Mathematics – Numerical Analysis
Scientific paper
2010-08-27
Mathematics in Engineering, Science and Aerospace. Vol. 1, No. 4 (2010) 407-426
Mathematics
Numerical Analysis
25 pages, 4 figures (typos fixed)
Scientific paper
On computers, discrete problems are solved instead of continuous ones. One must be sure that the solutions of the former problems, obtained in real time (i.e., when the stepsize h is not infinitesimal) are good approximations of the solutions of the latter ones. However, since the discrete world is much richer than the continuous one (the latter being a limit case of the former), the classical definitions and techniques, devised to analyze the behaviors of continuous problems, are often insufficient to handle the discrete case, and new specific tools are needed. Often, the insistence in following a path already traced in the continuous setting, has caused waste of time and efforts, whereas new specific tools have solved the problems both more easily and elegantly. In this paper we survey three of the main difficulties encountered in the numerical solutions of ODEs, along with the novel solutions proposed.
Brugnano Luigi
Iavernaro Felice
Trigiante Donato
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