An unconditionally stable scheme for the finite-difference time-domain method.pdf
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1、IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 51, NO. 3, MARCH 2003697 An Unconditionally Stable Scheme for the Finite-Difference Time-Domain Method Young-Seek Chung, Member, IEEE, Tapan K. Sarkar, Fellow, IEEE, Baek Ho Jung, and Magdalena Salazar-Palma, Senior Member, IEEE AbstractIn t
2、his work, we propose a numerical method to obtain an unconditionally stable solution for the finite-differ- ence time-domain (FDTD) method for the?case. This new method does not utilize the customary explicit leapfrog time scheme of the conventional FDTD method. Instead we solve the time-domain Maxw
3、ells equations by expressing the transient behaviors in terms of weighted Laguerre polynomials. By using these orthonormal basis functions for the temporal variation, the time derivatives can be handled analytically, which results in an implicit relation. In this way, the time variable is eliminated
4、 from the computations. By introducing the Galerkin temporal testing procedure, the marching-on in time method is replaced by a recursive relation between the different orders of the weighted Laguerre polynomials if the input waveform is of arbitrary shape. Since the weighted Laguerre polynomials co
5、nverge to zero as time progresses, the electric and magnetic fields when expanded in a series of weighted Laguerre polynomials also converge to zero. The other novelty of this approach is that, through the use of the entire domain-weighted Laguerre polynomials for the expansion of the temporal varia
6、tion of the fields, the spatial and the temporal variables can be separated. To verify the accuracy and the efficiency of the proposed method, we compare the results of the conventional FDTD method with the proposed method. Index TermsFinite difference time domain (FDTD), Laguerre polynomials, uncon
7、ditionally stable scheme. I. INTRODUCTION T HE finite-difference time-domain (FDTD) method has been widely used for the numerical analysis of tran- sient electromagnetic problems because it is conditionally stable and very easy to implement 1. Moreover, since it is a time-domain technique, one singl
8、e run of simulation can provide much information overa wide-band using abroad-band excitation. However, since the FDTD method is an explicit time-marching technique, its time step size should be limited by the well-known CourantFriedrichLecy (CFL) stability condition. Since the time step is dependen
9、t on the smallest lengthofthecellinacomputationaldomain,thisCFLcondition Manuscript received May 30, 2002; September 18, 2002. Y.-S.ChungiswiththeDepartmentofCommunicationEngineering,Myongji University, Kyunggi 449-728, Korea (e-mail: ychung05mju.ac.kr). T. K. Sarkar is with the Department of Electr
10、ical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244-1240 USA (e-mail: tksarkarsyr.edu). B. H. Jung is with the Department of Information and Communication Engineering,HoseoUniversity,Chungnam336-795,Korea(e-mail: bhjungoffice.hoseo.ac.kr). M. Salazar-Palma is with the Depa
11、rtamento de Seales, Sistemas y Radiocomunicaciones, Universidad Politcnica de Madrid, Madrid 28040, Spain (e-mail: Salazargmr.ssr.upm.es). Digital Object Identifier 10.1109/TMTT.2003.808732 may be too restrictive to solve problems with fine structures, such as thin material, slot, and via. In recent
12、 times, to eliminate the CFL stability condition, the alternating-direction-implicit (ADI) method was proposed in order to formulate the implicit FDTD scheme 26. The resulting ADI-FDTD method was found to be unconditionally stable. Therefore, one can use the larger value of the time step rather than
13、 the CFL limit. However, the larger value of the time step rather than the CFL limit results in a larger dispersion error. In this paper, we propose an unconditionally stable solution procedure for the FDTD method for the two-dimensional (2-D) case using weighted Laguerre polynomials as temporal bas
14、is and testing functions. Since Laguerre polynomials are defined fromto, they are suitable for a causal system 7, 8. Laguerre polynomials of higher orders can be generated recursively and are orthogonal with respect to a weighting function in a function space defined through the inner product of two
15、 continuous functions. Using the Laguerre polynomials and the weighting function, one can construct a set of orthogonal basis functions, which we call the weighted Laguerre polynomials. Physical quantities that are functions of time can be spanned in terms of these orthogonal basis functionsweighted
16、 Laguerre polynomials. Note that the weighted Laguerre polynomials are completely convergent to zero as. Therefore, arbitrary quantities or functions spanned by these basis functions are also convergent to zero as time progresses. Using the Galerkins method, we introduce a temporal testing procedure
17、, which results in an implicit FDTD formulation. By applying the temporal testing procedure to the FDTD, one can eliminate the time-step limitation that is the hallmark of the explicit time-domain technique. Instead of the leapfrog procedure, we introduce a marching-on-in-order of the basis function
18、s. Therefore, we can obtain the unknown coefficients for the basis functions from the zeroth order to theorder by solving recursively the FDTD with weighted Laguerre polynomials. The minimum order or number of basis functions is dependent on the time duration and the frequencybandwidth product of th
19、e problem. When employing the conventional FDTD method, there is no matrix inversion involved with this computation procedure. However, the proposed method produces a banded sparse system matrix and is independent of the time step. However, this method also uses the same system matrix regardless of
20、the order of basis functions to recursively solve for the unknowns. Therefore, one can assemble this sparse system matrix only once. 0018-9480/03$17.00 2003 IEEE 698IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 51, NO. 3, MARCH 2003 The paper isorganized in thefollowingmanner. In Sectio
21、nII, the formulations of the proposed FDTD are described. In Sec- tion III, the numerical results are presented. Finally, in Sec- tion IV, we summarize some conclusions. II. FDTD USINGWEIGHTEDLAGUERREPOLYNOMIALS AS BASISFUNCTIONS A. FDTD With theCase With simple and lossless media, themodel formulat
22、ion of the time-domain Maxwells equations is (1) (2) (3) whereis the electric permittivity andis the magnetic perme- ability. An upper dot denotes the derivative with respect to time. Using the central difference scheme both on time and space, (1)(3) are discretized as (4) (5) (6) where (7) (8) For
23、the above difference equations,is not a real position but an array index of each field variable, as shown in Fig. 1. Fig. 1 shows the position of the electric and magnetic field vector components over the 2-D cells.andare the lengths of the edge where the electric fields are located. andare the dist
24、ances between the center nodes where the magnetic fields are located. In this paper, we use the dispersive boundary condition (DBC) as an absorbing boundary condition (ABC) 9. The first-order DBC atoris given by (9) Fig. 1.Position of the transient electric and magnetic field vector components on th
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