Model Cascades for Stochastic Particle Simulations of Rarefied Polyatomic Gases

小型装置或稀薄条件下的气体流动可能明显偏离热平衡，因此传统的流体动力学Navier-Stokes-Fourier模型无法正确描述它们。另一方面，玻尔兹曼方程通过提供分子相互作用的统计说明，为在这种情况下研究气体流动提供了适当的起点。然而，由于玻尔兹曼碰撞算子的复杂性和其解域的高维性，需要替代的简化模型和计算方法。除了计算上昂贵的直接离散玻尔兹曼型方程外，矩方程通过模型简化程序设计偏微分方程，使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。这些方法被广泛使用，具有很高的物理精度，但在低速和接近平衡流动时，计算成本很高。在接近平衡状态下，波尔兹曼碰撞算子的刚度导致了大量的计算成本，可以通过传递到波尔兹曼方程的扩散极限来处理。在这种情况下，福克-普朗克（FP）方程自然地作为扩散近似出现。FP算符可以看作是波尔兹曼算符的近似，其中二元碰撞的效应由漂移扩散机制描述。换句话说，由于连续碰撞作用在粒子上的净力被分解为与粒子速度（漂移）相关的部分和纯粹随机贡献（扩散）的其余部分。将玻尔兹曼方程的不同矩的演化与FP模型的矩的演化相匹配，得到漂移系数和扩散系数的闭合方程。最后，通过实现粒子系综的基本朗万型随机微分方程，有效地求解了FP方程。在这个项目中，我们将把FP方法看作是一个级联模型，在这个级联模型中，操作员的层次结构允许将玻尔兹曼方程近似到更高的程度。该方法将应用于需要扩大相空间的多原子玻尔兹曼碰撞算子。随机微分方程的高阶和熵稳定实现将被研究。

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.