On late-time stability of time domain integral equations for electromagnetics

TL;DR

This paper addresses the longstanding issue of late-time instability in time domain integral equations for electromagnetics by proposing a stable bilinear form construction based on energy minimization, with theoretical error bounds and demonstration results.

Contribution

It introduces a new bilinear form based on energy minimization that ensures stability and provides rigorous error bounds, advancing the theoretical foundation of stable time domain integral equations.

Findings

The proposed scheme is stable over long simulation times.

Theoretical bounds on approximation errors are established.

Sample scattering results confirm stability and effectiveness.

Abstract

The problem of late time instability in time domain integral equations for electromagnetics is longstanding. While several techniques have been suggested for addressing this problem, they either require impractically high degrees of freedom in the basis function or an analytical computation of matrix elements. The authors recently proposed a method that demonstrates stability without requiring either of these. The paper, however, does not present theoretical foundations for the choice of, or a rigorous error bounds on the approximation of, the bilinear form. This paper complements the authors' previous work by presenting a construction of the bilinear form based on the minimization of the energy in the system and a proof for the bounds on the approximation. We present results on the bounds developed and few sample scattering results that demonstrate the stability of the proposed scheme.