The general synthetic iterative scheme (GSIS) is extended to find the steady-state solution of the nonlinear gas kinetic equation, resolving the long-standing problems of slow convergence and requirement of ultra-fine grids in near-continuum flows. The key ingredient of GSIS is the tight coupling of gas kinetic and macroscopic synthetic equations, where the constitutive relations explicitly contain Newton's law of shear stress and Fourier's law of heat conduction. The higher-order constitutive relations describing rarefaction effects are calculated from the velocity distribution function; however, their constructions are simpler than our previous work (Su et al., 2020 ) for linearized gas kinetic equations. On the other hand, solutions of macroscopic synthetic equations are used to accelerate the evolution of gas kinetic equation at the next iteration step. A rigorous linear Fourier stability analysis of the present schemes in periodic system shows that the error decay rate of GSIS can be smaller than 0.5, which means that the deviation to steady-state solution can be reduced by 3 orders of magnitude in 10 iterations. Other important advantages of the GSIS are: (i) it does not rely on the specific form of Boltzmann collision operator, and (ii) it can be solved by sophisticated techniques in computational fluid dynamics, making it amenable to large scale engineering applications. In this paper, the efficiency and accuracy of GSIS are demonstrated by a number of canonical test cases in rarefied gas dynamics, covering different flow regimes.
- fast convergence
- Fourier stability analysis
- gas kinetic equation
- general synthetic iterative scheme
- multi-scale rarefied gas flows