Finite difference delay modeling of potential time integrals

The temporal discretization of the time-domain integral equations (TDIE) is commonly accomplished by either the implicit marching-on-in-time (MOT) schemes using subdomain Lagrange polynomial interpolations or the always-stable marching on-in-order/degrees (MOD) of the entire-domain weighted Laguerre basis functions. An alternative approach for discretizing the time convolution integrals in the TDIE, competitive to the time basis expansion in the MOT or MOD recipes, is the Lubich´s convolution quadrature methods (CQM), using the (first or) second order backward finite difference (BFD) approximations in the Laplace domain. The underlying physics describing the wave scattering process is time invariant, as the material properties do not change over time. The CQM are utilized to transform continuous-time representation of the time-invariant integral kernel (system transfer function) to discrete-time domain. The CQM are called finite difference delay modeling (FDDM) when the scattering analysis of arbitrarily shaped three-dimensional (3D) structures is carried out in a marching style.