Title: Stable Numerical Methods for Acoustic and Electromagnetic Scattering

Abstract:
We discuss the conditioning properties of classical boundary perturbation
methods for scattering from diffraction gratings. We will demonstrate that
significant cancellations are present in the recursions generated by these
methods, resulting in numerical algorithms which deteriorate with
decreasing regularity of the grating profile. These cancellations prevent
a direct proof of the convergence of these recursions, however, such a
proof can be realized if a change of variables is performed before the
derivation of the perturbation series. These observations also provide
guiding principles for the design of efficient and highly accurate low-
(first- and second-) and high-order algorithms.  The low-order recurrences
are stabilized by explicitly accounting for cancellations, while in the
high-order setting we advocate several methods for the implicit management
of these cancellations. Numerical results illustrating the advantages of
these approaches are presented.