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

This paper considers the 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 considered scattering problems as the boundary value problems and prove that each boundary value problem has a unique solution by utilizing the integral equation method associated with the two-layered Green function. Moreover, we develop the Nystr\"{o}m method for numerically solving the considered boundary value problems, based on the proposed integral equation formulations. We establish the convergence results of the Nystr\"{o}m 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 Nystr\"{o}m 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 Nystr\"{o}m method.