Fast Algorithms for Models of Incompressible Flow

不可压缩流动模型的快速算法C。Elman计算机科学系和高级计算机业务研究所马里兰大学帕克学院，MD 20742本项目涉及开发，分析和测试有效的算法，用于解决不可压缩流体流动模型产生的方程组。 这种算法的发展对于理解科学现象和构建涉及流体流动的工程工具具有广泛的潜在用途。 例子包括生物流动，如心脏或静脉和动脉中的血流;环境污染物在河流或地下水中的扩散;微流体装置的设计，用于诊断疾病和识别流体中的病原体;以及大气和海洋现象。通过纯实验技术来了解这些过程是极其昂贵的，而建模与计算解一起使用通过提供关于诸如流速、压力和溶剂浓度的量的信息使得能够基本理解物理学。 本研究涉及快速计算算法的发展，使模型能够通过计算机模拟快速和廉价地求解。研究集中在两类问题上：代数特征值问题，由稳态不可压缩Navier-Stokes方程解的稳定性分析引起;以及当热效应和化学效应与不可压缩流模型耦合时产生的线性和非线性方程组。 对于这两类问题，离散化导致需要求解一系列大型线性方程组。 我们研究有效的预处理技术，利用这些系统的结构的问题。 除了对应用科学的影响外，高效求解器的开发还解决了基本的数学和计算问题，例如数学结构对算法开发的影响以及复杂几何形状中稳定流动的识别。

FAST ALGORITHMS FOR MODELS OF INCOMPRESSIBLE FLOWHoward C. ElmanDepartment of Computer Science andInstitute for Advanced Computer StudiesUniversity of MarylandCollege Park, MD 20742This project concerns development, analysis and testing of efficient algorithms for solving systems of equations arising from models of flow of incompressible fluids. The development of such algorithms is of broad potential use for understanding scientific phenomena and constructing engineering tools involving fluid flows. Examples include biological flows such as blood flow in the heart or veins and arteries; dispersal of environmental pollutants in rivers or groundwater; design of microfluidics devices, which are used in diagnosis of diseases and identification of pathogens in fluids; and atmospheric and oceanic phenomena.Understanding such processes through purely experimental techniques is prohibitively expensive or impossible, whereas the use of modelling together with computational solution enables basic understanding of the physics by providing information about quantities such as flow rates, pressures, and concentrations of solvents. This research involves the development of fast computational algorithms that allows models to be resolved quickly and inexpensively by computer simulation.The research is focused on two classes of problems: algebraic eigenvalue problems that arise from analysis of the stability of solutions of the steady-state incompressible Navier-Stokes equations; and linear and nonlinear systems of equations that arise when thermal and chemical effects are coupled with models of incompressible flow. For both problem classes, discretization leads to the requirement to solve a sequence of large-scale linear systems of equations. We study efficient preconditioning techniques for these systems that take advantage of the structure of the problems. In addition to the impact on applied science, development of efficient solvers addresses fundamental mathematical and computational questions such as the impact of mathematical structure on algorithm development and the identification of stable flows in complex geometries.