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

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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.