The Concept of Stability in Numerical Mathematics için kapak resmi
Başlık:
The Concept of Stability in Numerical Mathematics
Dil:
English
ISBN:
9783642393860
Yayın Bilgileri:
Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2014.
Fiziksel Tanımlama:
XV, 188 p. online resource.
Seri:
Springer Series in Computational Mathematics, 45
İçerik:
Preface -- Introduction -- Stability of Finite Algorithms -- Quadrature -- Interpolation -- Ordinary Differential Equations -- Instationary Partial Difference Equations -- Stability for Discretisations of Elliptic Problems -- Stability for Discretisations of Integral Equations -- Index.
Özet:
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.  .

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Özet

Özet

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.


Yazar Notları

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".


İncelemeler 1

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All mathematicians seek perfect solutions to interesting problems. Failing that, whereas pure mathematicians freely abstract away from the original problem to seek more tractable challenges, applied mathematicians, lacking that luxury due to their client's requirements, must aim for merely adequate solutions. Thus, basic pure mathematics pedagogy emphasizes abstraction, while applied pedagogy habitually pushes "cookbook" methods. Actually, at least in a narrow way, applied mathematicians trade problems, too, constructing sequences of easier numerical problems that approximate the given theoretical one. The efficacy (or not) of this approach depends on the presence of stability, namely, the tendency of solutions of the artificial approximating problems to furnish effective approximations to the true theoretical solution. So rather than present a particular class of problems or methods, this nontraditional book by Hackbusch (Max Planck Institute for Mathematics in the Sciences, Germany) headlines the abstract stability concept. (P. Bürgisser and F. Cucker's Condition, CH, Jul'14, 51-6222, similarly spotlights that abstract applied mathematic concept.) Because stability requires possibly subtle technical reformulation in each special context (recurrence relations; quadrature; interpolation; ordinary differential equations; hyperbolic, parabolic, and elliptic partial differential equations; and integral equations), this monograph (but not textbook--no exercises) ultimately serves a broad but unusually thoughtful introduction to (or reexamination of) numerical analysis. --David V. Feldman, University of New Hampshire