A combination of Spectral Methods and Finite Differences will be used to solve the Navier-Stokes equations for a viscous flow past a circular cylinder and past symmetric Joukowski airfoils. Different discretizations of the physical problem will be explored, and the solution of the equations will be analyzed for different geometries and boundary conditions. This project is the continuation of our research started as a Master Thesis at Wichita State University under the advising of Professor Alan Elcrat; the project is a deep exploration of the solution of Navier-Stokes equations by implementing new methods of discretization including spectral differentiation. We will compare results previously obtained by Gauss-Seidel/Successive Over-Relaxation Methods (SOR) together with Finite Differences, with results using Newton's Method, based on work by Bengt Fornberg, but implementing spectral differentiation. As we will see, due to the nature of the physical domain and the conformal map involved to transform it to a more tractable domain, the use of spectral methods in both directions of our two dimensional problem proved to be inefficient due to unnecessary concentration of points in areas of the domain of low gradients. However, to take advantage of spectral methods, we combined spectral methods in one direction with high order finite vii differences on the other direction, where different mesh densities were designed to have higher concentration of points where required. With this discretizations, spectral methods were approached as the limiting order of finite differences as presented in A Practical Guide to Pseudospectral Methods We will explore the solution for flows past more general geometries, symmetric Joukowski airfoils. Then we will study the implementation and effect of suction boundary conditions on the obstacle. In this text I have decided to include part of the introduction and theoretical background shown in my Master thesis to allow new readers to get familiarized with the subject, but the solution scheme, the different discretizations and results are all new explorations that we are proud to present.

Following up the seminal Spectral Methods in Fluid Dynamics, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics contains an extensive survey of the essential algorithmic and theoretical aspects of spectral methods for complex geometries. These types of spectral methods were only just emerging at the time the earlier book was published. The discussion of spectral algorithms for linear and nonlinear fluid dynamics stability analyses is greatly expanded. The chapter on spectral algorithms for incompressible flow focuses on algorithms that have proven most useful in practice, has much greater coverage of algorithms for two or more non-periodic directions, and shows how to treat outflow boundaries. Material on spectral methods for compressible flow emphasizes boundary conditions for hyperbolic systems, algorithms for simulation of homogeneous turbulence, and improved methods for shock fitting. This book is a companion to Spectral Methods: Fundamentals in Single Domains.

This well-written book explains the theory of spectral methods and their application to the computation of viscous incompressible fluid flow, in clear and elementary terms. With many examples throughout, the work will be useful to those teaching at the graduate level, as well as to researchers working in the area.

This book presents the basic algorithms, the main theoretical results, and some applications of spectral methods. Particular attention is paid to the applications of spectral methods to nonlinear problems arising in fluid dynamics, quantum mechanics, weather prediction, heat conduction and other fields.The book consists of three parts. The first part deals with orthogonal approximations in Sobolev spaces and the stability and convergence of approximations for nonlinear problems, as the mathematical foundation of spectral methods. In the second part, various spectral methods are described, with some applications. It includes Fourier spectral method, Legendre spectral method, Chebyshev spectral method, spectral penalty method, spectral vanishing viscosity method, spectral approximation of isolated solutions, multi-dimensional spectral method, spectral method for high-order equations, spectral-domain decomposition method and spectral multigrid method. The third part is devoted to some recent developments of spectral methods, such as mixed spectral methods, combined spectral methods and spectral methods on the surface.

In developing this book, we decided to emphasize applications and to provide methods for solving problems. As a result, we limited the mathematical devel opments and we tried as far as possible to get insight into the behavior of numerical methods by considering simple mathematical models. The text contains three sections. The first is intended to give the fundamen tals of most types of numerical approaches employed to solve fluid-mechanics problems. The topics of finite differences, finite elements, and spectral meth ods are included, as well as a number of special techniques. The second section is devoted to the solution of incompressible flows by the various numerical approaches. We have included solutions of laminar and turbulent-flow prob lems using finite difference, finite element, and spectral methods. The third section of the book is concerned with compressible flows. We divided this last section into inviscid and viscous flows and attempted to outline the methods for each area and give examples.

Since the publication of "Spectral Methods in Fluid Dynamics" 1988, spectral methods have become firmly established as a mainstream tool for scientific and engineering computation. The authors of that book have incorporated into this new edition the many improvements in the algorithms and the theory of spectral methods that have been made since then. This latest book retains the tight integration between the theoretical and practical aspects of spectral methods, and the chapters are enhanced with material on the Galerkin with numerical integration version of spectral methods. The discussion of direct and iterative solution methods is also greatly expanded.