Dispersion-relation-preserving FDTD algorithms for large-scale three ...

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Dispersion-Relation-Preserving FDTD Algorithms for Large-Scale Three-Dimensional Problems Shumin Wang and Fernando L. Teixeira, Member, IEEE

Abstract—We introduce dispersion-relation-preserving (DRP) algorithms to minimize the numerical dispersion error in large-scale three-dimensional (3-D) finite-difference time-domain (FDTD) simulations. The dispersion error is first expanded in spherical harmonics in terms of the propagation angle and the leading order terms of the series are made equal to zero. Frequency-dependent FDTD coefficients are then obtained and subsequently expanded in a polynomial (Taylor) series in the frequency variable. An inverse Fourier transforation is used to allow for the incorporation of the new coefficients into the FDTD updates. Butterworth or Chebyshev filters are subsequently employed to fine-tune the FDTD coefficients for a given narrowband or broadband range of frequencies of interest. Numerical results are used to compare the proposed 3-D DRP-FDTD schemes against traditional high-order FDTD schemes. Index Terms—Finite-difference time-domain (FDTD) method, numerical dispersion, optimization.

I. INTRODUCTION

N

UMERICAL (grid) dispersion is a major limiting factor for the overall accuracy of the finite-difference time-domain (FDTD) method in large-scale problems (several tens or hundreds of wavelengths) [1], [2]. Several approaches have been proposed to reduce dispersion error [3]–[17]. Among them, higher order FDTD schemes with fourth order of accuracy in space and time, or (4,4) schemes [3]–[5], have been shown to provide a particularly attractive trade-off between accuracy and computational cost for broadband simulations [7]–[13]. The application of dispersion-relation-preserving (DRP) schemes to reduce the local dispersion error in two-dimensional (2-D) FDTD simulations has been discussed in [14]. In this paper, we extend the DRP approach to reduce the local dispersion error and accumulated phase error of large-scale three-dimensional (3-D) FDTD simulations for all angles (in a minimax sense) and for a given (adjustable) frequency range. The DRP procedure here starts by expanding the local dispersion error (in the frequency domain) in spherical harmonics in terms of the propagation angle. The leading order terms of this series are made equal to zero and frequency-dependent DRP FDTD coefficients are obtained. These coefficients are then expanded into a polynomial (Taylor) series in terms of the frequency variable and inverse Fourier transformed for direct incorporation into the 3-D DRP FDTD update equations. Manuscript received February 20, 2002; revised July 16, 2002. The authors are with the ElectroScience Laboratory and Department of Electrical Engineering The Ohio State University, Columbus, OH 43212 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2003.815435

Butterworth (maximally flat) or Chebyshev filters are subsequently used to fine tune the performance in frequency. By employing such filters, the dispersion relation characteristics can be adjusted to minimize the dispersion error in a particular (possibly broad) frequency range of interest. Because the high frequency components are subject to larger cumulative phase error than low frequency components (for a given FDTD computational domain size), such filters can also be designed to produce FDTD schemes with smaller (local) dispersion error at high frequencies than at low frequencies (in traditional FDTD schemes usually the opposite is true) and reduce the maximum phase error accumulated in the FDTD grid. This paper is organized as follows. In Section II, we describe the DRP methodology and present the theoretical analysis for both filtered DRP and nonfiltered DRP schemes. In Section III, we compare numerical dispersion from the proposed 3-D DRP FDTD schemes against various traditional FDTD schemes. In Section IV, we summarize the main conclusions. II. METHODOLOGY A. 3-D DRP FDTD Equations The FDTD equations using a (2,4) stencil can be written as (1) (2) where the superscripts indicate the time step. In the above, reciprocity is explicitly enforced in order to obtain a conditionally stable scheme [15], [16]. The operator is defined as

(3) and , where and mined, and fined as

are coefficients to be deterare displacement operators [10] de-

(4)

(5) and similarly for the other components. In the above, the subscripts indicate the spatial location. The displacement operators

0018-926X/03$17.00 © 2003 IEEE

WANG AND TEIXEIRA: DISPERSION-RELATION-PRESERVING FDTD ALGORITHMS FOR LARGE-SCALE 3-D PROBLEMS

should be understood as being applied to field components collocated to the components on the left hand side of (1) and (2), so that their image produces field components at the usual positions on the staggered FDTD grid. For example, the explicit update is given by

(6) In the FDTD grid, the and fields can be expanded into a discrete set of Fourier modes as

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Either or can be eliminated from (9) and (10) to obtain the (discrete) dispersion relation. For a given set of coefficients and , , (recall that in a traditional (2,4) scheme, and ), this relation is used to analyze the numerical dispersion in the FDTD grid. In this work we will adopt an inverse standpoint and use this relation to design DRP FDTD schemes with minimized dispersion error over a range of frequencies. Specifically, we will enforce the exact (continuum) , in relation between frequency and wavenumber, viz. the dispersion relation, and then solve (in an approximate sense) and as the unknowns. for We consider a plane wave (Fourier mode) propagating in the ) direction such that , , ( . A possible way to obtain and is to and directly in (9) and (10) and then solve for the enforce and are given in terms of the coefficients. In this way, ), for given , , frequency and propagation angle ( , and . An obvious limitation of this approach is that zero ), dispersion error is approached only for a particular and ( with no guarantee of a small dispersion error at other angles. Instead, we search here for DRP coefficients which minimize the maximum dispersion error for all angles. This is done by expanding the dispersion error in a series in terms of the elevation angle (with respect to the -axis), , and the azimuth angle, , and by enforcing the dominant terms to be zero. polarization so that We consider the update for the (11) (12) Substituting (11) and (12) into (10), we obtain two independent equations

(7) (13) where

(8) Substituting (7) and (8) into (1) and (2), we obtain

with

(9)

(14)

(10)

The other polarization gives similar equations for the update. and , and (14) to find and We can use (13) to find . Notice that (14) involves four unknowns and depends on both and , while (13) involves two coefficients and depends on only. This is a consequence of the symmetry of the problem, which allows us to work solely with (13) to obtain and .1 In a Cartesian grid, the remaining coefficients are simultaneously determined because they obey identical equations with respect to the elevation angles of the associated axes. 1Note that the dispersion error retains the same functional dependency with respect to  (for fixed C and C ) regardless of the choice for C and C .

