High‐order high‐frequency solutions of rough surface scattering problems

Abstract

[1] A new method is introduced for the solution of problems of scattering by rough surfaces in the high-frequency regime. It is shown that high-order summations of expansions in inverse powers of the wave number can be used within an integral equation framework to produce highly accurate results for surfaces and wavelengths of interest in applications. Our algorithm is based on systematic use and manipulation of certain TaylorFourier series representations and explicit asymptotic expansions of oscillatory integrals. Results with machine precision accuracy are presented which were obtained from computations involving expansions of order as high as 20.

High-order numerical solutions in frequency-independent computational times for scattering applications associated with surfaces with composite roughness

“…This low-order nature of standard asymptotic approximations was largely due to the mathematical and implementational complications that arise in attempts at increasing their order. As we have recently shown [16,17] (see also [18]), however, these complexities can be effectively dealt with in the context of highfrequency applications by resorting to integral-equation formulations and high-order versions of the classical theory of oscillatory integrals [19]. In this manner, we have demonstrated that high-frequency problems can be resolved with significantly improved accuracy in computational times that are comparable to those required by classical low-order approaches.…”

“…In contrast with the standard low-order versions, however, here we systematically derive a complete perturbation series for the scattered field [23,24], and we identify each term with the solution of a scattering problem off the slow portion of the surface. These problems derive their incidences from combinations of the high-frequency components of the scattering geometry and they are therefore amenable to a treatment based on (suitable extensions of) a class of high-order high-frequency solvers that we have recently designed [16][17][18]. Indeed, as we show, these solvers can be extended to allow for the efficient evaluation of high-order derivatives of the solutions, as needed in the recursive calculation of successive terms of the perturbation series.…”

Local solutions to high-frequency 2D scattering problems

“…The functions a j (t) are C 1 and independent of k. They can be constructed explicitly for all sufficiently smooth and convex scatterers. Examples of such computations are given in [17] and in [8,Ch. 1].…”

On the O(1) solution of multiple-scattering problems

“…In this case (1) takes on the form where , and in . Alternatively, this equation can be written as (4) where and the operators and are defined as…”

“…Our recent work [2] (see also [3] and [4]) has demonstrated the feasibility of such an numerical scheme for surfacescattering problems by convex obstacles. This algorithm evaluates scattering at arbitrarily high-frequencies, with a prescribed accuracy, and in a frequency-independent (thus ) computational time.…”