in the literature. Several numerical examples are used to validate the performance of the IUGKS. Accurate solutions have been obtained efficiently in all cases from rarefied to continuum regimes and from low to hypersonic speed.

In order to further enhance the computational efficiency of the implicit unified gas-kinetic scheme (IUGKS, JCP 315 (2016) 16-38) for multi-scale flow simulation, a two-step IUGKS is proposed in this paper. The multiscale solution of the UGKS is determined by the integral solution of the kinetic model equation, which is composed of the Lagrangian integration of the equilibrium and the free particle transport of the nonequilibrium state. With the implicit evaluation of the macroscopic variables in the first iterative step, the integration of the equilibrium can be directly used in the flux calculation of macroscopic flow variables in the second iterative step. This is equivalent to include the viscous flux in the implicit scheme to accelerate the convergence of the solution instead of using the Euler flux in the iterative process of the original IUGKS. In the present IUGKS, the update of macroscopic flow variables are closely coupled with the implicit evolution of the gas distribution function. At the same time, in order to get the more accurate solution, the full Boltzmann collision term is incorporated into the current scheme through the penalty method. Different iterative techniques, such as LU-SGS and multi-grid, are used for solving the linear algebraic system of coupled macroscopic and microscopic equations. The efficiency of the IUGKS has reached a favorable level among all implicit schemes for the kinetic equations in the literature. Several numerical examples are used to validate the performance of the IUGKS. Accurate solutions have been obtained efficiently in all flow regimes from low speed to hypersonic ones.

The non-equilibrium gas dynamics is described by the Boltzmann equation, which can be solved numerically through the deterministic and stochastic methods. Due to the complicated collision term of the Boltzmann equation, many kinetic relaxation models have been proposed and used in the past seventy years for the study of rarefied flow. In order to develop a multiscale method for the rarefied and continuum flow simulation, by adopting the integral solution of the kinetic model equation a DVM-type unified gas-kinetic scheme (UGKS) has been constructed. The UGKS models the gas dynamics on the cell size and time step scales while the accumulating effect from particle transport and collision has been taken into account within a time step. Under the UGKS framework, a unified gas-kinetic wave-particle (UGKWP) method has been further developed for non-equilibrium flow simulation, where the time evolution of gas distribution function is composed of analytical wave and individual particle. In the highly rarefied regime, particle transport and collision will play a dominant role. Due to the single relaxation time model for particle collision, there is a noticeable discrepancy between the UGKWP solution and the full Boltzmann or DSMC result, especially in the high Mach and Knudsen number cases. In this paper, besides the kinetic relaxation model, a modification of particle collision time according to the particle velocity will be implemented in UGKWP. As a result, the new model greatly improves the performance of UGKWP in the capturing of non-equilibrium flow. There is a perfect match between UGKWP and DSMC or Boltzmann solution in the highly rarefied regime. In the near continuum and continuum flow regime, the UGKWP will gradually get back to the macroscopic variables based Navier-Stokes flow solver at small cell Knudsen number.