Finite difference schemes and partial differential equations.

*(English)*Zbl 0681.65064
Wadsworth & Brooks/Cole Mathematics Series. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software. xii, 386 p. (1989).

This is an excellent monography on the analysis of finite difference methods for partial differential equations, one of the few existing. The author has succeeded in representing the basic and important material necessary to do numerical computations jointly with the rigorous theory to understand the performance of the methods.

The selection of the material represented is a little bit biased towards time dependent problems (11 of 14 chapters) which of course contain the two classes of hyperbolic and parabolic equations resp. systems. A thorough treatment of consistency and stability is given where as a unifying tool Fourier analysis has been used throughout the text. From the beginning the fundamental concepts of the Lax-Richtmyer theory are used which contributes very much to clarify the main ideas and to make the diversity of different schemes understandable. Many new ideas come in at this and other points which is a feature hoped for in books written by experts working themselves in the field.

While the contents of the book are already outlined it is worth especially mentioning two chapters on the “analysis of well-posed and stable problems” including the Kreiss matrix theorem and “well-posed and stable initial-boundary value problems” presenting the theory pertaining to the well-posedness of boundary conditions for parabolic and hyperbolic equations and its discretizations. It is a characteristic of the book to provide the reader also with the basic facts on the theory of partial differential equations which makes it very suitable for using it in courses for students which frequently are not so familiar with PDEs. In this respect the large number of exercises included is also helpful. One chapter is dedicated to the conjugate gradient method including preconditioning but other sophisticated methods for solving elliptic difference equations are not treated. Nonlinear equations and adaptive grids have been omitted from the text, explainable already by the limitations in space where a volume of nearly 400 pages has been reached.

This book should be available in all libraries.

The selection of the material represented is a little bit biased towards time dependent problems (11 of 14 chapters) which of course contain the two classes of hyperbolic and parabolic equations resp. systems. A thorough treatment of consistency and stability is given where as a unifying tool Fourier analysis has been used throughout the text. From the beginning the fundamental concepts of the Lax-Richtmyer theory are used which contributes very much to clarify the main ideas and to make the diversity of different schemes understandable. Many new ideas come in at this and other points which is a feature hoped for in books written by experts working themselves in the field.

While the contents of the book are already outlined it is worth especially mentioning two chapters on the “analysis of well-posed and stable problems” including the Kreiss matrix theorem and “well-posed and stable initial-boundary value problems” presenting the theory pertaining to the well-posedness of boundary conditions for parabolic and hyperbolic equations and its discretizations. It is a characteristic of the book to provide the reader also with the basic facts on the theory of partial differential equations which makes it very suitable for using it in courses for students which frequently are not so familiar with PDEs. In this respect the large number of exercises included is also helpful. One chapter is dedicated to the conjugate gradient method including preconditioning but other sophisticated methods for solving elliptic difference equations are not treated. Nonlinear equations and adaptive grids have been omitted from the text, explainable already by the limitations in space where a volume of nearly 400 pages has been reached.

This book should be available in all libraries.

Reviewer: R.D.Grigorieff

##### MSC:

65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |

65F10 | Iterative numerical methods for linear systems |

65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |

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

65Nxx | Numerical methods for partial differential equations, boundary value problems |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65N06 | Finite difference methods for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

35Jxx | Elliptic equations and elliptic systems |

35Kxx | Parabolic equations and parabolic systems |

35Lxx | Hyperbolic equations and hyperbolic systems |