Discrete Green's function diakoptics for stable FDTD interaction between multiply-connected domains
We have developed FDTD boundary conditions based on discrete Green's function diakoptics for arbitrary multiply-connected 2D domains. The associated Z-domain boundary operator is symmetric, with an imaginary part that can be proved to be positive semi-definite on the upper half of the unit circle in the complex Z-plane. Through Schwarz' exterior formula an integral representation of this operator is obtained that is analytic outside that unit circle. A quadrature-rule based rational approximation of the operator corresponds to a self-consistent finite-lookback scheme in the discretised time domain. This scheme is demonstrably stable, in that only secular, non-growing, source-free solutions remain, which may be suppressed.