Spectral Methods for Incompressible Viscous Flow

This well-written book explains the theory of spectral methods and their aplication to incompressible fluid flow in clear and elementary terms. It begins with an introduction to the fundamentals of spectral methods and then moves on to cover, in particular, the Fourier and Chebyshev methods. Examples and exercises are included. Chapters 4 and 5 handle streamfunction-vorticity and velocity-pressure for Navier-Stokes equations. Chapter 6 addresses special topics such as self-adaptive coordinate transform, domain decomposition, treatment of singularities, and free-surface flow. The work will be useful to those teaching in the field at the graduate level, as well as to researchers working in the area.

The objective of this book is to provide a comprehensive discussion of Fourier and Chebyshev spectral methods for the computation of incom pressible viscous flows, based on the Navier-Stokes equations. and confidence in the numerical results, the re For reasons of efficiency searchers and practitioners involved in computational fluid dynamics must be able to master the numerical methods they use. Therefore, in writing this book, beyond the description of the algorithms, I have also tried to provide information on the mathematical and computational, as well as implementational characteristics of the methods. The book contains three parts. The first is intended to present the fun damentals of the Fourier and Chebyshev methods for the solution of differ ential problems. The second part is entirely devoted to the solution of the N avier-Stokes equations, considered in vorticity-streamfunction and velocity-pressure formulations. The third part is concerned with the so lution of stiff and singular problems, and with the domain decomposition method. In writing this book, lowe a great debt to the joint contribution of several people to whom I wish to express my deep gratitude. First, I express my friendly thanks to L. Sirovich, editor of the series "Applied Mathematical Sciences," who suggested that I write the book. Many thanks are also addressed to my colleagues and former students who contributed to the completion of the book in various ways. I am happy to thank P. Bontoux, O. Botella, J.A. Desideri, U. Ehrenstein, M.Y. Forestier, J. Frohlich, S.

Includes supplementary material: sn.pub/extras

Texte du rabat
This book provides a comprehensive discussion of Fourier and Chebyshev spectral methods for the computation of incompressible viscous flows, based on the Navier-Stokes equations. The book is in three parts. The first part presents the fundamentals of the Fourier and Chebyshev methods for the solution of the Navier-Stokes equations considered in vorticity-streamfunction and velocity-pressure formulations. The third part of the book is concerned with the solution of stiff and singular problems, and with the domain decomposition method.
Every topic is accompanied by numerical examples, which further illustrate and assess the methods. Graduate students and researchers in applied mathematics and engineering working in fluid dxnamics, scientific computing, and numerical analysis will find this book of interest.

Contenu
I Basic spectral methods.- 1 Fundamentals of spectral methods.- 2 Fourier Method.- 3 Chebyshev method.- 4 Time-dependent equations.- II. Navier-Stokes equations.- 5 Navier-Stokes equations for incompressible fluids.- 6 Vorticity-Streamfunction Equations.- 7 Velocity-Pressure Equations.- III Special topics.- 8 Stiff and singular problems.- 9 Domain Decomposition Method.- Appendix A Formulas on Chebyshev polynomials.- A.1 Definition and general properties.- A.2 Differentiation.- A.3 Collocation points.- A.4 Truncated series expansion.- A.5 Lagrange interpolation polynomial.- A.6 Derivatives at Gauss-Lobatto points.- A.7 Integration.- A.8 Numerical integration based on Gauss-Lobatto points.- Appendix B Solution of a quasi-tridiagonal system.- Appendix C Theorems on the zeros of a polynomial.- References.