RUI: Development of Fast Scalable Adaptive High Order Methods for Solving the Boltzmann Equation

这个项目的目标是提高一个人使用计算机模拟来应对科学和技术挑战的能力，方法是使用动力学玻尔兹曼方程在微观尺度上进行建模。这一提议的应用跨越了气体动力学、等离子体动力学、自组织系统、网络和细菌动力学。该项目将专注于动力学建模中的一个瓶颈问题-开发用于高保真模拟稀薄气体中粒子相互作用的快速方法。该项目最直接的影响是开发新的航空航天技术，以及美国在清洁能源、生物技术和新材料开发方面的重要举措。这将通过将其应用于在稀薄气体中运行或在真空中制造的设备的计算机模拟来实现。该项目将通过让学生参与研究来为STEM劳动力提供培训。尽管在过去的几十年里，人们对Boltzmann方程的确定性数值解进行了深入的研究，但其数值解仍然是含糊不清的。为了获得适合于应用的全三维解，需要设计快速可伸缩的自适应数值方法来计算五重Boltzmann碰撞积分。这一建议将通过发展基于速度变量中节点不连续Galerkin(NODAL-DG)离散化的Boltzmann碰撞积分的卷积公式，通过在八叉树网格上发展碰撞算子的自适应节点DG小波离散，以及通过开发基于傅立叶变换的应用来计算碰撞积分的卷积形式，来解决这些缺点。新方法最多需要O(n^6)次运算才能对Boltzmann碰撞积分进行完全确定的计算，并且需要O(n^5)个存储单元来存储预先计算的碰撞核，其中n是速度空间中一维离散化点数。新的方法将在并行体系结构上实现，并且将是可扩展的。这一提议的实施将导致发展产生波尔兹曼方程的高保真解的能力、产生基准解的能力和验证动力学模型的方法。这些研究活动将导致节点DG小波在玻尔兹曼碰撞积分近似中的新应用。

This project's goal is to advance one's ability to use computer simulations to address scientific and technological challenges by employing modeling at microscopic scales using the kinetic Boltzmann equation. Applications of this proposal span the dynamics of gas, plasma, self-organizing systems, networks, and bacterial dynamics. The project will focus on a bottleneck issue in kinetic modeling --- the development of fast methods for high fidelity simulations of particle interactions in rarefied gases. The project's most immediate impact is in the development of novel aerospace technologies and in important U.S. initiatives in the development of clean energy, biotechnology, and new materials. This will be through its applications to computer simulation of devices that either operate in rarefied gas or are manufactured in vacuum. The project will provide training for the STEM workforce by engaging students in research. Despite of being studied intensely in the last decades, deterministic numerical solutions of the Boltzmann equation continue to be evasive. To achieve a full three-dimensional solution suitable for use in applications, fast scalable adaptive numerical approaches for evaluating the five-fold Boltzmann collision integral need to be devised. This proposal will address these shortcomings by developing convolution formulations of the Boltzmann collision integral based on nodal discontinuous Galerkin (nodal-DG) discretizations in the velocity variable, by developing adaptable nodal-DG wavelet discretizations of the collision operator on octree meshes, and by developing fast algorithms for evaluating the convolution form of the collision integral based on an application of the Fourier transform. The new methods will require at most O(n^6) operations for a fully deterministic evaluation of the Boltzmann collision integral, and will require O(n^5) memory units to store the pre-computed collision kernels, where n is the number of discretization points in one dimension in the velocity space. The new methods will be implemented on parallel architectures and will be scalable. Implementation of this proposal will result in the development of capabilities for producing high-fidelity solutions to the Boltzmann equation, capabilities for producing benchmark solutions and methods for validation of kinetic models. The research activities will result in a new application of nodal-DG wavelets to the approximation of the Boltzmann collision integral.