Model Cascades for Stochastic Particle Simulations of Rarefied Polyatomic Gases

小尺度或稀薄条件下的气体流动可能严重偏离热平衡，传统的Navier-Stokes-Fourier流体动力学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分子相互作用的统计说明，为研究这种情况下的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，替代的简化模型和计算方法是可取的。除了计算昂贵的直接离散化的玻尔兹曼型方程，矩方程设计偏微分方程的模型简化程序，使流体动力学模型可以扩展到非平衡。另一方面，在粒子蒙特-卡罗方法（DSMC）中，流场由代表分布函数的计算粒子系综来描述。这些方法被广泛使用，并提供高的物理精度，但在显着的计算成本，特别是在低速和近平衡流。玻尔兹曼碰撞算子的刚度导致在近平衡状态下的大量计算成本，可以通过传递到玻尔兹曼方程的扩散极限来处理。在这种情况下，Fokker-Planck（FP）方程自然地作为扩散近似而出现。FP算符可以被看作是玻尔兹曼算符的近似，其中二元碰撞的效应由漂移扩散机制描述。换句话说，由于连续碰撞而作用在粒子上的净力被分解为与粒子速度相关的部分（漂移）和纯粹随机贡献的剩余部分（扩散）。匹配的玻尔兹曼方程的不同时刻的演变与所产生的FP模型，导致封闭方程的漂移和扩散系数。最终，FP方程有效地解决了执行底层Langevin型随机微分方程的粒子系综。在这个项目中，我们将FP方法视为模型级联，其中一个层次的运营商允许近似玻尔兹曼方程到更高的程度。该方法将被应用到多原子玻尔兹曼碰撞算子，这需要一个扩大的相空间。随机微分方程的高阶和熵稳定的实现将被研究。

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.