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

这项工作涉及的发展和分析方法的计算线性和非线性代数系统出现在粘性不可压缩流体流动模型的数值解。重点是迭代方法，特别是预处理的Krylov子空间方法，用于求解由不可压缩性约束下的稳态Navier-Stokes方程的离散化和线性化引起的线性系统。主要目标包括制定和分析对大雷诺数有效的预调节器，构建并行架构的有效方法，以及研究线性化策略对性能和总体成本的影响。研究者研究了各种稳定有限元和有限差分离散化方案的这些问题，并考虑了离散化对离散问题的代数性质和性能的影响。在基准问题的计算实验中，用收敛速度的解析界和性能评价对这些方法进行了研究和比较。这些求解算法所应用的方程是计算流体动力学的基础，用于模拟各种物理环境下流动的影响（即估计速度、压力和温度等量）。它们可以应用的领域包括车辆空气动力学和航空航天模型、环境流动模型、生物医学模型（血液流动）、核反应堆的冷却模型和薄膜涂层（例如粘合剂或光纤）。这些过程的数学模型使研究不同物理参数的影响成为可能，例如飞机的机翼形状或薄膜的化学成分。使用纯粹的实验技术，即通过构建原型和比例模型来确定这种影响，是非常昂贵的，耗时的，并且由于测量设备的存在而导致不准确。数学模型不能取代实验工作，而是对实验工作的补充，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用于这些模型的算法。

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.