High Order Methods for Kinetic Transport Models

动力学输运模型的精确、鲁棒和高效的模拟对于物理和工程中的许多应用（例如稀薄气体动力学、等离子体物理、核和生物医学工程以及网络动力学）具有根本重要性。 虽然流体模型通常提供低维近似，但它们仅在系统接近其平衡时才有效，因此动力学输运模型需要考虑更广泛的现象，包括那些在空间和/或时间中显示多个尺度的现象或那些涉及来自多个能量群的粒子的现象。 本项目旨在从算法和数学两个方面为含时多尺度动力学输运模型提供高阶精度的计算工具。动力学输运方程是对中子、光子、分子等粒子的输运及其与宿主介质或它们之间相互作用的数学描述，它们有着广泛的应用。数值计算的挑战在于相空间的高维性，小尺度，空间（甚至时间）的多尺度，非线性相互作用，多重积分的碰撞算子，以及解决方案的守恒性或正性。 因此，该项目的目标是在理解和模拟随时间变化的多尺度动力学输运模型方面迈出重要的一步，目的是：1）为动力学输运界提供具有良好稳定性的精确确定性模拟工具，2）开发新的分析和数值技术，以解决与精确多尺度动力学输运模拟相关的一些挑战。长期目标是开发算法，以高分辨率和高效率稳健地模拟更真实的动力学模型。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Accurate, robust and efficient simulations of kinetic transport models are of fundamental importance to many applications in physics and engineering, such as rarefied gas dynamics, plasma physics, nuclear and biomedical engineering, and network dynamics. While fluid models often provide lower dimensional approximations, they are valid only when the systems are closed to their equilibria, and therefore kinetic transport models are necessary to account for a wider range of phenomena, including those displaying multiple scales in space and/or time or those involving particles from multiple energy groups. This project aims at advancing computational tools of high order accuracy for time-dependent multi-scale kinetic transport models, both algorithmically and mathematically.Kinetic transport equations are mathematical descriptions of the transport of particles such as neutrons, photons, molecules as well as their interaction with a host medium or among themselves, and they arise in a broad range of applications. The numerical challenges lie in the high dimensionality of the phase space, small scales, multiple scales in space (and even in time), nonlinear interactions, collision operators with multi-fold integrals, as well as the conservation or positivity property of the solutions. The objective of this project is therefore to take important steps in the understanding and simulations of the time-dependent multi-scale kinetic transport models, with the aim of: 1) providing accurate deterministic simulation tools with excellent stability to the kinetic transport community, and 2) developing novel analytical and numerical techniques to address some challenges related to accurate multi-scale kinetic transport simulations. The longer term goal is to develop algorithms robustly simulating more realistic kinetic models with high resolution and good efficiency.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.