Uniform Asymptotic Expansions of Multiple Scattering Iterations

Although every implementation of a recent high frequency multiple scattering solver has displayed a frequency independent operation count, its numerical analysis yet remains as a challenging open problem. This is, in part, due to the absence of detailed information on the uniform asymptotic expansions of multiple scattering iterations. Here we address precisely this issue for a collection of convex obstacles in both two and three space dimensions and further, as an application, we present a generalized geometrical optics solver. Introduction Although every implementation of a recent high frequency multiple scattering solver [1] has displayed a frequency independent operation count to attain a prescribed accuracy, its numerical analysis yet remains as a challenging open problem. This is, in part, due to the absence of detailed information on the uniform asymptotic expansions of multiple scattering iterations. Indeed, asymptotic expansions, in their full generality, are only known for a single convex obstacle illuminated by a plane-wave incidence [2] and this, in turn, has given rise to the development of asymptotically O(1) single-scattering solvers [1], [3], [4]. Here we extend the results in [2] to encompass a collection of compact strictly convex obstacles and thereby enable a straightforward extension of the single-scattering algorithms [3], [4] to accompany the multiple scattering solver in [1]. Further, as an application of our derivations, we present a generalized geometrical optics solver. 1 Multiple scattering We consider here the sound soft acoustic scattering problem [5] from a smooth compact obstacle K in Rn, n = 2, 3, whose solution can be expressed as a single-layer potential with unknown density η, the normal derivative of the total field on ∂K. Although a variety of integral equations exist for η, for simplicity, we use here