Spectral Methods

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 their 1988 book, Canuto et al. now incorporate the many improvements in the algorithms and the theory of spectral methods that have been made since then.

This second new treatment, Evolution to Complex Geometries and Applications to Fluid Dynamics, provides an extensive overview of the essential algorithmic and theoretical aspects of spectral methods for complex geometries, in addition to detailed discussions of spectral algorithms for fluid dynamics in simple and complex geometries. Modern strategies for constructing spectral approximations in complex domains, such as spectral elements, mortar elements, and discontinuous Galerkin methods, as well as patching collocation, are introduced, analyzed, and demonstrated by means of numerous numerical examples. Representative simulations from continuum mechanics are also shown. Efficient domain decomposition preconditioners (of both Schwarz and Schur type) that are amenable to parallel implementation are surveyed. The discussion of spectral algorithms for fluid dynamics in single domains focuses on proven algorithms for the boundary-layer equations, linear and nonlinear stability analyses, incompressible Navier-Stokes problems, and both inviscid and viscous compressible flows. An overview of the modern approach to computing incompressible flows in general geometries using high-order, spectral discretizations is also provided.

The recent companion book 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 essential concepts and formulas from this book are included in thecurrent text for the reader’s convenience.

• Compressible Flows
• Fluid Dynamics
• Fourier Approximation
• Galerkin Approximation
• High-Order Methods
• Incompressible Flows
• Navier-Stokes equation
• Numerical Analysis
• Scientific Computing
• Spectral Algorithms
• Spectral Methods
• Spectral Multigrid Methods
• Stability
• fluid- and aerodynamics

From the reviews:

"In 2006 Canuto, Quarteroni and Zang presented us on 550 pages a new book on spectral methods … . Now the second new book (‘Evolution of complex geometrics and application to fluid dynamics’, CHQZ3) is published and it contains further 600 pages on spectral methods. … the book presents the actual state-of-the-art of spectral methods and yields for the active researcher a nice comprehensive survey on the modern theory and an excellent bibliography." (H.-G. Roos, Zentralblatt für Angewandte Mathematik und Mechanik, Vol. 88 (1), 2008)

"This book is … dedicated to the application of spectral methods in fluid dynamics. … the main objective of the new book is to modernize the classical spectral methods, accounting for advances in the theory and extensive application of multidomain spectral methods in fluid dynamics. … This is a very useful book for graduate students, research mathematicians, geoscientists, physicists, and engineers interested in modern methods of numerical analysis, mathematical modeling and computational fluid dynamics." (Yuri N. Skiba, Mathematical Reviews, Issue 2009 d)

• Dipartimento di Matematica, Politecnico di Torino, Torino, Italy

  Claudio Canuto

• SB-IACS-CMCS, EPFL, Lausanne, Switzerland

  Alfio Quarteroni

• MOX, Politecnico di Milano, Milano, Italy

  Alfio Quarteroni

• School of Computational Science, Florida State University, Tallahassee, USA

  M. Yousuff Hussaini

• NASA Langley Research Center, Hampton, USA

  Thomas A. Zang

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