一个多尺度流输运的不确定性量化随机动力学格式
A stochastic kinetic scheme for multi-scale flow transport with uncertainty quantification
摘要：
气体流动在不同的特征尺度上表现出一系列不同的行为。考虑到流体理论中的粗粒度建模，流场解与真实物理之间可能存在相当大的不确定性。从分子水平到流体动力学水平研究不确定性的产生、传播和演化，为发展合理的理论和可靠的多尺度数值算法带来了巨大的机遇和挑战。在本文中，将开发一个新的随机动力学计划，包括通过随机Galerkin和配置方法的杂交的不确定性。该格式基于Boltzmann-BGK模型方程，采用尺度相关的演化解，在时空域上离散控制方程。因此，典型的流动物理可以恢复相对于不同的物理特征尺度和数值分辨率在一个自适应的方式。我们证明了该格式在包含随机变量的不同流态下形式上是渐近保持的，从而可用于研究不确定性影响下的多尺度非平衡气体动力学.数值实验验证了该方案的有效性。我们进行了新的物理观测，如从连续到稀薄状态的不确定性的波传播模式。这些现象将被提出和定量分析。目前的方法提供了一种新的工具来量化多尺度流动演变的不确定性。
英文摘要：
Gaseous flows show a diverse set of behaviors on different characteristic scales. Given the coarse-grained modeling in theories of fluids, considerable uncertainties may exist between the flow-field solutions and the real physics. To study the emergence, propagation and evolution of uncertainties from molecular to hydrodynamic level poses great opportunities and challenges to develop both sound theories and reliable multi-scale numerical algorithms. In this paper, a new stochastic kinetic scheme will be developed that includes uncertainties via a hybridization of stochastic Galerkin and collocation methods. Based on the Boltzmann-BGK model equation, a scale-dependent evolving solution is employed in the scheme to construct governing equations in the discretized temporal-spatial domain. Therefore typical flow physics can be recovered with respect to different physical characteristic scales and numerical resolutions in a self-adaptive manner. We prove that the scheme is formally asymptotic-preserving in different flow regimes with the inclusion of random variables, so that it can be used for the study of multi-scale non-equilibrium gas dynamics under the effect of uncertainties. Several numerical experiments are shown to validate the scheme. We make new physical observations, such as the wave-propagation patterns of uncertainties from continuum to rarefied regimes. These phenomena will be presented and analyzed quantitatively. The current method provides a novel tool to quantify the uncertainties within multi-scale flow evolutions.
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