  The general synthetic iteration scheme (GSIS) is extended to find the
steady-state solution of nonlinear gas kinetic equation, removing the
long-standing problems of slow convergence and requirement of ultra-fine grids
in near-continuum flows. The key ingredients of GSIS are that the gas kinetic
equation and macroscopic synthetic equations are tightly coupled, and the
constitutive relations in macroscopic synthetic equations 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. Journal of
Computational Physics 407 (2020) 109245) for linearized gas kinetic equations.
On the other hand, solutions of macroscopic synthetic equations are used to
inform the evolution of gas kinetic equation at the next iteration step. A
rigorous linear Fourier stability analysis 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 is demonstrated by a number of canonical test cases in
rarefied gas dynamics.
