Chebyshev & Fourier spectral methods.

*(English)*Zbl 0681.65079
Lecture Notes in Engineering, 49. Berlin etc.: Springer-Verlag. XVI, 798 p. DM 168.00 (1989).

This is a course on spectral methods which is aimed at graduate students but the required background of it is limited. The book discusses Hermite, Laguerre, rational Chebyshev, sinc and spherical harmonic functions. The book contains 19 chapters and 7 appendices. Chapters 1-5 consider questions of Fourier and Chebyshev theory and also Galerkin interpolation and collocation methods. Chapter 1 is an introduction to the methods, while chapters 2-5 are a selfcontained treatment of basic convergence and interpolation theory. All of this is a preparation for Chapter 6: pseudospectral methods for linear boundary value problems.

Chapters 7-11 are devoted to practical applications of pseudospectral time-integration methods. Iterative methods for solving linear matrix equations in chapter 12 are described with many examples. It is shown how these iterations can be directly extended to nonlinear problems. Also, the domain decomposition method is discussed. In chapter 13 mostly one- dimensional coordinate transformations are considered and it is demonstrated how they simplify computer programs for solving differential equations. In chapter 14 a variety of special tools for dealing with problems in infinite or semi-infinite domains is discussed. In chapter 15 particularities at the solution of problems in spherical geometry are discussed, and it is explained why “spherical harmonics” have been so popular. In chapter 16 brief descriptions of spectral tricks are given that are useful under special circumstances, and in chapter 17 symbolic spectral methods are treated.

The appendices contain large spectra of formulas and methods, basis functions, cardinal functions, matrix methods, the Newton-Kantorovich method for nonlinear boundary value and eigenvalue problems, and the continuation method. This makes the book selfcontained. The subject in the book is expressed in a simple way and often is demonstrated with program realizations. Although it is a graduate course, this book is also useful to researchers.

Chapters 7-11 are devoted to practical applications of pseudospectral time-integration methods. Iterative methods for solving linear matrix equations in chapter 12 are described with many examples. It is shown how these iterations can be directly extended to nonlinear problems. Also, the domain decomposition method is discussed. In chapter 13 mostly one- dimensional coordinate transformations are considered and it is demonstrated how they simplify computer programs for solving differential equations. In chapter 14 a variety of special tools for dealing with problems in infinite or semi-infinite domains is discussed. In chapter 15 particularities at the solution of problems in spherical geometry are discussed, and it is explained why “spherical harmonics” have been so popular. In chapter 16 brief descriptions of spectral tricks are given that are useful under special circumstances, and in chapter 17 symbolic spectral methods are treated.

The appendices contain large spectra of formulas and methods, basis functions, cardinal functions, matrix methods, the Newton-Kantorovich method for nonlinear boundary value and eigenvalue problems, and the continuation method. This makes the book selfcontained. The subject in the book is expressed in a simple way and often is demonstrated with program realizations. Although it is a graduate course, this book is also useful to researchers.

Reviewer: L.G.Vulkov

##### MSC:

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

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

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |

35-04 | Software, source code, etc. for problems pertaining to partial differential equations |