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**Spectral methods. Evolution to complex geometries and applications to fluid dynamics.**
*(English)*
Zbl 1121.76001

Scientific Computation. Berlin: Springer (ISBN 978-3-540-30727-3/hbk). xxx, 596 p. (2007).

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 theoretical and practical aspects of spectral methods was the hallmark of the 1988 book [the first edition: C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral methods in fluid dynamics. Springer Series in Computational Physics. New York etc.: Springer-Verlag (1988; Zbl 0658.76001); 2nd corr. printing: C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral methods in fluid dynamics. Springer Series in Computational Physics. Berlin etc.: Springer-Verlag (1991; Zbl 0717.76004)], now the authors incorporate many improvements in the algorithms and in the theory of spectral methods that have been made since 1988.

This second edition provides an extensive overview of essential algorithmic and theoretical aspects of spectral methods for complex geometries, in addition to detailed discussions of spectral algorithms for fluid dynamics in simple 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 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 in single domains by expansions in smooth global basis functions. The essential concepts and formulas from this book are included in the current text for the reader’s convenience.

This second edition provides an extensive overview of essential algorithmic and theoretical aspects of spectral methods for complex geometries, in addition to detailed discussions of spectral algorithms for fluid dynamics in simple 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 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 in single domains by expansions in smooth global basis functions. The essential concepts and formulas from this book are included in the current text for the reader’s convenience.

Reviewer: Nina Shokina (Freiburg)

### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76M22 | Spectral methods applied to problems in fluid mechanics |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |