Systematic Lagrangian Methods for Transport Problems

这项工作的重点是发展的高阶拉格朗日粒子方法的问题，具有远程自力表现出代数衰减。拉格朗日粒子框架的优点是，需要表征的区域（即网格点）正好位于有测试粒子的地方。大多数粒子方法最多是一阶的；特别是，它们的获取方式使得系统地推导高阶边界条件变得极具挑战性。长程力问题的一个重要子类是等离子体物理学中出现的问题。基本的控制方程组是Boltzmann-Maxwell （BM）系统，在一些关键的假设条件下可以简化为Vlasov-Poisson （VP）系统。这些问题是6D +时间，并表现出一系列复杂的动力学。通常，粒子单元（PIC）用于解决由这些系统控制的计算问题。提案人一直致力于通过开发系统的粒子公式来解决PIC（和其他方案）的问题。最初的框架是基于拉格朗日流图的演变，其中使用快速求和算法评估远程力。提出的工作旨在发展具有四维或更高维度的高阶拉格朗日方法。需要解决的关键问题包括：1)严格分析基于网格的粒子方法中的数值加热，2)提高拉格朗日粒子方法的空间精度，3)分析正则化的使用及其对边界积分方法精度的影响，4)使用高阶校正加速显式/隐式光谱延迟校正（SDC）。该项目将创建比现有方法更准确、更可靠、更广泛适用于应用物理、化学和材料科学中感兴趣的一系列问题的新的模拟工具。更高的精度和更好的预测能力在应用科学中对减少实验设计过程的巨大费用产生了真正的影响，通过在施工之前提供精确的设计。与空军研究实验室的合作正在进行中，旨在解决现实世界的问题，如模拟航天器羽流相互作用。

This work focuses on the development of higher-order Lagrangian particle methods for problems which have long-range self forces exhibiting algebraic decay. The advantage of a Lagrangian particle framework is that regions that require characterization (i.e., mesh points) are precisely where there are test particles. Most particle methods are first order at best; in particular, the ad hoc fashion in which they are obtained makes it extremely challenging to systematically derive higher-order boundary conditions. An important sub-class of problems with long range forces are those arising in plasma physics. The fundamental governing set of equations is the Boltzmann-Maxwell (BM) system, which may be reduced to the Vlasov-Poisson (VP) system under key assumptions. These problems are 6D plus time and exhibit a range of complex dynamics. Typically, Particle-In-Cell (PIC) is used to address computational problems governed by these systems. The proposer has been engaged in addressing issues with PIC (and other schemes) by developing a systematic particle formulation. The initial framework is based on the evolution of the Lagrangian flow map, where long range forces are evaluated using fast summation algorithms. The proposed work seeks to develop high-order Lagrangian methods with four or higher dimensions. Critical issues to be addressed include: I) a rigorous analysis of numerical heating in mesh-based particle methods, II) increasing the spatial accuracy of Lagrangian particle methods, III) an analysis of the use of regularization and its impact on the accuracy of boundary integral methods, and IV) accelerating explicit/implicit Spectral Deferred Correction (SDC) using high-order correction.The project will create new simulation tools that are more accurate, more reliable, and of broader applicability than existing methods for a range of problems of interest in applied physics, chemistry and materials science. Higher accuracy and better predictive capability have a real impact in the applied sciences on reducing the huge expense of experimental design procedure by providing a refined design prior to construction. Collaboration with the Air Force Research Laboratory is under way, aimed at real world problems such as modeling spacecraft plume interactions.