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2011 | OriginalPaper | Buchkapitel

1. Fifty Years of Stiffness

verfasst von : Luigi Brugnano, Francesca Mazzia, Donato Trigiante

Erschienen in: Recent Advances in Computational and Applied Mathematics

Verlag: Springer Netherlands

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Abstract

The notion of stiffness, which originated in several applications of a different nature, has dominated the activities related to the numerical treatment of differential problems for the last fifty years. Contrary to what usually happens in Mathematics, its definition has been, for a long time, not formally precise (actually, there are too many of them). Again, the needs of applications, especially those arising in the construction of robust and general purpose codes, require nowadays a formally precise definition. In this paper, we review the evolution of such a notion and we also provide a precise definition which encompasses all the previous ones.

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Nächstes Kapitel Efficient Global Methods for the Numerical Solution of Nonlinear Systems of Two Point Boundary Value Problems
Fußnoten
1
Moreover, his concept of structure, i.e. events which are able to accelerate the normal flow of time, is also interesting from our point of view, because it somehow recalls the mathematical concept of large variation in small intervals of time (see later).
 
2
Even Finance makes the distinction between short time and long time traders.
 
3
It is not clear if one is enough: in principle the definition may require to apply all of them.
 
4
Only in particular circumstances, for example in the linear case, it is sometimes allowed the language abuse: the nonlinear case may contain simultaneously stable and unstable solutions.
 
5
A great deal of this improvement is due to the author of the previous sentence.
 
6
We omit, for simplicity, the other fact which could affect new definitions, i.e., the fact that the solutions of the linear equation can be integrated over any large interval because of the equivalence, in this case, between asymptotic and exponential stability.
 
7
It is interesting to observe that the same theorem is known as the Ostrowsky’s Theorem, in the theory of iterative methods.
 
8
Often, it appears under the name of one-sided Lipschitz condition.
 
9
This problem has been suggested by J.I. Montijano.
 
10
Observe that, in the case of IVPs, B 0=I and B 1=O, so that Q=I.
 
11
It is both defined by the used method and by the considered mesh.
 
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Metadaten
Titel
Fifty Years of Stiffness
verfasst von
Luigi Brugnano
Francesca Mazzia
Donato Trigiante
Copyright-Jahr
2011
Verlag
Springer Netherlands
Buch
Recent Advances in Computational and Applied Mathematics

Print ISBN: 978-90-481-9980-8

Electronic ISBN: 978-90-481-9981-5

Copyright-Jahr: 2011

DOI
https://doi.org/10.1007/978-90-481-9981-5_1

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