Temporal Multi-Scale Simulation Tools Kinetic Plasma Equations

在这项工作中，派和他的学生考虑了动力学运输问题的新方法的发展。目标是开发能够连接多个时间尺度的方法。例如，等离子体物理中的许多问题，如聚变系统中的边缘等离子体问题的建模，可以同时表现出在同一问题中既符合扩散主导输运又符合无碰撞流动的行为。一般而言，这类问题可以归结为具有刚性松弛项的双曲组。通常，刚性弛豫项可以是无量纲化的，导致在形式1/d的弛豫项前面有一个标度常数。在感兴趣的问题中，当d接近零时，系统从双曲转变为抛物线。解决这类问题的一种方法是使用区域分解方法，并结合各种物理行为的适当模型。然而，这种方法的一个主要困难是定义域间双向耦合的边界条件。本文所采用的方法是发展动力学系统的数值方法，以便在系统成为碰撞支配的极限时恢复正确的极限行为。我们在这里关注的特定方法类被称为渐近保持(AP)方法。设计AP方法的目标是开发对任何d保持其精度顺序的时间步进策略。特别地，当d接近零时，AP方法应该为极限行为恢复一致的离散化。然而，发展高阶的AP方法已被证明是很难构造的。此外，对于一系列重要的测试问题，AP方法的CFL被限制为小于空间离散的平方的时间步长。在这项工作中，PI和他的学生研究了一种基于AP框架内的伪上卷法的新方法，该方法具有与d无关的收敛速度，并且时间步长的表观CFL与空间离散化成正比。此外，PI提出了一种基于积分延迟校正的将低阶AP方法提升到高阶的新方法，这是PI及其合作者开发的一种缺陷校正方法。该方法可推广到一类具有刚性松弛项的动力学方程。科学中的许多重要问题都具有多重尺度的特点。这包括研究航天器发射和再入的空气动力学，微/纳米机械系统的特性，以及等离子体照明、微芯片设计和未来的清洁能源系统(如聚变)中的电荷粒子传输的研究，如离解电子和离子。在这些例子中，在最小的尺度上，组成气体的单个原子可以被认为是弹跳的台球，每个台球都有自己的速度和方向。气体分子相互碰撞，以及流动中的障碍物边界，彼此之间以及环境中交换能量和动量。在这个尺度上，这个系统被称为动力学方程的模型很好地描述了，这些模型从概率的角度描述了气体的行为。动力学方程解释了单个原子相互碰撞的时间尺度。在最大长度的尺度上，气体表现出集体行为，比如我们认为只有一个速度的风。当气体的密度从低密度变为高密度时，系统的行为从单个粒子变为集体平均行为。这种转变发生在许多系统中，一个有趣的例子是航天器的再入，在高海拔，大气是一种密度非常低的气体，而在地面，大气的密度高出20个数量级。在低密度下，这些体系表现出仅由动力学模型描述的效应。在高密度下，系统表现出我所描述的简单得多的模型的集体行为。用粒子间碰撞来描述的临界动力学时间尺度是以密度为单位的。这项工作的重要性是开发一类新的模拟工具，它可以处理与经历这种非常急剧的密度转变的系统相关的非常严格的时间尺度，从而可以有效地模拟稀有和密集气体状态，以及密度的转变。该框架将允许模拟以前超出标准动力学解算器范围的问题，允许解算器恢复正确的限制行为，并使效率比现有方法提高数量级。

In this work the PI and his student consider the development of novel methods for problems in kinetic transport. The goal is to develop methods that are capable of bridging multiple time scales. For instance, many problems in plasma physics, such as modeling edge plasma problems in fusion systems, can exhibit behavior which is consistent with both diffusion dominated transport as well as collisionless flow in the same problem at the same time. Generally speaking, this class of problems can be summarized as hyperbolic systems with stiff relaxation terms. Typically the stiff relaxation term can be nondimensionalized, leading to a scaling constant out in front of the relaxation term of the form 1/d. In the problems of interest, as d approaches zero, the system transitions from hyperbolic to parabolic. One approach to such problems is to use domain decomposition methods coupled with appropriate models for the various physical behaviors. However, a major difficulty with this approach is defining the boundary conditions for the two way coupling between the domains. The approach taken in this work is to develop numerical methods for the kinetic systems which can recover the correct limiting behavior in the limit of the system becoming collision dominated. The particular class of methods we focus on here are referred to as Asymptotic Preserving (AP) methods. The goal in designing an AP method is to develop time stepping strategies that maintain their order of accuracy for any d. In particular, as d approaches zero, the AP method should recover a consistent discretization for the limiting behavior. However, developing AP methods which are high order have proven difficult to construct. Further, for a range of important test problems, the CFL for the AP method is restricted to time steps less than the square of the spatial discretization. In this work, the PI and his student investigate a new method based on a pseudo upwinding method inside of the AP framework, which gives rise to a method which has a convergence rate independent of d with an apparent CFL of the time step proportional to the spatial discretization. Further, the PI proposes a novel method for lifting low order AP methods to high order based on integral deferred correction, a defect correction methodology developed by the PI and his collaborators. The approach is generalizable to a wide class of kinetic equations with stiff relaxation terms. A large number of important problems in science are characterized by multiple length scales. This includes studying the aerodynamics of spacecraft launch and reentry, the characterization of micro/nano mechanical systems and the study of charge particle transport, such as disassociated electrons and ions, in plasma lighting, micro chip design, and clean energy systems of the future, such as fusion, to name a few. In these examples, on the smallest scales, the individual atoms which make up the gas can be thought of as billiard balls bouncing around, each billiard ball having its own speed and direction. The gas molecules collide with each other, as well as the boundaries of obstacles in the flow, exchanging energy and momentum with each other as well as the environment. On this scale the system is well characterized by models know as kinetic equations, which describe the behavior of the gas from a probabilistic perspective. Kinetic equations account for time scales of individual atoms colliding with each other. On the largest length scales, the gas exhibits collective behavior such as wind, which we think of as having a single speed. As the density of a gas changes from low density to high density, the system behavior changes from individual particles to a collective average behavior. This transition happens in many systems, one interesting example is the reentry of a spacecraft, where at high altitude the atmosphere is a very low density gas and at ground level the atmosphere is 20 orders of magnitude higher in density. At low densities, these systems exhibit effects only described by kinetic models. At high densities, the systems exhibit collective behavior described my much simpler models. The critical kinetic time scale, described by inter-particle collisions, scales as one over the density. The importance of this work is to develop a new class of simulation tools that can handle the very stiff time scales associated with systems that undergo this very sharp transition in densities that can efficiently simulate both the rarified and dense gas regimes, as well as the transition in density. This framework will allow for the simulation of problems previously outside the scope of standard kinetic solvers, allowing the solvers to recover the correct limiting behavior with orders of magnitude increase in efficiency over existing methods.