IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 11, NO. 11, NOVEMBER 2001 465 A New Temporal Basis Function for the Time-Domain Integral Equation Method Jin-Lin Hu, Chi H. Chan, Senior Member, IEEE, and Yuan Xu Abstract—A new temporal basis function that has all-order continuous derivative has been constructed using a nonlinear optimization scheme. This new basis function provides a much more stable explicit marching-on-in-time (MOT) solution, based on the time-domain integral equation (TDIE) method, than what is presently available. Two examples are presented to illustrate the superior stability of the proposed temporal basis function. Index Terms—Nonlinear optimization, temporal basis function, time-domain integral equation method. I. INTRODUCTION N UMERICALLY rigorous transient analyses are based either on the differential or integral equation approach. A time-domain integral equation (TDIE) method that requires only a surface discretization is sometimes preferred over the differential one using a volumetric discretization. Furthermore, TDIE implicitly imposes the radiation condition and there exists no grid dispersion. While TDIE has been around for over 30 years [1], its widespread use as an engineering tool has been deterred by three factors, namely, i) computational complexity of the algorithm, ii) availability of the required spatial-time domain Green’s functions of inhomogeneous medium such as the layered media, and iii) stability of the marching-on-in-time (MOT) process. The first two factors have been addressed recently in [2]–[4], and further improvements are likely to be developed. In contrast, there is a continuous effort in searching for stable MOT schemes [5]–[8]. Each of these schemes in [5]–[8] pushes the late-time instability further down in time but could not eliminate it completely unless an implicit scheme, such as the one proposed in [9], which requires solving a large matrix equation, is employed. In [8], there is evidence that TDIE that employs a temporal basis function with a continuous derivative would provide a more stable MOT scheme. In this paper, we introduce a new temporal basis function that has continuous all-order derivative which leads to a much more stable explicit MOT solution than what are presently available. Manuscript received May 10, 2001; revised September 4, 2001. This work was supported by the Hong Kong Research Grant Council under Grant 9 040 623. The review of this letter was arranged by Associate Editor Dr. Arvind Sharma. J.-L. Hu is with the Wireless Communications Research Center, City Univer- sity of Hong Kong, Kowloon, Hong Kong. He is also with the Department of Applied Mathematics, Northwestern Polytechnical University, Xian, Shaanxi, China. C. H. Chan is with the Wireless Communications Research Center, City Uni- versity of Hong Kong, Kowloon, Hong Kong. Y. Xu is with the Millimeter Wave Technique Laboratory, Nanjing University of Science & Technology, Nanjing, China. Publisher Item Identifier S 1531-1309(01)10794-4. Fig. 1. Comparison of the spectral content of the triangular temporal basis function and our new basis function (real line: spectral content of our new basis function; dotted line: spectral content of triangular basis function). II. NEW TEMPORAL BASIS FUNCTION To have a temporal basis function that has all-order contin- uous derivative, we choose the following form otherwise (1) [shown in (1) at the top of the next page]. For simplicity, we choose and the function spans from to 1. The un- known coefficients and are determined using the optimal construction scheme proposed in [10]. In [10], a nonlinear op- timization scheme is adopted to minimize an objective function which requires that . Through this construction process, we obtain, from (1), the new temporal basis function as otherwise. (2) An explanation for the cause of instability is that if the temporal basis function has a rich high-frequency content, the spatial dis- cretization may not be fine enough for these high frequencies. The accumulated error will lead to late-time instability. Fig. 1 shows the comparison of the spectral content of the triangular temporal basis function and the proposed basis function. It is clear that this new function has less high-frequency contents 1531–1309/01$10.00 © 2001 IEEE