Acceleration of two-dimensional time domain integral equation solvers using a Hilbert transform

In EM scattering theory we introduce a fast method for evaluating the [B/sub m/] vector in the equation [A/sub mk/][J/sub k/(t/sub n/)]=[B/sub m/] where [J/sub k/(t/sub n/)] is a vector whose element J/sub k/(t/sub n/) denotes the current density on the kth spatial patch at time t/sub n/, [A/sub mk/] is a matrix describing instantaneous interactions between current elements, [B/sub m/] is a vector whose elements are calculated from the known incident field and that produced by currents prior to time t/sub n/. This method relies on an expression of the vector potential in terms of a time-gated Hilbert transform and its computational complexity scales as O(N/sub t/logN/sub t/). The proposed method is a subset of the plane wave time domain methods that our group developed earlier. In contrast to the latter, the current technique only results in improved scaling of the computational cost w.r.t. the number of time steps in the analysis and does not change the scaling of the cost w.r.t. the number of spatial unknowns. Nonetheless, the proposed scheme is far more simple to implement than full-fledged plane wave time domain solvers and therefore deserves separate attention.