Fokas integral equations for three dimensional layered-media scattering

The scattering of acoustic waves by periodic structures is of central importance in a wide range of problems of scientific and technological interest. This paper describes a rapid, high-order numerical algorithm for simulating solutions of Helmholtz equations coupled across irregular (non-trivial) interfaces meant to model acoustic waves incident upon a multiply layered medium. Building upon an interfacial formulation from previous work, we describe an Integral Equation strategy inspired by recent developments of Fokas and collaborators for its numerical approximation. The method requires only the discretization of the layer interfaces (so that the number of unknowns is an order of magnitude smaller than volumetric approaches), while it requires neither specialized quadrature rules nor periodized fundamental solutions characteristic of many popular Boundary Integral/Element Methods. As with previous contributions by the authors on this formulation, this approach is efficient and spectrally accurate for smooth interfaces.