Analysis of multiple scattering iterations for high-frequency scattering problems. II: The three-dimensional scalar case

In this paper, we continue our analysis of the treatment of multiple scattering effects within a recently proposed methodology, based on integral-equations, for the numerical solution of scattering problems at high frequencies. In more detail, here we extend the two-dimensional results in part I of this work to fully three-dimensional geometries. As in the former case, our concern here is the determination of the rate of convergence of the multiple-scattering iterations for a collection of three-dimensional convex obstacles that are inherent in the aforementioned high-frequency schemes. To this end, we follow a similar strategy to that we devised in part I: first, we recast the (iterated, Neumann) multiple-scattering series in the form of a sum of periodic orbits (of increasing period) corresponding to multiple reflections that periodically bounce off a series of scattering sub-structures; then, we proceed to derive a high-frequency recurrence that relates the normal derivatives of the fields induced on these structures as the waves reflect periodically; and, finally, we analyze this recurrence to provide an explicit rate of convergence associated with each orbit. While the procedure is analogous to its two-dimensional counterpart, the actual analysis is significantly more involved and, perhaps more interestingly, it uncovers new phenomena that cannot be distinguished in two-dimensional configurations (e.g. the further dependence of the convergence rate on the relative orientation of interacting structures). As in the two-dimensional case, and beyond their intrinsic interest, we also explain here how the results of our analysis can be used to accelerate the convergence of the multiple-scattering series and, thus, to provide significant savings in computational times.     

Wood's anomalies in the scattering problem of a plane wave on a smooth periodic boundary

A series of multiple diffracted fields in a high-frequency asymptotic expansion of the solution to the problem of grazing scattering of a plane wave by a smooth periodic boundary is summed in the case of appearance of Wood's anomalies. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 87–100, 1990. Translated by V. V. Zalipaev.     

Asymptotic expansions of multiply scattered surface currents

We have recently uncovered the convergence characteristics of multiple scattering iterations for “two-dimensional” as well as “three-dimensional scalar (acoustics)” scattering models in the high-frequency regime. As we have demonstrated, a most distinctive property of these latermodels, compared to their two-dimensional counterparts, is the dependence of corresponding asymptotic expansions on the relative angle of rotation between the principal axes of the successive reflection points of the optical rays. Concerning the case of fully “three-dimensional vector (electromagnetic)” scattering problems, here we show that the vectorial nature of the problem, in turn, gives rise to new additional complex structure. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical methods for scattering problems by curved gratings at high frequencies

A new numerical method is derived for the calculation of high-frequency asymptotic expansions of the scalar wave scattered by curved periodic structures. Optimal error estimates for this method are established. © 1997 B. G. Teubner Stuttgart-John Wiley & Sons Ltd.     

A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering

We propose a new robust method for the computation of scattering of high-frequency acoustic plane waves by smooth convex objects in 2D. We formulate this problem by the direct boundary integral method, using the classical combined potential approach. By exploiting the known asymptotics of the solution, we devise particular expansions, valid in various zones of the boundary, which express the solution of the integral equation as a product of explicit oscillatory functions and more slowly varying unknown amplitudes. The amplitudes are approximated by polynomials (of minimum degree d) in each zone using a Galerkin scheme. We prove that the underlying bilinear form is continuous in L ₂, with a continuity constant that grows mildly in the wavenumber k. We also show that the bilinear form is uniformly L ₂-coercive, independent of k, for all k sufficiently large. (The latter result depends on rather delicate Fourier analysis and is restricted in 2D to circular domains, but it also applies to spheres in higher dimensions.) Using these results and the asymptotic expansion of the solution, we prove superalgebraic convergence of our numerical method as d → ∞ for fixed k. We also prove that, as k → ∞, d has to increase only very modestly to maintain a fixed error bound (d ∼ k ^1/9 is a typical behaviour). Numerical experiments show that the method suffers minimal loss of accuracy as k →∞, for a fixed number of degrees of freedom. Numerical solutions with a relative error of about 10⁻⁵ are obtained on domains of size for k up to 800 using about 60 degrees of freedom.     

