Accurate high performance computing for nonlinear collisional kinetic theory

这项工作将实现、分析和基准非线性碰撞动力学方程的精确数值格式，并将其扩展到高性能计算。动力学方程描述了介观水平上的物理系统，其中粒子之间有大量的相互作用，但它们不够频繁，不能被描述为一个连续体。在相空间中的每个点上，一个非线性积分项模拟与系统中所有其他粒子相互作用的影响。由于相互作用及其维度的精细守恒结构，这种积分的计算既困难又昂贵，但它具有许多性质，使其非常适合于并行计算。这项工作试图克服动力学模型中碰撞积分项的计算瓶颈，将保守的谱数值方法应用于大规模并行计算机体系结构，并将研究这种方法如何随着计算节点的增加而扩展。这项工作将通过对系统中的传输项应用高阶空间和时间离散化来进一步增强该方法，该方法以前由于计算碰撞项的代价而不可行，并研究了降低积分复杂性以进一步提高效率的方法。它还将仔细研究关于速度域截止和求积的计算精度顺序。这项工作将把守恒谱方法推广到角碰撞模型中各向异性的情况，特别是掠过碰撞(Landau)极限的情况，并将其应用于碰撞等离子体模型。最后，这项工作将尝试扩展渐近保持格式的思想，以有效地计算刚性粒子的碰撞项。许多重要的科学问题可以在分子水平上描述为单个粒子在周围反弹，每个粒子都有自己的速度和方向。例如，这些粒子通过交换能量和动量的碰撞相互作用，以及与其他物体相互作用，这些相互作用的平均行为可以感觉到像风一样。动力学模型描述的问题中，粒子密度足够低，碰撞效应在问题的动力学中很重要，但又不是很强，人们可以简单地使用平均行为来描述演化。动力学模型可以表征各种应用，如纳米和微尺度设备的设计，半导体和核聚变等等离子体物理中的能量应用，以及航天器和卫星的大气层再入。除了物理应用外，这些模型还可用于生物和社会动态的建模，例如有限空间中的人群动态，或对动物的成群和放牧进行建模，这可能具有安全和环境意义。这项工作将开发新的模拟工具，能够利用高性能计算资源对以前在计算上不可行的问题进行高精度模拟，从而提供更广泛的动力学应用视角。

This work will implement, analyze, and benchmark accurate numerical schemes for nonlinear collisional kinetic equations and their extension to high performance computing. Kinetic equations describe physical systems at the mesoscopic level, where there are large numbers of interactions between particles but they are not frequent enough to be described as a continuum. At each point in phase space a nonlinear integral term models the effect of interactions with all of the other particles in the system. This integral is difficult and expensive to evaluate due to the delicate conservation structure of the interactions and its dimensionality, however it has many properties that makes it well-suited for parallel computation. This work seeks to overcome the computational bottleneck of the collisional integral term in kinetic models by adapting a conservative spectral numerical method to massively parallel computer architectures, and will investigate how this scales with increasing computational nodes. The proposed work will further enhance the method by applying high order space and time discretization to the transport terms in the system, which was previously infeasible due to the expense of evaluating the collision term, as well as investigating methods for reducing the complexity of the integration for further efficiency. It will also carefully investigate the order of accuracy of computation with regards to the velocity domain cutoff and quadrature. This work will extend the conservative spectral method to the case of anisotropic in angle collisional models, in particular the grazing collisions (Landau) limit and applications to collisional plasma models. Finally, this work will attempt to extend the ideas of asymptotic-preserving schemes to efficiently compute the collision term in stiff regimes.Many important problems in science can be described at the molecular level as individual particles bouncing around, each with its speed and direction. These particles interact with each other as well as other objects through collisions that exchange energy and momentum, and the average behavior of these interactions can be felt as wind, for example. Kinetic models describe problems where particle density is low enough that collisional effects matter in the dynamics of the problem, but is not so strong that one can simply use the average behavior to describe the evolution. Kinetic models can characterize a wide variety of applications such as design of nano- and micro-scale devices, energy applications in plasma physics such as semiconductors and nuclear fusion, and atmospheric re-entry for spacecraft and satellites. In addition to physical applications, these models can be used for the modeling of biological and social dynamics, such as the dynamics of crowds in confined spaces, or modeling the flocking and herding of animals, which could be of security and environmental interest. This work will develop new simulation tools that are able to leverage high performance computing resources to perform high accuracy simulation of problems that were previously computationally infeasible, which can provide a broader view of kinetic applications.