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**Implementing spectral methods for partial differential equations. Algorithms for scientists and engineers.**
*(English)*
Zbl 1172.65001

Scientific Computation. Berlin: Springer (ISBN 978-90-481-2260-8/hbk; 978-90-481-2261-5/ebook). xviii, 394 p. (2009).

This book offers a systematic and self-contained approach to solve partial differential equations (PDEs) numerically using single and multidomain spectral methods. It contains detailed algorithms in a pseudocode for the application of spectral approximations to both one and two dimensional PDEs of mathematical physics describing potentials, transport, and wave propagation. The author, a well-known researcher in the field with extensive practical experience, shows how only a few fundamental algorithms form the building blocks of any spectral code, even for problems with complex geometries.

The book addresses computational and applications scientists, as it emphasizes the practical derivation and implementation of spectral methods over abstract mathematics. It is divided into two parts: First comes a primer on spectral approximation and the basic algorithms, including fast Fourier transform algorithms, Gauss quadrature algorithms, and how to approximate derivatives. The second part shows how to use those algorithms to solve steady and time dependent PDEs in one and two space dimensions. Exercises and questions at the end of each chapter encourage the reader to experiment with the algorithms.

The book addresses computational and applications scientists, as it emphasizes the practical derivation and implementation of spectral methods over abstract mathematics. It is divided into two parts: First comes a primer on spectral approximation and the basic algorithms, including fast Fourier transform algorithms, Gauss quadrature algorithms, and how to approximate derivatives. The second part shows how to use those algorithms to solve steady and time dependent PDEs in one and two space dimensions. Exercises and questions at the end of each chapter encourage the reader to experiment with the algorithms.

Reviewer: Wilhelm Heinrichs (Essen)

### MSC:

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

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

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

65Y15 | Packaged methods for numerical algorithms |

65D32 | Numerical quadrature and cubature formulas |

35J25 | Boundary value problems for second-order elliptic equations |

35L15 | Initial value problems for second-order hyperbolic equations |

65T60 | Numerical methods for wavelets |