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**Finite difference methods for ordinary and partial differential equations. Steady-state and time-dependent problems.**
*(English)*
Zbl 1127.65080

Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-0-898716-29-0/pbk). xv, 341 p. (2007).

The book is based on the lecture notes of the author written mainly for students in applied mathematics at the University of Washington. It is organized into two main parts and some appendices. The first part deals with steady-state boundary value problems beginning with two-point boundary value problems followed by bi- and three dimensional elliptic equations. Part II concerns time-dependent problems, starting with initial value problems for ordinary differential equations. Part III consists of a set of appendices covering background material necessary at various points of the main text.

I: Boundary value problems and iterative methods. {Chapter 1.} Finite Difference Approximations. {Chapter 2}. Steady states and boundary value problems. {Chapter 3}. Elliptic equations. {Chapter 4.} Iterative methods for sparse linear systems.

II: Initial value problems. {Chapter 5.} The initial value problem for ordinary differential equations (ODEs). {Chapter 6.} Zero-stability and convergence for initial value problems. {Chapter 7.} Absolute stability for ODEs. {Chapter 8.} Stiff ODEs. {Chapter 9.} Diffusion equations and parabolic problems. {Chapter 10.} Advection equations and hyperbolic systems. {Chapter 11.} Mixed equations.

III. Appendices. A. Measuring errors. B. Polynomial interpolation and orthogonal polynomials. C. Eigenvalues and inner-product norms. D. Matrix powers and exponentials. E. Partial differential equations.

The emphasis is on building an understanding of the essential ideas that underlie the development analysis, and practical use of finite difference methods. Stability theory plays a large role, and the author explains several key concepts, their relation to one another. The author’s aim is to form a foundation from which the reader can approach the vast literature of more advanced topics and further explore and use the finite difference methods according to his interests and needs. The references contain 106 titles covering the years 1950–2006.

I: Boundary value problems and iterative methods. {Chapter 1.} Finite Difference Approximations. {Chapter 2}. Steady states and boundary value problems. {Chapter 3}. Elliptic equations. {Chapter 4.} Iterative methods for sparse linear systems.

II: Initial value problems. {Chapter 5.} The initial value problem for ordinary differential equations (ODEs). {Chapter 6.} Zero-stability and convergence for initial value problems. {Chapter 7.} Absolute stability for ODEs. {Chapter 8.} Stiff ODEs. {Chapter 9.} Diffusion equations and parabolic problems. {Chapter 10.} Advection equations and hyperbolic systems. {Chapter 11.} Mixed equations.

III. Appendices. A. Measuring errors. B. Polynomial interpolation and orthogonal polynomials. C. Eigenvalues and inner-product norms. D. Matrix powers and exponentials. E. Partial differential equations.

The emphasis is on building an understanding of the essential ideas that underlie the development analysis, and practical use of finite difference methods. Stability theory plays a large role, and the author explains several key concepts, their relation to one another. The author’s aim is to form a foundation from which the reader can approach the vast literature of more advanced topics and further explore and use the finite difference methods according to his interests and needs. The references contain 106 titles covering the years 1950–2006.

Reviewer: Erwin Schechter (Moers)

### MSC:

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

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

65L12 | Finite difference and finite volume methods for ordinary differential equations |

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

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

65L70 | Error bounds for numerical methods for ordinary differential equations |

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

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

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

65N15 | Error bounds for boundary value problems involving PDEs |