Model Cascades for Stochastic Particle Simulations of Rarefied Polyatomic Gases

气体在小规模装置或稀薄条件下的流动可能会严重偏离热平衡，因此传统的流体力学的Navier-Stokes-Fourier模型不能很好地描述它们。另一方面，玻尔兹曼方程通过提供对分子相互作用的统计描述，为研究这种情况下的气体流动提供了合适的起点。然而，由于玻尔兹曼碰撞算符的复杂性及其解域的高维性，需要替代的简化模型和计算方法。除了计算昂贵的Boltzmann型方程的直接离散化外，力矩方程还通过模型简化过程设计了偏微分方程组，使得流体动力学模型可以扩展到非平衡状态。另一方面，在粒子蒙特卡罗方法(DSMC)中，流场由表示分布函数的计算粒子系综来描述。这些方法应用广泛，具有很高的物理精度，但计算成本很高，特别是在低速和接近平衡的情况下。玻尔兹曼碰撞算符的刚性在近平衡区导致了大量的计算成本，可以通过传递到玻尔兹曼方程的扩散极限来处理。在此背景下，福克-普朗克(FP)方程自然作为扩散近似而产生。FP算符可以看作是Boltzmann算符的近似，其中双碰撞的影响是用漂移扩散机制来描述的。换句话说，由于连续碰撞而作用在粒子上的净力被分解成与粒子速度相关的部分(漂移)，其余部分是纯粹随机的贡献(扩散)。玻尔兹曼方程的不同矩与FP模型的不同矩的匹配演化导致了漂移和扩散系数的闭合方程。最后，通过实现粒子系综朗之万型随机微分方程组，有效地求解了FP方程。在这个项目中，我们将把FP方法看作是模型级联，在该模型中，算符的层次结构允许将Boltzmann方程逼近到更高的程度。这种方法将应用于多原子玻尔兹曼碰撞算符，它需要更大的相空间。我们将研究随机微分方程的高阶和熵稳定的实现。

Gas flows in small scale devises or rarefied conditions may depart significantly from thermal equilibrium, and thus the conventional Navier-Stokes-Fourier model of fluid dynamics fail to describe them properly. The Boltzmann equation on the other hand, gives the appropriate starting point for studying gas flows at such scenarios by providing a statistical account of molecular interactions. However, due to the complexity of the Boltzmann collision operator and high dimensionality of its solution domain, alternative reduced models and computational methods are desirable. Besides a computationally expensive direct discretization of a Boltzmann-type equations, moment equations devise partial differential equations by a model reduction procedure such that fluid dynamic models can be extended into non-equilibrium. On the other hand, in particle Monte-Carlo methods (DSMC) the flow field is described by a computational particles ensemble representing the distribution function. These methods are widely used and offer high physical accuracy, but at significant computational cost especially at low speed and near-equilibrium flows. The stiffness of the Boltzmann collision operator which results in massive computational costs in the near-equilibrium regimes, can be treated by passing to the diffusion limit of the Boltzmann equation. In this context, the Fokker-Planck (FP) equation naturally arises as a diffusion approximation. The FP operator can be regarded as an approximation of the Boltzmann operator, where the effect of binary collisions is described by a drift-diffusion mechanism. In other words, the net force acting on a particle due to successive collisions is decomposed into a part which is correlated to the particles velocity (drift) and the remainder which is a purely random contribution (diffusion). Matching evolution of different moments of the Boltzmann equation with those arising from the FP model, leads to closure equations for drift and diffusion coefficients. Ultimately, the FP equation is solved efficiently by implementing the underlying Langevin-type stochastic differential equations for a particle ensemble. In this project we will view the FP approach as model cascade in which a hierarchy of operators allows to approximate the Boltzmann equation to ever higher degree. The approach will be applied to the polyatomic Boltzmann collision operator, which requires an enlarged phase space. Higher order and entropy-stable implementations of the stochastic differential equations will be investigated.