Filtering Strategies for Radiation Transport Equations

辐射是许多物理过程中能量交换的基本机制，也是能量和医疗设备运行中的基本机制。此外，辐射还被用作探索材料基本结构的实验工具。改进的数学算法和对辐射输运方程的分析将使这些领域取得重要进展。更广泛地说，这些方程是用于描述稀薄气体、等离子体和多相流的各种动力学模型的数学原型。它们还展示了驱动耗散系统的基本多尺度特性。从历史上看，辐射输运方程一直是基本数值方法研究中许多基本发展的驱动力。这一趋势将在这个项目中继续下去。在教育方面，该项目将支持和培训一名学生在计算数学方面的职业生涯。这包括为参加会议和讲习班的旅行提供支持，以便分享研究成果和提供联网机会。这个项目旨在解决辐射动力学描述中的一个基本挑战：辐射传输方程的离散纵坐标近似中的射线效应的开始。本文提出的方法是基于过滤器的使用，它以特定的方式平滑这些方程的数学解，以提高欠分辨网格计算的数值解的保真度。该项目将推进辐射传输方程的数值方法，更广泛地说，将改进动力学方程和复杂的多物理系统的数值方法。它将促进谱逼近技术的发展，并将这些技术综合到大规模方程的求解中。它将引入分析工具，以更好地理解和提高欠分辨的数值逼近的准确性，并将与目前用于流体模拟的其他类型的过滤和次尺度建模方法建立联系。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Radiation is the fundamental mechanism of energy exchange in many physical processes and in the operations of devices for energy and medical applications. In addition, radiation is used as an experimental tool to explore the basic structure of materials. Improved mathematical algorithms and analysis of radiation transport equations will enable important advances in these areas. More generally, these equations serve as a mathematical prototype for a variety of kinetic models that are used to describe dilute gases, plasmas, and multiphase flows. They also demonstrate the fundamental multi-scale nature of driven-dissipative systems. Historically, radiation transport equations have been a driver for many fundamental developments in basic numerical methods research. That trend will continue with this project. On the education side, this project will support and train a student for a career in computational mathematics. This includes support for travel to attend conferences and workshops in order to share research and provide networking opportunities. This grant will support 1 graduate student per year for 3 years.This project seeks to address one of the fundamental challenges in the kinetic description of radiation: the onset of ray effects in the discrete ordinates approximation of the radiation transport equation. The approach proposed here is based on the use of filters, which smooth the mathematical solution of these equations in a specified manner, in order to improve the fidelity of numerical solutions computed with under-resolved meshes. The project will advance numerical methods for radiation transport equations and, more generally, for kinetic equations and complex multi-physics systems. It will enable to the development of spectral approximation techniques and the synthesis of such techniques into the solution of large scale equations. It will introduce analysis tools to better understand and improve the accuracy of numerical approximation that are under-resolved, and it will establish connections with other types of filtering and sub-scale modeling approaches that are currently used in the simulation of fluids.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.