Abstract The in-depth knowledge of rarefied gas dynamics is crucial to address challenges in a wide range of engineering problems, where gas flows are usually multiscale, i.e., covering a wide range of Knudsen numbers. As the traditional Navier–Stokes equations fail, gas kinetic equations are required to model the flows. So far, very few numerical methods are designed to efficiently solve the multiscale gas dynamics and reveal the role of internal degrees of freedom of gas molecules. In this work, a general synthetic iterative scheme (GSIS) is proposed to find steady-state solutions of the gas kinetic equations for molecular gas flows accurately and efficiently, where the gas kinetic equations are solved together with the macroscopic synthetic equations that expedite solutions towards the steady state. In the macroscopic synthetic equations, while the momentum equation is the same as that used in the GSIS for monatomic gas, two energy equations are introduced here for polyatomic gases: one is for the translational energy and the other for the internal energy; these equations are derived exactly from the gas kinetic equations hence no approximation is made in final solutions. The Fourier stability analysis is performed to show that the GSIS permits fast convergence to steady-state solutions in the entire flow regime; meanwhile the asymptotic analysis shows that the GSIS recovers the Navier–Stokes equations when the Knudsen number is small, even on the spatial grid with cell size much larger than the molecular mean free path. With all these unique features, several challenging numerical examples are given to show that the proposed GSIS is a promising tool to simulate multiscale molecular gas flows and investigate the effects of internal degrees of freedom.

中文翻译：

---

通过通用合成迭代方案对分子气体流动进行多尺度模拟

摘要 稀薄气体动力学的深入知识对于解决各种工程问题中的挑战至关重要，其中气体流动通常是多尺度的，即涵盖范围广泛的克努森数。由于传统的 Navier-Stokes 方程失败，因此需要气体动力学方程来模拟流动。到目前为止，很少有数值方法被设计来有效地解决多尺度气体动力学问题并揭示气体分子内部自由度的作用。在这项工作中，提出了一种通用的合成迭代方案 (GSIS) 来准确有效地找到分子气体流动的气体动力学方程的稳态解，其中气体动力学方程与宏观合成方程一起求解，加速求解稳态。在宏观综合方程中，虽然动量方程与 GSIS 中用于单原子气体的动量方程相同，但这里为多原子气体引入了两个能量方程：一个是平移能，另一个是内能；这些方程完全从气体动力学方程推导出来，因此在最终解中没有进行近似。傅里叶稳定性分析表明 GSIS 允许在整个流态中快速收敛到稳态解；同时渐近分析表明，即使在单元尺寸远大于分子平均自由程的空间网格上，当 Knudsen 数较小时，GSIS 也能恢复 Navier-Stokes 方程。凭借所有这些独特的功能，