Numerical Methods in Large-Scale Computation

9706866麦考密克拟议研究的主要目的是开发有效的数值技术来模拟各种物理过程。重要的应用包括空气动力学、气象学、弹性、电磁学、多孔介质和颗粒传输。总的目标是开发精确的离散化方法和快速的代数求解器，用于管理这些和其他应用的偏微分方程组。研究主题包括：多层一阶系统最小二乘，它涉及将偏微分方程组重新表述为适定的最小化原则，以允许稳健和有效的求解方法；多孔介质问题，将被成功地用于此类多方和反应流的欧拉-拉格朗日局部伴随方法；传输现象，将使用高效和稳健的最小二乘方法模拟三维Boltzmann输运方程；以及迭代方法，旨在为在许多应用中出现的方程开发有效的代数解算器。本项目的重点是计算数学领域的研究。其目的是提高我们对计算机模拟复杂物理现象背后的数学的理解。这类模拟是研究和控制许多重要过程的关键，包括地下水流动、全球变化、能源生产和材料科学。这类模拟中的挑战之一是开发改进的数学方法来求解这些模型中出现的方程。这项研究的基本目标是大幅提高我们以更高的准确性和效率对日益复杂和复杂的过程进行建模的能力。这将为模拟铺平道路，为科学家、工程师和政策制定者提供更强大的工具，以了解和改善我们的工业、科学和环境。

9706866 McCormick The main thrust of the proposed research is the development of efficient numerical techniques for simulation of a wide variety of physical processes. Important applications include aerodynamics, meteorology, elasticity, electromagnetics, porous media, and particle transport. The general goal is to develop accurate discretization methods and fast algebraic solvers for the partial differential equations that govern these and other applications. The research topics include: multilevel first-order system least squares, which involves reformulation of partial differential equations as well-posed minimization principles to allow for robust and efficient solution methods; porous media problems, which will be treated by Eulerian-Lagrangian localized adjoint methods that have been successful for such multiparty and reactive flows; transport phenomena, which will be simulated using efficient and robust least-squares methods for the three-dimensional Boltzmann transport equations; and iterative methods, which is aimed at developing effective algebraic solvers for the equations that arise in many applications. The focus of this project is research in the field of computational mathematics. The purpose is to improve our understanding of the mathematics behind computer simulation of complex physical phenomena. Such simulations are key to the study and control of many important processes, including groundwater flow, global change, energy production, and material science. One of the challenges in such simulations is the development of improved mathematical methods for solving the equations that arise in these models. The basic aim of this research is dramatic improvement in our ability to model increasingly more complicated and sophisticated processes with much greater accuracy and efficiency. This will pave the way for simulations that can provide scientists, engineers, and policy-makers with much more powerful tools to understand and improve our industry, science, and environm ent.