On Foldy-Lax models for time-domain scattering by small particles (Maryna Kachanovska, INRIA Saclay)

Foldy-Lax models [Foldy 1945; Lax 1951] describe wave scattering
by multiple obstacles, in the regime when their characteristic size tends to
zero. While frequency-domain Foldy-Lax models are now fairly well-studied
(see e.g. [Martin 2006; Challa and Sini 2014; 2015; 2016]), their time-domain
counterparts were considered only very recently, see [Barucq et al. 2021; Sini
et al. 2021].

This talk is dedicated to the derivation, stability and convergence analysis
of the time-domain Foldy-Lax model for sound-soft scattering by circular scat-
terers. We start with the time-domain counterpart of the respective frequency-
domain model from [Cassier and Hazard 2013] and prove that it is unstable for
some geometric configurations. To stabilize it, we propose its reinterpretation
as a perturbed Galerkin discretization of a single layer boundary integral equa-
tion. 

Its unperturbed Galerkin discretization is then automatically stable due
to a coercivity-like property of the underlying operator and thus serves as a
basis to derive the stabilized model.

We will present the convergence analysis of the new model, 

discuss its numerical implementation, and illustrate our findings with numerical experiments.