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Recent Advances in Computational and Applied Mathematics pp 1–21Cite as

Fifty Years of Stiffness

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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.

Frustra fit per plura quod potest per pauciora.

Razor of W. of Ockham, doctor invincibilis.

Work developed within the project “Numerical methods and software for differential equations”.

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Notes

  1. 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. 2.

    Even Finance makes the distinction between short time and long time traders.

  3. 3.

    It is not clear if one is enough: in principle the definition may require to apply all of them.

  4. 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. 5.

    A great deal of this improvement is due to the author of the previous sentence.

  6. 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. 7.

    It is interesting to observe that the same theorem is known as the Ostrowsky’s Theorem, in the theory of iterative methods.

  8. 8.

    Often, it appears under the name of one-sided Lipschitz condition.

  9. 9.

    This problem has been suggested by J.I. Montijano.

  10. 10.

    Observe that, in the case of IVPs, B 0=I and B 1=O, so that Q=I.

  11. 11.

    It is both defined by the used method and by the considered mesh.

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Acknowledgements

The authors wish to thank the reviewers, for their comments and suggestions.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Firenze, Viale Morgagni 67/A, 50134, Firenze, Italy

    Luigi Brugnano

  2. Dipartimento di Matematica, Università di Bari, Via Orabona 4, 70125, Bari, Italy

    Francesca Mazzia

  3. Dipartimento di Energetica, Università di Firenze, Via Lombroso 6/17, 50134, Firenze, Italy

    Donato Trigiante

Authors
  1. Luigi Brugnano
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  2. Francesca Mazzia
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  3. Donato Trigiante
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Correspondence to Luigi Brugnano .

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Editors and Affiliations

  1. Dpt. Computer Science and Technology, University of Peloponnese, Tripolis, GR-221 00, Greece

    Theodore E. Simos

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Brugnano, L., Mazzia, F., Trigiante, D. (2011). Fifty Years of Stiffness. In: Simos, T. (eds) Recent Advances in Computational and Applied Mathematics. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9981-5_1

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