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

研究人员继续努力开发数值方法来解决多孔介质中的流动问题，并开发和分析偏微分方程组、计算流体力学和几个线性代数问题的快速算法。具体地说，他们(A)开发了求解Boltzmann方程的快速并行数值方法，用于模拟一维和三维空间中的中性粒子和带电粒子的输运；(B)开发了用于在并行计算机上进行大规模计算流体力学应用的多层分区技术；(C)研究了多孔介质中流体流动的离散化和求解技术，涉及具有非均匀材料性质的复杂几何形状；(D)通过引入新的因变量并形成最小二乘泛函，开发了求解偏微分方程组的新方法。在每一种情况下，他们都利用和扩展了多层次计算技术。这些项目侧重于重要的应用，例如计算癌症治疗的辐射剂量、用于半导体开发的离子注入、全球天气模拟、翼型和火箭喷嘴上的流体流动、石油回收和地下供水中的污染物传输。它们反映了人们日益认识到，对物理现象进行有效的数值建模涉及几个阶段--建立数学模型，将模型转换为离散问题，以及在高级计算机上开发离散问题的数值解的算法--任何一个阶段都不应在不考虑其他因素的情况下进行尝试。每个项目检查过程的所有组成部分，重新制定模型，必要时进行离散化，以产生可以在并行计算机上高效地数值求解的离散问题。

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.