A Method to Reduce the Oscillations of the Solution of Time Domain Integral Equation Using Laguerre Polynomials

Abstract

In this paper, the reason of oscillations appeared in late time low energy region of time domain electric flled integral equation using laguerre polynomials is analyzed. And then, we propose an approach to reduce the oscillations by combining frequency and time domain. According to the Fourier transform relation between frequency responses and time responses, the solution of TDIE method with Laguerre expansion is absolutely convergent. Replace the spectrum of oscillation parts of time domain data with accurate frequency data by MOM. The stable time domain data can be obtained by performing IFFT to the revised frequency data. Numerical results are presented to illustrate the e-ciency of this approach. 1. INTRODUCTION For solving the time domain integral equation, the marching on in time (MOT) method (1) is usually employed. However, as pointed by many researchers, some of the TD-EFIE formulations associated with the MOT method sufiers from late-time instability, which usually takes the form of an expo- nentially increasing high frequency oscillation. In order to eliminate the instability of the MOT methods, the approximate averaging method (2) and the implicit scheme based on backward flnite difierence (3) approximation or central flnite difierence (4) methodology are proposed. Even though employing these methods, the solution obtained by using MOT has still exponentially increased late-time oscillations but only extend the span of the stable region. Young-seek Chung and Sarkar (5{7) proposed a marching on in order method of using the expansion by weighted Laguerre poly- nomials for temporal coe-cients of Hertz vector. The advantages of using the weighted Laguerre polynomials are the solution is independent of the time discretization, and all the computations are carried out using spatial variables. So the exponentially increasing solution will not emerge in late time. But there still exist some oscillations in late-time low energy region due to oscilla- tions characteristic of weighted Laguerre polynomials. Oscillation phenomenon efiects on obtaining other electromagnetic characteristic from the late time scattering data. In this paper, the reason of oscillations appeared in late time low energy region of time domain electric flled integral equation using laguerre polynomials is analyzed, and then, we consider absolute convergence of the time so- lution and the Fourier transform relation between temporal response and frequency response. An approach based on discrete Fourier transforms to improve the stabilities in late time is proposed.