The state of art of time domain integral equation (TDIE) solvers has grown by leaps and bounds over the past decade. During this time, advances have been made in (i) the development of accelerators that can be retrofitted with these solvers and (ii) understanding the stability properties of the electric field integral equation. As is well known, time domain electric field integral equation solvers have been notoriously difficult to stabilize. Research into methods for understanding and prescribing remedies have been on the uptick. The most recent of these efforts are (i) Lubich quadrature and (ii) exact integration. In this paper, we re-examine the solution to this equation using (i) the undifferentiated form of the TD-EFIE and (ii) a separable approximation to the spatio-temporal convolution. The proposed scheme can be constructed such that the spatial integrand over the source and observer domains is smooth and integrable. As several numerical results will demonstrate, the proposed scheme yields stable results for long simulation times and a variety of targets, both of which have proven extremely challenging in the past.

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Stability of time domain integral equation (TDIE) solvers has remained an elusive goal for many years. Advancement of this research has largely progressed on four fronts: (1) Exact integration, (2) Lubich quadrature, (3) smooth temporal basis functions, and (4) Space-time separation of convolutions with the retarded potential. The latter method was explored in [1]. This method's efficacy in stabilizing solutions to the time domain electric field integral equation (TD-EFIE) was demonstrated on first order surface descriptions (flat elements) in tandem with 0th order functions as the temporal basis. In this work, we develop the methodology necessary to extend to higher order surface descriptions as well as to enable its use with higher order temporal basis functions. These higher order temporal basis functions are used in a Galerkin framework. A number of results that demonstrate convergence, stability, and applicability are presented.

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