Shortwave scattering by a diffraction echelette-grating

The two-dimensional problem of the scattering of a plane wave by a periodic perfectly conducting grating, an echelette with a right angle, is considered in the case of a high-frequency approximation (the wavelength is assumed to be small as compared with the period of the grating). The situation where the incident plane wave glides along one of the faces of a wedge is discussed. A ray-optical solution of the problem (a shortwave asymptotic result) is derived by the method of summing multiple diffracted fields, which is well known in the geometric theory of diffraction. The main result of this paper consists of obtaining simple formulas for the efficiency e_n of a diffraction order with maximal value of e_n, derived in the shortwave approximation. Numerical results are presented, and important optical properties obtained by asymptotic analysis are described. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 109–135. Translated by V. V. Zalipaev.     

Scattering Poles near the Real Axis for Two Strictly Convex Obstacles

To study the location of poles for the acoustic scattering matrix for two strictly convex obstacles with smooth boundaries, one uses an approximation of the quantized billiard operator M along the trapped ray between the two obstacles. Using this method Gérard (cf. [7]) obtained complete asymptotic expansions for the poles in a strip Im z ≤ c as Re z tends to infinity. He established the existence of parallel rows of poles close to Assuming that the boundaries are analytic and the eigenvalues of Poincaré map are non-resonant we use the Birkhoff normal form for M to improve his result and to get the complete asymptotic expansions for the poles in any logarithmic neighborhood of real axis. Submitted: May 17, 2006. Accepted: September 19, 2006.     

A new integral representation for quasi-periodic scattering problems in two dimensions

Boundary integral equations are an important class of methods for acoustic and electromagnetic scattering from periodic arrays of obstacles. For piecewise homogeneous materials, they discretize the interface alone and can achieve high order accuracy in complicated geometries. They also satisfy the radiation condition for the scattered field, avoiding the need for artificial boundary conditions on a truncated computational domain. By using the quasi-periodic Green’s function, appropriate boundary conditions are automatically satisfied on the boundary of the unit cell. There are two drawbacks to this approach: (i) the quasi-periodic Green’s function diverges for parameter families known as Wood’s anomalies, even though the scattering problem remains well-posed, and (ii) the lattice sum representation of the quasi-periodic Green’s function converges in a disc, becoming unwieldy when obstacles have high aspect ratio.     

Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves

In this paper we address several theoretical questions related to the numerical approximation of the scattering of acoustic waves in two or three dimensions by penetrable non-homogeneous obstacles using convolution quadrature (CQ) techniques for the time variable and coupled boundary element method/finite element method for the space variable. The applicability of CQ to waves requires polynomial type bounds for operators related to the operator Δ − s ² in the right half complex plane. We propose a new systematic way of dealing with this problem, both at the continuous and semidiscrete-in-space cases. We apply the technique to three different situations: scattering by a group of sound-soft and -hard obstacles, by homogeneous and non-homogeneous obstacles.     

Periodic Homogenization of Green and Neumann Functions

For a family of second‐order elliptic operators with rapidly oscillating periodic coefficients, we study the asymptotic behavior of the Green and Neumann functions, using Dirichlet and Neumann correctors. As a result we obtain asymptotic expansions of Poisson kernels and the Dirichlet‐to‐Neumann maps as well as optimal convergence rates in L^p and W^1,p for solutions with Dirichlet or Neumann boundary conditions. © 2014 Wiley Periodicals, Inc.     

Local Expansions of Periodic Spline Interpolation with Some Applications

In this paper we prove some asymptotic expansions of the error of interpolation on equally spaced nodes with periodic smoothest splines of arbitrary degree on a uniform partition. We obtain a local expansion in terms of derivatives of the interpolate. Afterwards we apply this result to the asymptotic study of the numerical solution of periodic integral equations of the second kind by means of ϵ – collocation methods. We show some new superconvergence results and give particular forms of these expansions depending on the choices of the parameter ϵ. We finally give some numerical experiments, which corroborate the theory.     

Numerical methods and asymptotic error expansions for the Emden-Fowler equations

In the present paper we analyse a numerical method for computing the solution of some boundary-value problems for the Emden-Fowler equations. The differential equations are discretized by a finite-difference method and we derive asymptotic expansions for the discretization error. Based on these asymptotic expansions, we use an extrapolation algorithm to accelerate the convergence of the numerical method.     

