Boundary-layer structures arising in linear transport theory.

We consider boundary-layer structures that arise in connection with the transport of neutral particles (e.g., photons or neutrons) through a participating medium.

Three-dimensional quasi-periodic shifted Green function throughout the spectrum, including Wood anomalies.

This work, part II in a series, presents an efficient method for evaluation of wave scattering by doubly periodic diffraction gratings at or near what are commonly called 'Wood anomaly frequencies'. At these frequencies, there is a grazing Rayleigh wave, and the quasi-periodic Green function ceases to exist. We present a modification of the Green function by adding two types of terms to its lattice sum.

Rapidly convergent quasi-periodic Green functions for scattering by arrays of cylinders-including Wood anomalies.

This paper presents a full-spectrum Green-function methodology (which is valid, in particular, at and around Wood-anomaly frequencies) for evaluation of scattering by periodic arrays of cylinders of arbitrary cross section-with application to wire gratings, particle arrays and reflectarrays and, indeed, general arrays of conducting or dielectric bounded obstacles under both transverse electric and transverse magnetic polarized illumination. The proposed method, which, for definiteness, is demonstrated here for arrays of perfectly conducting particles under transverse electric polarization, is based on the use of the shifted Green-function method introduced in a recent contribution (Bruno & Delourme 2014 , 262-290 (doi:10.1016/j.

Superalgebraically convergent smoothly windowed lattice sums for doubly periodic Green functions in three-dimensional space.

This work, part I in a two-part series, presents: (i) a simple and highly efficient algorithm for evaluation of quasi-periodic Green functions, as well as (ii) an associated boundary-integral equation method for the numerical solution of problems of scattering of waves by doubly periodic arrays of scatterers in three-dimensional space. Except for certain 'Wood frequencies' at which the quasi-periodic Green function ceases to exist, the proposed approach, which is based on smooth windowing functions, gives rise to tapered lattice sums which converge superalgebraically fast to the Green function-that is, faster than any power of the number of terms used. This is in sharp contrast to the extremely slow convergence exhibited by the lattice sums in the absence of smooth windowing.

High-order integral equation methods for problems of scattering by bumps and cavities on half-planes.

This paper presents high-order integral equation methods for the evaluation of electromagnetic wave scattering by dielectric bumps and dielectric cavities on perfectly conducting or dielectric half-planes. In detail, the algorithms introduced in this paper apply to eight classical scattering problems, namely, scattering by a dielectric bump on a perfectly conducting or a dielectric half-plane, and scattering by a filled, overfilled, or void dielectric cavity on a perfectly conducting or a dielectric half-plane. In all cases field representations based on single-layer potentials for appropriately chosen Green functions are used.

Fourier continuation methods for high-fidelity simulation of nonlinear acoustic beams.

On the basis of recently developed Fourier continuation (FC) methods and associated efficient parallelization techniques, this text introduces numerical algorithms that, due to very low dispersive errors, can accurately and efficiently solve the types of nonlinear partial differential equation (PDE) models of nonlinear acoustics in hundred-wavelength domains as arise in the simulation of focused medical ultrasound. As demonstrated in the examples presented in this text, the FC approach can be used to produce solutions to nonlinear acoustics PDEs models with significantly reduced discretization requirements over those associated with finite-difference, finite-element and finite-volume methods, especially in cases involving waves that travel distances that are orders of magnitude longer than their respective wavelengths. In these examples, the FC methodology is shown to lead to improvements in computing times by factors of hundreds and even thousands over those required by the standard approaches.

Efficient high-order evaluation of scattering by periodic surfaces: deep gratings, high frequencies, and glancing incidences.

We present a superalgebraically convergent integral equation algorithm for evaluation of TE and TM electromagnetic scattering by smooth perfectly conducting periodic surfaces z=f(x). For grating-diffraction problems in the resonance regime (heights and periods up to a few wavelengths) the proposed algorithm produces solutions with full double-precision accuracy in single-processor computing times of the order of a few seconds. The algorithm can also produce, in reasonable computing times, highly accurate solutions for very challenging problems, such as (a) a problem of diffraction by a grating for which the peak-to-trough distance equals 40 times its period that, in turn, equals 20 times the wavelength; and (b) a high-frequency problem with very small incidence, up to 0.

Prescribed error tolerances within fixed computational times for scattering problems of arbitrarily high frequency: the convex case.

We present a new algorithm for the numerical solution of problems of electromagnetic or acoustic scattering by large, convex obstacles. This algorithm combines the use of an ansatz for the unknown density in a boundary-integral formulation of the scattering problem with an extension of the ideas of the method of stationary phase. We include numerical results illustrating the high-order convergence of our algorithm as well as its asymptotically bounded computational cost as the frequency increases.

Inverse scattering problem for optical coherence tomography.

We deal with the imaging problem of determining the internal structure of a body from backscattered laser light and low-coherence interferometry. Specifically, using the interference fringes that result when the backscattering of low-coherence light is made to interfere with the reference beam, we obtain maps detailing the values of the refractive index within the sample. Our approach accounts fully for the statistical nature of the coherence phenomenon; the numerical experiments that we present, which show image reconstructions of high quality, were obtained under noise floors exceeding those present for various experimental setups reported in the literature.