Mathematical Sciences: Numerical Solution of Algebraic Problems Arising in Fluids Models

这项工作涉及的发展和分析的方法计算的数值解的线性和非线性代数系统模型中产生的粘性不可压缩流体流动。 重点是迭代方法，特别是，预处理Krylov子空间方法，求解线性系统所产生的离散化和线性化的稳态Navier-Stokes方程不可压缩约束。主要目标包括制定和分析的预处理器是有效的大雷诺数，建设高效的方法并行架构，和线性化策略的性能和总成本的影响的研究。 调查研究这些问题的各种稳定的有限元和有限差分离散计划，并考虑离散化的代数性质的离散问题和性能的影响。 的方法进行了研究和比较，使用两个分析界的收敛速度和性能评估的基准问题的计算实验。 应用这些求解算法的方程是计算流体动力学的基础，用于模拟各种物理设置中的流动效应（即，估计速度，压力和温度等量）。 它们可以应用的领域的例子包括车辆空气动力学和航空航天模型、环境流模型、生物医学模型（血流）、核反应堆冷却模型和薄膜涂层（例如，粘合剂或光纤）。 这种过程的数学模型使得能够研究不同物理参数的影响，例如飞机的机翼形状或薄膜的化学成分。 使用纯实验技术确定这种影响，即，通过构造原型和比例模型，是非常昂贵、耗时的，并且容易受到测量装置的存在所引入的不准确性的影响。 数学模型并不能取代实验工作，而是对它们的补充，并允许有效地识别有前途的原型。 然而，只有在可靠和快速的计算算法可用的情况下，数学模型的精确解才是可行的。 该项目的目标是开发与这些模型一起使用的算法。

This work concerns the development and analysis of methods for computing the numerical solution of the linear and nonlinear algebraic systems arising in models of viscous incompressible fluid flow. The emphasis is on iterative methods, specifically, preconditioned Krylov subspace methods, for solving the linear systems arising from discretization and linearization of the steady-state Navier-Stokes equations subject to incompressibility constraints. Primary goals include the formulation and analysis of preconditioners that are effective for large Reynolds numbers, the construction of efficient methods for parallel architectures, and the study of effects of linearization strategy on performance and overall costs. The investigator studies these issues for a variety of stable finite element and finite difference discretization schemes and considers the influence of discretization on the algebraic properties of the discrete problems and on performance. The methods are studied and compared using both analytic bounds on convergence rates and assessment of performance in computational experiments with benchmark problems. The equations to which these solution algorithms are applied are fundamental in computational fluid dynamics, for simulating the effects of flow (that is, estimating quantities such as velocities, pressures and temperatures) in a wide variety of physical settings. Examples of areas where they can be applied include vehicle aerodynamics and aerospace models, models of environmental flows, biomedical models (blood flow), cooling models for nuclear reactors, and thin film coating (e.g., of adhesives or optical fibers). Mathematical models of such processes enable the study of the effects of different physical parameters, for example, wing shape in an airplane or chemical content of a thin film. Determination of such effects using purely experimental techniques, i.e., through the construction of prototypes and scale models, is prohibitively expensive, time consuming and subject to inaccuracies introduced by the presence of measuring devices. Mathematical models do not replace experimental work but complement them and allow for efficient identification of promising prototypes. Accurate solution of the mathematical models is only feasible, however, if reliable and fast computational algorithms are available. The goal of the project is to develop such algorithms for use with these models.