The general synthetic iterative scheme (GSIS) is extended to find the steady-state solution of 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. Journal of Computational Physics 407 (2020) 109245) 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 is demonstrated by a number of canonical test cases in rarefied gas dynamics, covering different flow regimes.

扩展了通用合成迭代方案（GSIS），以找到非线性气体动力学方程的稳态解，从而解决了长期收敛缓慢的问题以及在近连续流中对超细网格的要求。GSIS的关键要素是气体动力学方程和宏观合成方程的紧密耦合，其中本构关系明确包含牛顿剪应力定律和傅立叶热传导定律。描述稀疏效应的高阶本构关系是从速度分布函数计算出来的，但是，它们的构造比我们以前的工作更简单（Su等。Journal of Computational Physics 407（2020）109245）中的线性气体动力学方程。另一方面，宏观合成方程的解用于在下一个迭代步骤中加速气体动力学方程的演化。在周期系统中对这些方案进行严格的线性傅里叶稳定性分析表明，GSIS的误差衰减率可以小于0.5，这意味着在10次迭代中，稳态求解的偏差可以减少3个数量级。GSIS的其他重要优点是：（i）不依赖于玻尔兹曼碰撞算符的特定形式，并且（ii）可以通过复杂的计算流体动力学技术来解决它，从而使其适合大规模工程应用。在本文中，