IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 49, NO. 7, JULY 2001 1079 The Time-Domain Discrete Green’s Function Method (GFM) Characterizing the FDTD Grid Boundary Ronen Holtzman and Raphael Kastner, Senior Member, IEEE Abstract—For a given FDTD simulation space with an arbi- trarily shaped boundary and an arbitrary exterior region, most exisitng absorbing boundary conditions become inapplicable. A Green’s function method (GFM) is presented which accommodates arbitrarily shaped boundaries in close proximity to a scattering ob- ject and an arbitrary composition in the exterior of the simulation space. Central to this method is the numerical precomputation of a Green’s function tailored to each problem which represents the effects of the boundary and the external region. This function be- comes the kernel for a single-layer absorbing boundary operator. It is formulated in a manner which naturally incorporates numer- ically induced effects, such as the numerical dispersion associated with the FDTD scheme. The Green’s function is an exact absorber in the discretized space. This property should be contrasted with other methods which are initially designed for the continuum and are subsequently discretized, thereby incurring inherent errors in the discrete space which cannot be eliminated unless the contiuum limit is recovered. In terms of accuracy, the GFM results have been shown to be of a similar quality to the PML, and decidedly superior to the Mur condition. The properties of the GFM are substantiated by a number of numerical examples in one, two, and three dimen- sions. Index Terms—Absorbing boundary conditions, FDTD methods. I. INTRODUCTION W ITH THE advent of newly introduced absorbing boundary conditions (ABCs) for mesh truncation in the context of the finite-difference–time-domain (FDTD) computations, it has been recognized that the boundaries of the computational domain can be defined in close proximity to scatterers and yet produce very small reflections. The most successful methods can be classified under two distinct categories: 1) approximations to the continuous one-way wave equation at the boundary, e.g., the Engquist-Majda-Mur conditions [1] and 2) the use of artificial or physical absorbing materials near the boundary, such as the PML [2], [3]. It is typically assumed that the computational domain is bounded by a simple shape, e.g., a rectangular box or a sphere, which is embedded in free space. In some cases, it is also assumed that the waves impinging on the boundary are propagating, rather than evanescent waves. Most of these ABCs are initially formulated in the continuous domain, and then discretized for use in the FDTD scheme. Traditional ABCs of category 1), such as the Engquist-Majda-Mur conditions, are a discretized generalization of the continuous one-way wave equation. The one-way wave equation is basically equivalent in Manuscript received August 19, 2000; revised August 28, 2000. This work was supported in part by the U.S.–Israel Binational Science Foundation, Jerusalem, Israel, under Grant 95-00399. The authors are with the Department of Electrical Engineering—Physical Electronics, Tel-Aviv University, Tel-Aviv 69978, Israel. Publisher Item Identifier S 0018-926X(01)01279-0. the continuum to an impedance relationship between the electric and magnetic fields. It is well known, though, that impedance relationships in the discretized world are different from the continuous ones, in view of the numerical dispersion that characterizes the discretized case. For this reason, adaptation of these conditions has been only partially successful, even when higher order approximations have been used. Significant experience has been generated recently with the application of category 2) ABCs such as the PML, which also relies on an initial continuous derivation. It is now recognized that several PML layers must be employed for sufficiently accurate results. This extra computational region imposes additional burden on the computational resources, particularly in 3-D, compared with simpler methods that only need a small stencil close to the boundary. The Johns matrix [4] was developed in the context of the TLM method as a Green’s function representation of the external do- main and can be considered as an early attempt toward the for- mulation of an ABC along lines similar to this work. At the time of its development, though, it did not translate well into an ac- curate and efficient FDTD formulation. One can consider it as a spectral domain method in the sense that the waves are sepa- rated into incident and reflected waves at each removed branch leading to the boundary. The external medium is then viewed as a linear system, whose input and output are the incident and re- flected waves, respectively. The impulse response of this system is prerecorded for subsequent use in conjunction with the con- ventional TLM procedure. This method also features a means for a relaxation of the memory requirements by the usage of spatial interpolation in order to decrease the number of recorded branches. Further developments of this method have been car- ried out primarily in the context of the TLM. Examples are [5] and [6], where the Johns matrix has been used to extract fre- quency-domain -parameters of a specific region, [7], where it was applied in 2-D and 3-D TLM formulations with artificial losses which provide an exponential drop-off as a means for fur- ther reducing the numerical reflections, and [8], where the Johns matrix was developed as the inverse discrete Fourier transform (IDFT) of the frequency domain modal -parameters of a given network. Green’s functions in 1-D and 2-D have also been ap- plied in the context of the FDTD algorithm in [9] as ABCs at the excitation plane. In this application, the temporal duration of the Green’s functions is set equal to the temporal length of the computations. Waveguide problems are treated on a modal basis for added efficiency in [10] and [11]. Discrete and analytic Green’s functions are compared in [12], where digital filters are used to eliminate unwanted high-frequency components. An ad- ditional degree of efficiency, using Laguerre polynomials, was 0018–926X/01$10.00 © 2001 IEEE