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Discrete-time and continuous-time modelling: some bridges and gaps

Published online by Cambridge University Press:  01 April 2007

HUBERT KRIVINE
Affiliation:
LPTMS, Bâtiment 100, Université Paris-Sud, F-91405 Orsay Email: hubert.krivine@u-psud.fr
ANNICK LESNE
Affiliation:
LPTMC UMR7600, Université Pierre et Marie Curie-Paris 6, 4 place Jussieu, F-75252 Paris Email: lesne@ihes.fr Institut des Hautes Études Scientifiques, 35 route de Chartres, F-91440 Bures-sur-Yvette, France Email: treiner@ccr.jussieu.fr
JACQUES TREINER
Affiliation:
LPTMS, Bâtiment 100, Université Paris-Sud, F-91405 Orsay Email: hubert.krivine@u-psud.fr

Extract

The relationship between continuous-time dynamics and the corresponding discrete schemes, and its generally limited validity, is an important and widely acknowledged field within numerical analysis. In this paper, we propose another, more physical, viewpoint on this topic in order to understand the possible failure of discretisation procedures and the way to fix it. Three basic examples, the logistic equation, the Lotka–Volterra predator–prey model and Newton's law for planetary motion, are worked out. They illustrate the deep difference between continuous-time evolutions and discrete-time mappings, hence shedding some light on the more general duality between continuous descriptions of natural phenomena and discrete numerical computations.

Type
Paper
Copyright
Copyright © Cambridge University Press 2007

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