Overview
- Offers a self-contained presentation of aspects of stability in numerical mathematics
- Compares and characterizes stability in different subfields of numerical mathematics
- Covers numerical treatment of ordinary differential equations, discretisation of partial differential equations, discretisation of integral equations and more
- Includes supplementary material: sn.pub/extras
Part of the book series: Springer Series in Computational Mathematics (SSCM, volume 45)
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Table of contents (8 chapters)
Keywords
About this book
In this book, the author compares the meaning of stability in different subfields of numerical mathematics.
Concept of Stability in numerical mathematics opens by examining the stability of finite algorithms. A more precise definition of stability holds for quadrature and interpolation methods, which the following chapters focus on. The discussion then progresses to the numerical treatment of ordinary differential equations (ODEs). While one-step methods for ODEs are always stable, this is not the case for hyperbolic or parabolic differential equations, which are investigated next. The final chapters discuss stability for discretisations of elliptic differential equations and integral equations.
In comparison among the subfields we discuss the practical importance of stability and the possible conflict between higher consistency order and stability.
Reviews
“The contents are presented in a way that is accessible to graduate students who may use the book for self-study of the topic, and it can easily be used as a textbook for a corresponding lecture series. Moreover, advanced researchers in numerical mathematics are likely to benefit from reading it, in particular because the book provides interesting insight into how stability relates to areas other than their own particular specialization field. … also useful reading material for numerical software developers.” (Kai Diethelm, Computing Reviews, October, 2015)
“This book is concerned with stability properties in various areas of numerical mathematics, and their strong connection with convergence of numerical algorithms. As a side effect, any parts of numerical analysis are reviewed in the course of the stability discussions. The book aims in particular at master and Ph.D. students.” (M. Plum, zbMATH 1321.65139, 2015)
“This nontraditional book by Hackbusch (Max Planck Institute for Mathematics in the Sciences, Germany) headlines the abstract stability concept. … ultimately serves a broad but unusually thoughtful introduction to (or reexamination of) numerical analysis. Summing Up: Recommended. Upper-division undergraduates and above.” (D. V. Feldman, Choice, Vol. 52 (4), December, 2014)
“It is the perfect complement to a lecture series on numerical analysis, starting with stability of finite arithmetic, quadrature and interpolation, followed by ODE, time-dependent PDE, Elliptic PDE, and integral equations. … All chapters are presented self-contained with separate reference list, so that they can be studied independently. … it is highly recommended for all lectures and all students in applied and numerical mathematics.” (Christian Wieners, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 94 (9), 2014)
Authors and Affiliations
About the author
Bibliographic Information
Book Title: The Concept of Stability in Numerical Mathematics
Authors: Wolfgang Hackbusch
Series Title: Springer Series in Computational Mathematics
DOI: https://doi.org/10.1007/978-3-642-39386-0
Publisher: Springer Berlin, Heidelberg
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer-Verlag Berlin Heidelberg 2014
Hardcover ISBN: 978-3-642-39385-3Published: 20 February 2014
Softcover ISBN: 978-3-662-51371-2Published: 23 August 2016
eBook ISBN: 978-3-642-39386-0Published: 06 February 2014
Series ISSN: 0179-3632
Series E-ISSN: 2198-3712
Edition Number: 1
Number of Pages: XV, 188
Topics: Numerical Analysis, Partial Differential Equations, Integral Equations