Numerical Applications of the Scaling Concept

Author: Fazio R.

Source: Acta Applicandae Mathematicae, Volume 55, Number 1, January 1999 , pp. 1-25(25)

Publisher: Springer

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Abstract:

We present a survey of recent developments in the applications of the scaling concept to numerical analysis. In addition, we report on some relevant topics not covered in existing surveys. Therefore, the present work updates and complements the existing surveys on the subject concerned.

Applications of the scaling concept are useful in the numerical treatment of both ordinary and partial differential problems. Applications to boundary-value problems governed by ordinary differential equations are mainly related to their transformation into initial-value problems. Within this context, special emphasis is placed on systems of governing equations, eigenvalue, and free boundary-value problems. An error analysis for a truncated boundary formulation of the Blasius problem is also reported. As far as initial-value problems governed by ordinary differential equations are concerned, we discuss the development of adaptive mesh methods. Applications to partial differential problems considered herein are related to the construction of finite-difference schemes for conservations laws, the solution structure of the Riemann problem, rescaling schemes and adaptive schemes for blow-up problems.

In writing this paper, our aim was to promote further and more important numerical applications of the scaling concept. Meanwhile, the pertinent bibliography is highlighted and is available on internet as the BIB file sc-gita.bib from the anonymous ftp area at the URL ftp://dipmat.unime.it/pub/papers/fazio/surveys.

Keywords: scaling concept; numerical analysis

Language: English

Document Type: Regular paper

Affiliations: 1: Department of Mathematics, University of Messina, Salita Sperone, 31, 98166 Messina, Italy e-mail: rfazio@dipmat.unime.it

Publication date: 1999-01-01