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

该项目的目标是通过使用动力学玻尔兹曼方程在微观尺度上进行建模，提高使用计算机模拟应对科学和技术挑战的能力。这个建议的应用范围包括气体动力学、等离子体动力学、自组织系统、网络和细菌动力学。该项目将集中在动力学建模的一个瓶颈问题-稀薄气体中粒子相互作用高保真模拟的快速方法的发展。该项目最直接的影响是发展新的航空航天技术，以及美国在发展清洁能源、生物技术和新材料方面的重要举措。这将是通过其应用程序的计算机模拟设备，无论是在稀薄气体或真空中制造。该项目将通过让学生参与研究，为STEM劳动力提供培训。 尽管在过去的几十年里得到了深入的研究，玻尔兹曼方程的确定性数值解仍然是回避。为了实现一个完整的三维解决方案，适用于应用程序中，快速可扩展的自适应数值方法来评估五重玻尔兹曼碰撞积分需要设计。该建议将解决这些缺点，通过开发卷积公式的玻尔兹曼碰撞积分的基础上节点不连续的伽辽金（dual-DG）离散的速度变量，通过开发自适应dual-DG小波离散的八叉树网格上的碰撞算子，并通过开发快速算法，用于评估卷积形式的碰撞积分的基础上的傅立叶变换的应用。新的方法将需要最多O（n^6）的操作来完全确定性地计算玻尔兹曼碰撞积分，并且将需要O（n^5）的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的数量。新的方法将在并行架构上实现，并具有可扩展性。执行这一提议将导致发展对玻尔兹曼方程产生高保真解的能力、产生基准解的能力和验证动力学模型的方法。研究活动将导致一个新的应用程序的martical-DG小波的玻尔兹曼碰撞积分的近似。

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.