Extraction of Uniformly Accurate Phase Functions Across Smooth Shadow Boundaries in High Frequency Scattering Problems

Abstract

Several recent numerical schemes for high frequency scattering simulations are based on the extraction of known phase functions from an oscillatory solution. The remaining function is typically no longer oscillatory, and as such it can be approximated numerically with a number of degrees of freedom that does not depend on the frequency of the original problem. Knowledge of the phase of a solution typically comes from asymptotic analysis, for example, from geometrical optics. We consider integral equation formulations of time-harmonic scattering by a smooth and convex obstacle and focus on the so-called shadow boundaries. They are the points where the incoming waves are tangential to the boundary of the scatterer. We devise a numerical method that incorporates advanced results from asymptotic analysis which describe the frequency-dependent transitional behavior of the solution uniformly across these points. We describe and resolve an apparent conflict between two theories that describe the asymptotic behavior of this problem. They are the well-known geometric theory of diffraction and the rigorous asymptotic analysis by Melrose and Taylor, based on microlocal analysis.