Numerical applications of the scaling concept

We present a survey of recent developments in the applications of the scali ng concept to numerical analysis. In addition, we report on some relevant t opics not covered in existing surveys. Therefore, the present work updates and complements the existing surveys on the subject concerned. Applications df the scaling concept are useful in the numerical treatment of both ordin ary and partial differential problems. Applications to boundary-value problems governed by ordinary differential e quations are mainly related to their transformation into initial-value prob lems. Within;this context, special emphasis is placed on systems of governi ng equations, eigenvalue, and free boundary-value problems. An error analys is for a truncated boundary formulation of the Blasius problem is also repo rted. As far as initial-value problems governed by ordinary differential eq uations are: concerned, we discuss the development of adaptive mesh methods . Applications to partial differential problems considered herein are relat ed to the construction of finite-difference schemes for conservations laws, the solution structure of the Riemann problem, rescaling schemes and adapt ive schemes for blow-up problems. In writing this paper, our aim was to promote further and more important nu merical applications of the scaling concept. Meanwhile, the pertinent bibli ography 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/pap ers/fazio/surveys.