1R40. Spectral Methods for Incompressible Viscous Flow. Applied Mathematical Sciences, Vol 148. - Edited by R Peyret (Lab JA Dieudonne, Univ de Nice-Sophia Antipolis, Parc Valrose, Nice, 06000, France). Springer-Verlag, New York. 2002. 432 pp. ISBN 0-387-95221-7. $59.95.
Reviewed by G de Vahl Davis (Sch of Mech and Manuf Eng, Univ of New S Wales, Sydney, 2052 NSW, Australia).
The foundations of spectral methods are not new. They lie in the use of series expansions, typically a Fourier series, to attack problems in mathematical physics, and especially fluid mechanics, and have so done since the 19th century. However, computational limitations in the BC (before computer) era—and even in the early AC years—limited their application. It was not until the 1970s, as Peyret points out, that a revival of the Fourier method was applied to the direct simulation of turbulence. Historically, spectral methods have formed the cornerstone of numerical methods used to study the physics of turbulence. Their resurgence can be attributed to the continually increasing power of computers and to the development of the fast Fourier transform.
Three principal methods are available for the solution of a system of coupled nonlinear partial differential equations such as those which describe fluid motion: finite difference (FD), finite element (FE), and spectral. FD and FE methods are based on a subdivision of the solution region into a number of small elements or control volumes, within which local representations of functions, usually by low-order polynomials, are made. Spectral methods, on the other hand, use global representations—now, typically, Fourier series and Chebyshev polynomials—which cover the entire computational domain.
FDM and FEM generally require fine grids and, therefore, a greater computational effort than spectral methods to achieve a given accuracy. To increase accuracy, both FDM and FEM use global or local grid refinement. Spectral methods (as well as some FEM) achieve increased accuracy by using higher order polynomials.
FDM and FEM lead to matrix equations which are sparse because of the local nature of the basis functions. Spectral methods lead to algebraic equations with full matrices.
FDM and FEM have little difficulty in handling conditions at the boundaries of the solution region. Spectral methods, in contrast, often suffer from stability problems which demand much smaller time steps to overcome, especially in two and three dimensions. They are most useful when the geometry of the solution region is fairly smooth and regular.
Spectral methods are usually more difficult to program, certainly in comparison with FDM. They are more costly, per degree of freedom, and they accommodate irregular geometries less happily. They thus enjoy some advantages, and suffer from some disadvantages, in comparison with FD and FE methods. Trefethen has stated that “difficulties with boundaries… are probably the primary reason why spectral methods have not replaced their lower-accuracy competition in most applications.”
In summary, spectral methods constitute a class of highly accurate numerical techniques generally only suitable for simple geometries. In this book, the author sets out “to provide a comprehensive discussion of Fourier and Chebyshev spectral methods for the computation of incompressible viscous flows, based on the Navier-Stokes equations.” He has succeeded admirably. The book encompasses the necessary mathematical background, provides a clear exposition of the associated computational techniques, and gives information on the implementation of these techniques.
This book, like Peyret’s Gaul, is divided into three parts. The first part (four chapters and almost 150 pages) discusses basic spectral methods: fundamentals, the Fourier method, the Chebyshev method, and time-dependent equations. Part II (three chapters and about 140 pages) covers the Navier-Stokes equations (in both velocity-pressure and vorticity-streamfunction formulations) and spectral methods for their solution in each formulation. Details such as the Boussinesq approximation and possible treatments of a semi-infinite domain are examined. Examples are given of the solution of Rayleigh-Be´nard convection, axisymmetric flow in a rotating annulus and three-dimensional flow in a rotating annulus. Part III (90 pages) has chapters on stiff and singular problems, and on domain decomposition (or spectral element methods). Short appendices cover formulas on Chebyshev polynomials, an algorithm for the solution of a quasi-tridiagonal system, and theorems on the zeros of a polynomial.
There are about 300 references; Peyret is author or co-author of some 10% of these, unquestionably establishing his credentials on this subject.
Spectral Methods for Incompressible Viscous Flow is an advanced text. It will appeal to applied mathematicians and CFD-oriented engineers at the post-graduate level and to anyone teaching or undertaking research on problems described by the Navier-Stokes equations. This book is highly recommended.