Multiscale Computation in Kinetic Theory

动力学理论中的多尺度计算

• 批准号: 1619778
• 负责人: Qin Li
• 金额: $ 25万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1619778&HistoricalAwards=false
• 关键词: Multiscale; Computation; Kinetic; Theory

Physical systems are modeled at different scales and to different approximations with different equations. The Schrodinger equation at the quantum level models the molecular and atomic scale. Newton's laws in classical mechanics model the macroscopic scale. The Boltzmann equation applies at the statistical level, where systems of many particles are studied, and the Navier-Stokes equation and others model distributed systems such as fluids in the continuum regime. A central question in applied mathematics and physics is to understand the relationships between the different models, and many tools (both analytical and numerical) have been developed for this task through the years. However, most of them idealize the systems under study and cannot tackle practical problems that have emerged in the study of complicated systems in chemistry, physics, and engineering. This project focuses on two longstanding challenges concerning these connections: the characterization of quantum information in the classical regime when chemical reactions are present, and the coupling between the statistical and the fluid description. The project aims to develop improved methods for the modeling of multiscale systems. Despite their fundamental importance in physics and engineering, effective mathematical analysis and computational techniques for multiscale problems in kinetic theory have remained rather elusive. The multiple scales inherent in many physical systems have posed notorious computational challenges. This project concerns development of multiscale numerical methods in kinetic theory, including numerical capture of the hydrodynamic limit of Boltzmann-type equations and the semi-classical limit of the Schrodinger equation. Both have long been regarded as fundamental problems in kinetic theory. More specifically, the project focuses on capturing the non-adiabatic transition in the classical regime derived from quantum mechanics, and boundary layer effects that connect the fluid description with the statistical mechanical description. Both problems emerge in transition regimes, the multi-physics phenomena can be captured by none of currently available mathematical treatments, and the computation is far from being efficient. The project aims to develop and analyze efficient computational tools for these problems, focusing on treatment of boundary layers and interfaces and design of asymptotic-preserving schemes. Besides leading to improved understanding of physical systems of these types, it is expected that the new tools under development could inspire treatments of similar problems emerging in other areas, for example, hyperbolic type problems with random media.

物理系统以不同的尺度和不同的近似用不同的方程建模。量子水平上的薛定谔方程模拟了分子和原子尺度。经典力学中的牛顿定律是宏观尺度的模型。玻尔兹曼方程适用于统计水平，在统计水平上研究许多粒子的系统，而纳维-斯托克斯方程和其他模型分布系统，如连续统状态下的流体。应用数学和物理学的一个核心问题是理解不同模型之间的关系，多年来已经开发了许多工具（分析和数值）来完成这项任务。然而，它们中的大多数将所研究的系统理想化，而不能解决在化学、物理和工程等复杂系统研究中出现的实际问题。该项目侧重于两个长期存在的挑战：当化学反应存在时，经典状态下量子信息的表征，以及统计和流体描述之间的耦合。该项目旨在开发改进的多尺度系统建模方法。尽管它们在物理和工程中具有重要的基础意义，但动力学理论中多尺度问题的有效数学分析和计算技术仍然相当难以捉摸。许多物理系统中固有的多尺度已经带来了臭名昭著的计算挑战。本项目涉及动力学理论中多尺度数值方法的发展，包括玻尔兹曼型方程的水动力极限和薛定谔方程的半经典极限的数值捕获。长期以来，这两个问题都被视为动力学理论的基本问题。更具体地说，该项目侧重于捕捉由量子力学导出的经典状态中的非绝热转变，以及将流体描述与统计力学描述联系起来的边界层效应。这两个问题都出现在过渡状态中，多物理场现象无法被目前可用的数学处理所捕获，而且计算效率还远远不够。该项目旨在为这些问题开发和分析有效的计算工具，重点是边界层和界面的处理以及渐近保持格式的设计。除了提高对这些类型的物理系统的理解外，预计正在开发的新工具可以启发对其他领域出现的类似问题的处理，例如随机介质的双曲型问题。

项目成果

Inverse Problems for the Stationary Transport Equation in the Diffusion Scaling

扩散标度中稳态输运方程的反演问题

• DOI: 10.1137/18m1207582
• 发表时间: 2019
• 期刊: SIAM Journal on Applied Mathematics
• 影响因子: 1.9
• 作者: Lai, Ru-Yu;Li, Qin;Uhlmann, Gunther
• 通讯作者: Uhlmann, Gunther