One of the central problems in the study of rarefied gas dynamics is to find the steady-state solution of the Boltzmann equation quickly. When the Knudsen number is large, i.e. the system is highly rarefied, the conventional iterative scheme can lead to convergence within a few iterations. However, when the Knudsen number is small, i.e. the flow falls in the near-continuum regime, hundreds of thousands iterations are needed, and yet the “converged” solutions are prone to be contaminated by accumulated error and large numerical dissipation. Recently, based on the gas kinetic models, the implicit unified gas kinetic scheme (UGKS) and its variants have significantly reduced the number of iterations in the near-continuum flow regime, but still much higher than that of the highly rarefied gas flows. In this paper, we put forward a general synthetic iterative scheme (GSIS) to find the steady-state solutions of rarefied gas flows within dozens of iterations at any Knudsen number. The key ingredient of our scheme is that the macroscopic equations, which are solved together with the Boltzmann equation and help to adjust the velocity distribution function, not only asymptotically preserve the Navier-Stokes limit in the framework of Chapman-Enskog expansion, but also contain the Newton's law for stress and the Fourier's law for heat conduction explicitly. For this reason, like the implicit UGKS, the constraint that the spatial cell size should be smaller than the mean free path of gas molecules is removed, but we do not need the complex evaluation of numerical flux at cell interfaces. What's more, as the GSIS does not rely on the specific collision operator, it can be naturally extended to quickly find converged solutions for mixture flows and even flows involving chemical reactions. These two superior advantages are expected to accelerate the slow convergence in the simulation of near-continuum flows via the direct simulation Monte Carlo method and its low-variance version.

稀有气体动力学研究的中心问题之一是快速找到玻尔兹曼方程的稳态解。当Knudsen数很大时，即系统非常稀少，常规的迭代方案可能会导致几次迭代收敛。但是，当Knudsen数小时，即流量落在近连续状态下，则需要成千上万次迭代，但是“收敛”的解决方案容易受到累积误差和较大数值耗散的污染。最近，基于气体动力学模型，隐式统一气体动力学方案（UGKS）及其变体已大大减少了近连续流态下的迭代次数，但仍远高于高度稀疏的气体流。在本文中，我们提出了一种一般的合成迭代方案（GSIS），以在任何Knudsen数的数十次迭代中找到稀薄气流的稳态解。该方案的关键要素是与Boltzmann方程一起求解并有助于调节速度分布函数的宏观方程，不仅在Chapman-Enskog展开框架中渐近地保留了Navier-Stokes极限，而且还包含牛顿应力定律和傅立叶热传导定律。因此，像隐含的UGKS一样，消除了空间像元大小应小于气体分子的平均自由程的约束，但是我们不需要对像元界面处的数值通量进行复杂的评估。更重要的是，由于GSIS并不依赖于特定的碰撞算子，因此可以自然地扩展它，以快速找到混合流甚至化学反应流的收敛解。通过直接模拟蒙特卡洛方法及其低方差版本，这两个优越的优势有望加速近连续流模拟的缓慢收敛。