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The general synthetic iteration scheme (GSIS) is extended to find thesteady-state solution of nonlinear gas kinetic equation, removing thelong-standing problems of slow convergence and requirement of ultra-fine gridsin near-continuum flows. The key ingredients of GSIS are that the gas kineticequation and macroscopic synthetic equations are tightly coupled, and theconstitutive relations in macroscopic synthetic equations explicitly containNewton's law of shear stress and Fourier's law of heat conduction. Thehigher-order constitutive relations describing rarefaction effects arecalculated from the velocity distribution function, however, theirconstructions are simpler than our previous work (Su et al. Journal ofComputational Physics 407 (2020) 109245) for linearized gas kinetic equations.On the other hand, solutions of macroscopic synthetic equations are used toinform the evolution of gas kinetic equation at the next iteration step. Arigorous linear Fourier stability analysis in periodic system shows that theerror decay rate of GSIS can be smaller than 0.5, which means that thedeviation to steady-state solution can be reduced by 3 orders of magnitude in10 iterations. Other important advantages of the GSIS are (i) it does not relyon the specific form of Boltzmann collision operator and (ii) it can be solvedby sophisticated techniques in computational fluid dynamics, making it amenableto large scale engineering applications. In this paper, the efficiency andaccuracy of GSIS is demonstrated by a number of canonical test cases inrarefied gas dynamics.

General synthetic iteration scheme for non-linear gas kinetic simulation of multi-scale rarefied gas flows