Overview

Since the publication of ""Spectral Methods in Fluid Dynamics"", spectral methods, particularly in their multidomain version, have become firmly established as a mainstream tool for scientific and engineering computation. While retaining the tight integration between the theoretical and practical aspects of spectral methods that was the hallmark of the earlier book, Canuto et al. now incorporate the many improvements in the algorithms and the theory of spectral methods that have been made since 1988. The initial treatment Fundamentals in Single Domains discusses the fundamentals of the approximation of solutions to ordinary and partial differential equations on single domains by expansions in smooth, global basis functions. The first half of the book provides the algorithmic details of orthogonal expansions, transform methods, spectral discretization of differential equations plus their boundary conditions, and solution of the discretized equations by direct and iterative methods. The second half furnishes a comprehensive discussion of the mathematical theory of spectral methods on single domains, including approximation theory, stability and convergence, and illustrative applications of the theory to model boundary-value problems. Both the algorithmic and theoretical discussions cover spectral methods on tensor-product domains, triangles and tetrahedra. All chapters are enhanced with material on the Galerkin with numerical integration version of spectral methods. The discussion of direct and iterative solution methods is greatly expanded as are the set of numerical examples that illustrate the key properties of the various types of spectral approximations and the solution algorithms. A companion book ""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 and provides detailed discussions of spectral algorithms forfluid dynamics in simple and complex geometries. 

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9783642068003

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Reviews

From the reviews: The main aim of the book is to discuss the approximations of solutions to ordinary and partial differential equations in single domains by expansions in smooth, global basis functions. ... furnishes a comprehensive discussion of the mathematical theory of spectral methods in single domains ... . All chapters are enhanced with material on Galerkin method ... . The discussion of direct and iterative solution methods is endowed with numerical examples that illustrate the key properties of various spectral approximations and solution algorithms. (Nina Shokina, Zentralblatt MATH, Vol. 1093 (19), 2006)

Aus den Rezensionen: ! Schon ist ! dass die Autoren klar Tendenzen der spektralen Methoden der letzten 20 Jahre aufzeigen ! Im Einzelnen findet der interessierte Leser ! viele interessante Details zu Fragen der polynomialen Approximation und ! eine ausfuhrliche Diskussion spektraler Methoden fur zahlreiche Modellprobleme ! Insgesamt ist es ! sichtlich eine lohnenswerte Quelle fur alle, die spektrale Methoden lieben. Und man kann gespannt sein, wie im Band 2 komplexere Gebiete und Probleme der Stromungsmechanik behandelt werden. (H.-G. Roos, in: ZAMM - Zeitschrift fur Angewandte Mathematik und Mechanik, 2007, Vol. 87, Issue 1, S. 69)

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The authors are among the leading researchers in the field of computational fluid dynamics and have pioneered and promoted the ""Spectral Methods in Fluid Dynamics"".

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