Homogenization in the scattering problem

The scattering problem is studied, which is described by the equation (-Δ _x +q(x,x/ɛ)−E)ψ = f(x), where ψ = ψ (x,ɛ) ∈ ℂ, x ℂ ℝ ^d , ɛ > 0, E > 0, the function q(x,y) is periodic with respect to y, and the function f is compactly supported. The solution satisfying radiation conditions at infinity is considered, and its asymptotic behavior as ɛ → O is described. The asymptotic behavior of the scattering amplitude of a plane wave is also considered. It is shown that in principal order both the solution and the scattering amplitude are described by the homogenized equation with potential

Asymptotic expansion of the coefficients of radiating plane waves in the scattering problem on a smooth periodic boundary

High-frequency asymptotic expansion of the coefficients of radiating plane waves in the problem of grazing scattering of a plane wave on a smooth periodic boundary is derived. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 71–86, 1990. Translated by V. V. Zalipaev.     

Generalized scattering phases for asymptotically hyperbolic manifolds

We prove asymptotic expansions of generalized scattering phases asssociated to pairs of Laplacians, for a class of noncompact manifolds with infinite volume and negative curvature near infinity. We use one of these expansions to define relative determinants which appear naturally in this context. To cite this article: J.-M. Bouclet, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On the convergence of expansions in polyharmonic eigenfunctions

We consider expansions of smooth, nonperiodic functions defined on compact intervals in eigenfunctions of polyharmonic operators equipped with homogeneous Neumann boundary conditions. Having determined asymptotic expressions for both the eigenvalues and eigenfunctions of these operators, we demonstrate how these results can be used in the efficient computation of expansions. Next, we consider the convergence. We establish the key advantage of such expansions over classical Fourier series–namely, both faster and higher-order convergence–and provide a full asymptotic expansion for the error incurred by the truncated expansion. Finally, we derive conditions that completely determine the convergence rate.     

The Eigenvalues of Mathieu's Equation and their Branch Points

A comprehensive account is given of the behavior of the eigenvalues of Mathieu's equation as functions of the complex variable q. The convergence of their small-q expansions is limited by an infinite sequence of rings of branch points of square-root type at which adjacent eigenvalues of the same type become equal. New asymptotic formulae are derived that account for how and where the eigenvalues become equal. Known asymptotic series for the eigenvalues apply beyond the rings of branch points; we show how they can now be identified with specific eigenvalues.     

Correlation between pole location and asymptotic behavior for Painlevé I solutions

We extend the technique of asymptotic series matching to exponential asymptotics expansions (transseries) and show by using asymptotic information that the extension provides a method of finding singularities of solutions of nonlinear differential equations. This transasymptotic matching method is applied to Painlevé's first equation, P1. The solutions of P1 that are bounded in some direction towards infinity can be expressed as series of functions obtained by generalized Borel summation of formal transseries solutions; the series converge in a neighborhood of infinity. We prove (under certain restrictions) that the boundary of the region of convergence contains actual poles of the associated solution. As a consequence, the position of these exterior poles is derived from asymptotic data. In particular, we prove that the location of the outermost pole x_p(C) on ℝ⁺ of a solution is monotonic in a parameter C describing its asymptotics on anti‐Stokes lines and obtain rigorous bounds for x_p(C). We also derive the behavior of x_p(C) for large C ∈ ℂ. The appendix gives a detailed classical proof that the only singularities of solutions of P1 are poles. © 1999 John Wiley & Sons, Inc.     

Study of the higher-order asymptotic behavior of quantum field expansions in the theory of two-dimensional fully developed turbulence

We use a simplified (0+1)-dimensional theory to develop approaches for studying the higher-order asymptotic behavior of quantum field expansions in the two-dimensional theory of fully developed turbulence. We consider the asymptotic behavior of the correlation function in the small-time limit in the theory of fully developed turbulence and derive and investigate the stationarity equation. We show that the perturbation series in this limit has a finite convergence radius.