Ebook: Spectral Methods in Fluid Dynamics

This is a book about spectral methods for partial differential equations: when to use them, how to implement them, and what can be learned from their of spectral methods has evolved rigorous theory. The computational side vigorously since the early 1970s, especially in computationally intensive of the more spectacular applications are applications in fluid dynamics. Some of the power of these discussed here, first in general terms as examples of the methods have been methods and later in great detail after the specifics covered. This book pays special attention to those algorithmic details which are essential to successful implementation of spectral methods. The focus is on algorithms for fluid dynamical problems in transition, turbulence, and aero­ dynamics. This book does not address specific applications in meteorology, partly because of the lack of experience of the authors in this field and partly because of the coverage provided by Haltiner and Williams (1980). The success of spectral methods in practical computations has led to an increasing interest in their theoretical aspects, especially since the mid-1970s. Although the theory does not yet cover the complete spectrum of applications, the analytical techniques which have been developed in recent years have facilitated the examination of an increasing number of problems of practical interest. In this book we present a unified theory of the mathematical analysis of spectral methods and apply it to many of the algorithms in current use.

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This textbook presents the modern unified theory of spectral methods and their implementation in the numerical analysis of partial differential equations occuring in fluid dynamical problems of transition, turbulence, and aerodynamics. It provides the engineer with the tools and guidance necessary to apply the methods successfully, and it furnishes the mathematician with a comprehensive, rigorous theory of the subject. All of the essential components of spectral algorithms currently employed for large-scale computations in fluid mechanics are described in detail. Some specific applications are linear stability, boundary layer calculations, direct simulations of transition and turbulence, and compressible Euler equations. The authors also present complete algorithms for Poisson's equation, linear hyperbolic systems, the advection diffusion equation, isotropic turbulence, and boundary layer transition. Some recent developments stressed in the book are iterative techniques (including the spectral multigrid method), spectral shock-fitting algorithms, and spectral multidomain methods. The book addresses graduate students and researchers in fluid dynamics and applied mathematics as well as engineers working on problems of practical importance.

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This textbook presents the modern unified theory of spectral methods and their implementation in the numerical analysis of partial differential equations occuring in fluid dynamical problems of transition, turbulence, and aerodynamics. It provides the engineer with the tools and guidance necessary to apply the methods successfully, and it furnishes the mathematician with a comprehensive, rigorous theory of the subject. All of the essential components of spectral algorithms currently employed for large-scale computations in fluid mechanics are described in detail. Some specific applications are linear stability, boundary layer calculations, direct simulations of transition and turbulence, and compressible Euler equations. The authors also present complete algorithms for Poisson's equation, linear hyperbolic systems, the advection diffusion equation, isotropic turbulence, and boundary layer transition. Some recent developments stressed in the book are iterative techniques (including the spectral multigrid method), spectral shock-fitting algorithms, and spectral multidomain methods. The book addresses graduate students and researchers in fluid dynamics and applied mathematics as well as engineers working on problems of practical importance.
Content:
Front Matter....Pages i-xv
Introduction....Pages 1-30
Spectral Approximation....Pages 31-75
Fundamentals of Spectral Methods for PDEs....Pages 76-93
Temporal Discretization....Pages 94-123
Solution Techniques for Implicit Spectral Equations....Pages 124-182
Simple Incompressible Flows....Pages 183-200
Some Algorithms for Unsteady Navier—Stokes Equations....Pages 201-239
Compressible Flow....Pages 240-274
Global Approximation Results....Pages 275-314
Theory of Stability and Convergence for Spectral Methods....Pages 315-374
Steady, Smooth Problems....Pages 375-414
Transient, Smooth Problems....Pages 415-443
Domain Decomposition Methods....Pages 444-476
Back Matter....Pages 477-567

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This textbook presents the modern unified theory of spectral methods and their implementation in the numerical analysis of partial differential equations occuring in fluid dynamical problems of transition, turbulence, and aerodynamics. It provides the engineer with the tools and guidance necessary to apply the methods successfully, and it furnishes the mathematician with a comprehensive, rigorous theory of the subject. All of the essential components of spectral algorithms currently employed for large-scale computations in fluid mechanics are described in detail. Some specific applications are linear stability, boundary layer calculations, direct simulations of transition and turbulence, and compressible Euler equations. The authors also present complete algorithms for Poisson's equation, linear hyperbolic systems, the advection diffusion equation, isotropic turbulence, and boundary layer transition. Some recent developments stressed in the book are iterative techniques (including the spectral multigrid method), spectral shock-fitting algorithms, and spectral multidomain methods. The book addresses graduate students and researchers in fluid dynamics and applied mathematics as well as engineers working on problems of practical importance.
Content:
Front Matter....Pages i-xv
Introduction....Pages 1-30
Spectral Approximation....Pages 31-75
Fundamentals of Spectral Methods for PDEs....Pages 76-93
Temporal Discretization....Pages 94-123
Solution Techniques for Implicit Spectral Equations....Pages 124-182
Simple Incompressible Flows....Pages 183-200
Some Algorithms for Unsteady Navier—Stokes Equations....Pages 201-239
Compressible Flow....Pages 240-274
Global Approximation Results....Pages 275-314
Theory of Stability and Convergence for Spectral Methods....Pages 315-374
Steady, Smooth Problems....Pages 375-414
Transient, Smooth Problems....Pages 415-443
Domain Decomposition Methods....Pages 444-476
Back Matter....Pages 477-567
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