A numerical method for scattering problems with unbounded interfaces

Abstract

We introduce a new class of computationally tractable scattering problems in unbounded domains, which we call decomposable problems. In these decomposable problems, the computational domain can be split into a finite collection of subdomains in which the scatterer has a "simple" structure. A subdomain is simple if the domain Green's function for this subdomain is either available analytically or can be computed numerically with arbitrary accuracy by a tractable method. These domain Green's functions are then used to reformulate the scattering problem as a system of boundary integral equations on the union of the subdomain boundaries. This reformulation gives a practical numerical method, as the resulting integral equations can then be solved, to any desired degree of accuracy, by using coordinate complexification over a finite interval, and standard discretization techniques.

Figures (21)

• Figure 1: Three dielectric channels meeting in a compact interaction zone, $D,$ showing sectors $S_1, S_2,S_3.$
• Figure 2: Two semi-infinite periodic boundaries meeting at along a common perpendicular line.
• Figure 3: The analytic solution test for our single wave-guide solver. We see that the solver is accurate to at least the adaptive integration tolerance $10^{-12}$ everywhere.
• Figure 4: The modulus of the solution densities from the analytic solution test in \ref{['fig:single-ana_test']}. The densities from the two halves of $\tilde{\gamma}$ overlap because of the symmetry of the problem and solution. The figure clearly shows that contour deformation was enough to ensure that the densities have decayed to machine precision.
• Figure 5: An example of $G_q$ and one of its derivatives.