A Highly Stable Marching-on-in-Time Volume Integral Equation Solver for Analyzing Transient Wave Interactions on High-Contrast Scatterers

Abstract

Time domain integral equation (TDIE) solvers represent an attractive alternative to finite difference (FDTD) and finite element (FEM) schemes for analyzing transient electromagnetic interactions on composite scatterers. Current induced on a scatterer, in response to a transient incident field, generates a scattered field. First, the scattered field is expressed as a spatio-temporal convolution of the current and the Green function of the background medium. Then, a TDIE is obtained by enforcing boundary conditions and/or fundamental field relations. TDIEs are often solved for the unknown current using marching on-in-time (MOT) schemes. MOT-TDIE solvers expand the current using local spatio-temporal basis functions. Inserting this expansion into the TDIE and testing the resulting equation in space and time yields a lower triangular system of equations (termed MOT system), which can be solved by marching in time for the coefficients of the current expansion. Stability of the MOT scheme often depends on how accurately the spatio-temporal convolution of the current and the Green function is discretized. In this work, band-limited prolate-based interpolation functions are used as temporal bases in expanding the current and discretizing the spatio-temporal convolution. Unfortunately, these functions are two sided, i.e., they require ”future” current samples for interpolation, resulting in a non-causal MOT system. To alleviate the effect of non-causality and restore the ability to march in time, an extrapolation scheme can be used to estimate the future values of the currents from their past values. Here, an accurate, stable and band-limited extrapolation scheme is developed for this purpose. This extrapolation scheme uses complex exponents, rather than commonly used harmonics, so that propagating and decaying mode fields inside the dielectric scatterers are accurately modeled. The resulting MOT scheme is applied to solving the time domain volume integral equation (VIE). Numerical results demonstrate that this new MOT-VIE solver maintains its stability and accuracy even when used in analyzing transient wave interactions on high-contrast scatterers.