Scattering by a perfect conductor in a waveguide: energy-preserving schemes for integral equations

The scattering matrix for a perfectly conducting electrical cylinder (or a sound hard obstacle) in a waveguide is unitary. This is a well-known result which is a consequence of the conservation of power. When a numerical method is employed to approximate the reflection and transmission coefficients of the cylinder, an approximate scattering matrix can be constructed. An integral equation of the second kind for an unknown density can be solved, and the density can then be used for computing the entries of the approximate scattering matrix. We show that this approximate matrix is unitary for cylinders of symmetric cross-section, regardless of the order of the approximation. In the nonsymmetric case, the approximate scattering matrix still satisfies a conservation of energy condition, albeit in an unfamiliar form. As the order of the approximation is increased, conservation of energy is also satisfied in the more familiar form to machine accuracy.