Stability Properties of the Time Domain Electric Field Integral Equation Using a Separable Approximation for the Convolution with the Retarded Potential

TL;DR

This paper introduces a new stable method for solving the time domain electric field integral equation using a separable approximation, improving long-term stability and applicability to complex targets.

Contribution

It proposes a separable approximation approach to the TD-EFIE that enhances stability and smoothness, addressing longstanding challenges in the field.

Findings

The scheme achieves stable long-term simulations.

It handles complex targets effectively.

Numerical results confirm improved stability.

Abstract

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.