Didier Sornette
Patrick Sebbah
Christian Vanneste
Lab. de Phys. de la Matiere Condensee, CNRS URA 190, Univ. de Nice---Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 2, France
The new model introduced recently [Vanneste et al., Europhys. Lett. 17, 715 (1992)] for the dynamical propagation of waves in arbitrary heterogeneous media, which is efficient for calculations on large systems (1024x1024) over long times (several 10[sup 6] inverse band widths) will be discussed. Instead of starting from a wave equation or a Hamiltonian that needs to be discretized for numerical implementation, the model is defined by the set of S matrices, one for each node, describing the interaction of the wave field with the scatterers. The different results are shown on wave packets in random media, which have been obtained using extensive numerical simulations on a parallel computer and these numerical results are compared with weak localization predictions. Finally, ``wave automaton'' is shown to be equivalent to a discretized version of the hyperbolic time-dependent wave and Klein--Gordon equations, when restricted to a suitable subclass of the control parameters, and the relationships between the two formulations of the wave propagation problem are made explicit. Compared to finite-difference versions of hyperbolic equations, the wave automaton is shown to be much more flexible for implementing arbitrary boundary conditions.