Pascal Redou, Laurent Gaubert, Gireg Desmeulles,
Pierre-Antoine Béal, Christophe Le Gal and Vincent Rodin.
Absolute stability of chaotic asynchronous multi-interactions schemes
for solving ODE.
CMES: Computer Modeling in Engineering & Sciences,
Tech Science Press, 70(1):11-40, 2010.
Abstract:
Multi Interaction Systems, used in the context of Virtual Reality, are
dedicated to real-time interactive simulations. They open the way to the
in virtuo experimentation, especially useful in the domain of
biochemical kinetics. To this purpose, chaotic and asynchronous scheduling
of autonomous processes is based upon desynchronization of phenomena
involved in the system. It permits interactivity, especially the capability
to add or remove phenomena in the course of a simulation. It provides
methods of resolution of ordinary differential systems and partial
derivative equations. Proofs of convergence for these methods have been
established, but the problem of absolute stability, although it is crucial
when considering multiscale or stiff problems, has not yet been treated.
The aim of this article is to present absolute stability conditions for
chaotic and asynchronous schemes. We give criteria so as to predict
instability thresholds, and study in details the significant example of a
damped spring-mass system. Our results, which make use of random matrices
products theory, stress the point that the desynchronization of phenomena,
and a random scheduling of their activations, can lead to instability.
Keywords:
Chaotic asynchronous scheduling, Multi-interaction systems, Ordinary
differential systems, Absolute stability, Random matrices products.
[doi:10.3970/cmes.2010.070.011]
[Redou10a.pdf]