Numerical methods for waves in layered media
This talk will present numerical issues evaluating the layered media Green's function for the Helmholtz equations. Usually, the layered media Green’s functions are represented by Sommerfeld-type integrals and they have to be numerically evaluated. One of the well-known challenges is the slow convergence of Sommerfeld integral when the source and target points are near the layer interface. We overcome the issue by using alternative direction formulas based on contour integrals that are equivalent to the original integrals. The alternative direction formulas converge exponentially when the Gauss-Laguerre quadrature is used. Then, a boundary integral equation for acoustic wave scattering from objects located close to the layer interface is set up and solved using Nystrom method with high precision. Finally, I will present a fast algorithm based on the fast multipole method to apply the layered media Green’s function to a given density function.