A Nyström Method for Scattering by a Two-layered Medium with a Rough Boundary

Abstract

This paper is concerned with problems of scattering of time-harmonic acoustic waves by a two-layered medium with a non-locally perturbed boundary (called a rough boundary in this paper) in two dimensions, where a Dirichlet or impedance boundary condition is imposed on the boundary. The two-layered medium is composed of two unbounded media with different physical properties and the interface between the two media is considered to be a planar surface. We formulate the scattering problems considered as boundary value problems and prove the result of the well-posedness of each boundary value problem by utilizing the integral equation method associated with the two-layered Green function. Moreover, we develop a Nyström method for numerically solving the boundary value problems considered, based on the proposed integral equation formulations. We establish the convergence results of the Nyström method with the convergence rates depending on the smoothness of the rough boundary. It is worth noting that in establishing the well-posedness of the boundary value problems as well as the convergence results of the Nyström method, an essential role is played by the investigation of the asymptotic properties of the two-layered Green function for small and large arguments. Finally, numerical experiments are carried out to show the effectiveness of the Nyström method.