Springer Series in Computational Mathematics

The Concept of Stability in Numerical Mathematics

Authors: Hackbusch, Wolfgang

  • 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
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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.

 

About the authors

The author is a very well-known author of Springer, working in the field of numerical mathematics for partial differential equations and integral equations. He has published numerous books in the SSCM series, e.g., about the multi-grid method, about the numerical analysis of elliptic pdes, about iterative solution of large systems of equation, and a book in German about the technique of hierarchical matrices. Hackbusch is also in the editorial board of Springer's book series "Advances in Numerical Mathematics" and "The International Cryogenics Monograph Series".

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)


Table of contents (8 chapters)

Buy this book

eBook $79.99
price for USA (gross)
  • ISBN 978-3-642-39386-0
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $109.00
price for USA
  • ISBN 978-3-642-39385-3
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $109.00
price for USA
  • ISBN 978-3-662-51371-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
The Concept of Stability in Numerical Mathematics
Authors
Series Title
Springer Series in Computational Mathematics
Series Volume
45
Copyright
2014
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-39386-0
DOI
10.1007/978-3-642-39386-0
Hardcover ISBN
978-3-642-39385-3
Softcover ISBN
978-3-662-51371-2
Series ISSN
0179-3632
Edition Number
1
Number of Pages
XV, 188
Topics