Mathematical Sciences: Multilevel and Algebraic Iterative Methods in Large-Scale Computation

数学科学：大规模计算中的多级代数迭代方法

• 批准号: 9312752
• 负责人: Thomas Manteuffel
• 金额: $ 45万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-08-01 至 1998-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9312752&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Multilevel; Algebraic; Iterative

The investigators continue work to develop numerical methods for solving problems of flows in porous media and to develop and analyze fast algorithms for partial differential equations, for computational fluid dynamics, and for several linear algebra problems. Specifically, they (a) develop fast parallel numerical methods for the solution of the Boltzmann equation used to model the transport of neutral and charged particles in 1 and 3 spatial dimensions; (b) develop multilevel zonal techniques for large scale computational fluid dynamics applications on parallel computers; (c) study discretization and solution techniques for fluid flows in porous media that involve complex geometry with heterogeneous material properties; (d) develop a new methodology for the solution of systems of partial differential equations by introducing new dependent variables and forming a least-squares functional. In each case, they exploit and extend multi-level computational techniques. These projects are focused on important applications, such as calculating radiation doses for cancer treatment, ion implantation for semiconductor development, global weather modeling, fluid flow over airfoils and through rocket nozzles, oil recovery and contaminant transport in subsurface water supplies. They reflect the growing recognition that effective numerical modeling of physical phenomena involves several stages - formulation of the mathematical model, transforming the model into a discrete problem, and developing algorithms for the numerical solution of the discrete problem on advanced computers - none of which should be attempted without consideration for the others. Each project examines all components of the process, reformulating the model and discretization when necessary to yield discrete problems that can be efficiently solved numerically on parallel computers.

研究人员继续致力于开发解决多孔介质中流动问题的数值方法，并开发和分析偏微分方程，计算流体动力学和几个线性代数问题的快速算法。 具体地说，他们（a）开发快速并行数值方法，用于求解玻尔兹曼方程，该方程用于模拟1维和3维空间中中性和带电粒子的传输;（B）开发用于并行计算机上的大规模计算流体动力学应用的多级分区技术;（c）研究多孔介质中流体流动的离散化和求解技术，这种介质涉及复杂的几何形状和不均匀的材料特性;（d）通过引入新的因变量和形成最小二乘泛函，发展一种新的偏微分方程组求解方法。 在每种情况下，他们都利用和扩展了多级计算技术。 这些项目的重点是重要的应用，如计算癌症治疗的辐射剂量，半导体开发的离子注入，全球天气建模，翼型和火箭喷嘴上的流体流动，石油回收和地下水供应中的污染物运输。 他们反映了越来越多的认识，有效的数值模拟的物理现象涉及几个阶段-制定的数学模型，将模型转化为一个离散的问题，并制定算法的数值解决方案的离散问题的先进计算机-其中任何一个都不应该尝试不考虑其他。 每个项目都会检查过程的所有组成部分，在必要时重新制定模型和离散化，以产生可以在并行计算机上有效解决的离散问题。