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**Spectral methods in fluid dynamics.**
*(English)*
Zbl 0658.76001

Springer Series in Computational Physics. New York etc.: Springer-Verlag. xiv, 557 p. DM 162.00 (1988).

This monograph is an extremely useful state of the art reference book addressing (i) computational fluid dynamicists who wish to use spectral methods and (ii) numerical analysts who are interested in their rigorous analysis. These methods employ sums \(\sum a_ k(t)\phi_ k(x)\) with \(k=1,\ldots, N\) as candidates for approximations of solutions of (non)linear PDEs with usual side conditions. Here, \(x\) stands for a spatial independent variable and \(t\) is time or a spatial coordinate in the case of steady-state problems. The \(\phi_ k\) are trial functions such as trigonometric or Chebyshev or Legendre polynomials, etc. If \(t\) is time, the \(a_ k\) are determined by use of any standard discretization method for initial value problems. Concerning the PDEs, the authors employ (a) Galerkin or tau- methods, both in terms of expansion coefficients or (b) collocation methods in terms of physical variables.

Chapter 1 gives an introduction. In chapters 2 (and 3), the authors discuss the fundamentals of spectral methods (for PDEs), making use of simple examples, particularly linear PDEs and the Burgers equation. Chapter 4 is concerned with pertinent temporal discretizations and stability considerations. Chapter 5 presents useful iteration methods for implicit spectral equations. Chapter 7 deals with several algorithms for the unsteady Navier-Stokes equations in the cases of laminar or turbulent flows. Chapter 8 presents algorithms for compressible flows, including a treatment of turbulence or shock waves. Chapter 9 deals with norm estimates for the global approximation error of the employed spectral methods. Chapter 10 is concerned with the theories of stability and convergence for linear PDEs. Applications with respect to the steady Navier-Stokes equations and hyperbolic problems appear in chapters 11 and 12, respectively. Chapter 13 presents recent algorithmic and theoretical developments concerning spectral methods in the case of more general geometries.

This monograph gives a critical evaluation of almost the total relevant literature consisting of 602 quoted references, approximately 40 of which are background material. The practical relevance of chapters 2–8 is shown by use of the following incomplete list of topics addresses in many places: comparisons with difference or finite element methods, numerical experience, explanations why algorithms work, aliasing, influence of round-off errors, convolution sums, useful tricks, the cost of algorithms, preconditioning of problems, multi-grid methods, etc. In Chapters 9–12, inequalities for inner products or norms are derived for very many important flow situations.

The monograph is self-contained since at least the basic theory is presented for every employed non-elementary mathematical domain or method. Because of the advanced nature of the subject, the reader should possess some experience (particularly concerning algorithms) in order to be able to appreciate this exceptionally thorough and practically useful presentation of one of the most active contemporary research domains.

Chapter 1 gives an introduction. In chapters 2 (and 3), the authors discuss the fundamentals of spectral methods (for PDEs), making use of simple examples, particularly linear PDEs and the Burgers equation. Chapter 4 is concerned with pertinent temporal discretizations and stability considerations. Chapter 5 presents useful iteration methods for implicit spectral equations. Chapter 7 deals with several algorithms for the unsteady Navier-Stokes equations in the cases of laminar or turbulent flows. Chapter 8 presents algorithms for compressible flows, including a treatment of turbulence or shock waves. Chapter 9 deals with norm estimates for the global approximation error of the employed spectral methods. Chapter 10 is concerned with the theories of stability and convergence for linear PDEs. Applications with respect to the steady Navier-Stokes equations and hyperbolic problems appear in chapters 11 and 12, respectively. Chapter 13 presents recent algorithmic and theoretical developments concerning spectral methods in the case of more general geometries.

This monograph gives a critical evaluation of almost the total relevant literature consisting of 602 quoted references, approximately 40 of which are background material. The practical relevance of chapters 2–8 is shown by use of the following incomplete list of topics addresses in many places: comparisons with difference or finite element methods, numerical experience, explanations why algorithms work, aliasing, influence of round-off errors, convolution sums, useful tricks, the cost of algorithms, preconditioning of problems, multi-grid methods, etc. In Chapters 9–12, inequalities for inner products or norms are derived for very many important flow situations.

The monograph is self-contained since at least the basic theory is presented for every employed non-elementary mathematical domain or method. Because of the advanced nature of the subject, the reader should possess some experience (particularly concerning algorithms) in order to be able to appreciate this exceptionally thorough and practically useful presentation of one of the most active contemporary research domains.

Reviewer: E.Adams

### MSC:

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

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

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

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