The Influence of Contrast and Temporal Expansion on the Marching-on-in-Time Contrast Current Density Volume Integral Equation

TL;DR

This paper introduces the MOT-JVIE, a stable and efficient time-domain integral equation method for high-contrast scatterers, demonstrating its stability, accuracy, and scalability through theoretical analysis and numerical experiments.

Contribution

The paper develops a new implicit marching-on-in-time scheme for contrast current density volume integral equations with stability independent of contrast for quadratic spline basis functions.

Findings

Stability of MOT-JVIE is independent of contrast for quadratic splines.

Lagrange and cubic splines show instabilities at high contrast.

MOT-JVIE accurately matches literature and frequency-domain results.

Abstract

The contrast current density volume integral equation, discretized with piecewise constant spatial basis and test functions and Dirac-delta temporal test functions and the piecewise polynomial temporal basis functions, results in a causal implicit marching-on-in-time scheme that we refer to as the marching-on-in-time contrast current density volume integral equation (MOT-JVIE). The companion matrix stability analysis of the MOT-JVIE solver shows that for a fixed spatial and temporal step size, the stability is independent of the scatterer's dielectric contrast for quadratic spline temporal basis functions. Whereas, Lagrange and cubic spline exhibit instabilities at higher contrast. We relate this stability performance to the expansion and testing procedure in time. We further illustrate the capabilities of the MOT-JVIE based on quadratic spline temporal basis functions by: comparing the MOT-JVIE solution to time-domain results from literature and frequency-domain results from a commercial combined field integral equation solver. Finally, we present a long time sequence for a high-constrast scatterer discretized with 24,000 spatial unknowns.