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By enforcing , and denoting the Courant–Friedrichs– , we can rewrite Levy (CFL) [1] number as (13) as

where are associated Legendre functions (with argument) with , and . As a result, we can rewrite the first term of the r.h.s. of (16) as

(15)

To expand the last two terms of the r.h.s. of (16), we use the identity

, or , and where, again, . of wavelengths per cell size

denotes the number

B. Non-Filtered DRP 3-D FDTD We define an error functional proportional to the difference between the left-hand side (LHS) and right-hand side (RHS) of (15), i.e., (17) (16) and in (15) is equivalent to letting . We expand in a series , with coefficients given of spherical harmonics by Solving for

and zero otherwise. Since , for for even. Bearing this in mind and using we have and the fact that , we calculate via integration by parts to obtain

(18) The first two terms are readily found as Since the functional in (16) is not a function of , the final spherical harmonics. equations will involve here only Nevertheless, we start from this most general form, which accommodates irregular grid cases. We use the following conventions for spherical harmonics All other terms can be evaluated recursively. So far, we have obtained the asymptotic series (19) shown at the bottom of the page. Next, we force the two leading terms in and terms, to be zero and solve (19), i.e., and . The solutions are given by (20)–(21) at the for bottom of the page where the denominator in (20) and (21) writes as

(19)

(20) (21)

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Substituting (20)–(21) into (19), the residue from (19) becomes (22), shown at the bottom of the page. and given in (20) and (21) The coefficients cannot be directly implemented in the time domain. Therefore, and in a Taylor series around we expand and retain the lowest order terms, i.e., and with (23) (24) The above can be easily transformed back to time domain . However, if straightforward time through discretization schemes are employed directly on the resulting equations with third order time derivatives, the update may become unconditionally unstable [4]. Alternatively, the second order time derivatives can be recast as spatial derivatives (Helmholtz equation) and discretized as such [5], [17], which is valid for staggered grids as long as and are uniform in the local stencil. In this manner, (23) and (24) becomes (25)

(26) Notice that if only the first order terms in (25) and (26) are considered, the traditional (2,4) scheme is recovered. The second terms in (25) and (26) are analogous to third order time derivative terms in traditional schemes with fourth order of accuracy in time [4]. The difference resides in multiplicative factors. We denote this scheme a (nonfiltered, minimax) DRP (4,4) scheme. and in staggered grids is The implementation of straightforward. The stability condition can be derived in a standard way [1] from the update equations and the final result is (27)

where

For the uniform grid case, , we drop all the coordinate subscripts for the coefficients in (27) so that the above upper bound reduces to (28) denotes the where , , and for all ( ). maximum magnitude of , , and (referring to (25) and Substituting , we have (26)) into

which takes the maximum when . Thus the CFL number with guaranteed stability is found to be 12/17. is always larger Note that this is a conservative bound, since was found to produce than zero in practice. In fact, stable results (including for schemes with filtering to be described next). C. Filtered 3-D DRP FDTD Schemes Rather than employing costly alternatives such as including additional terms in the Taylor expansion or more sophisticated time integration schemes, it is possible to improve on the results of the previous section by employing filters to further reduce the dispersion error over some preassigned frequency band. Since the procedure is essentially the same for all coefficients, we recase. strict the discussion here to the

(22)

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1) Butterworth Filters: Expanding (20) in a Taylor series to , and noting that , the fourth order, say we have

TABLE I COEFFICIENTS USED IN THE VARIOUS (4,4) SCHEMES

(29) in and above refer to ButterThe superscript worth coefficients. Previously, we have chosen and so as to make the first and second terms of the right hand side of the above identically zero (and the truncation error fourth order in ). Now we will treat these coefficients as unknowns for the moment. The above equation can be rewritten as

TABLE II COEFFICIENTS USED IN THE VARIOUS (4,4) SCHEMES (CONT’D)

(30) where

By expanding the above in a new basis { }, where refers to a center frequency of interest, the following relationship holds:

To solve for obtain

and

, we force

and

(34)

with

(35) (31)

(32)

and, therefore, (30) can be rewritten as

(33) where

to be zero and

which are functions of and In this manner

. Note that, if we let .

, we recover

At the center frequency, and . The remainder error. Around the center frequency, the corresponds to a term. Theoretically, we error is dominated by the can increase indefinitely the polynomial order above, and hence, make the response as close to an exact second-order Butterworth filter response as desired. This would have the . one-time cost of inverting increasingly larger matrices In practice, a fast convergence to the exact filter response is observed for larger . In the following numerical simulations, we throughout. Note that because of its maximally employ flat characteristics, the use of Butterworth filters is of interest not only to reduce the total phase error, but also to minimize the discrete group velocity error [18]. In practice, the design in (31) may be given by the highest frequency frequency of interest.

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Fig. 1. Comparison of the maximum (for all angles) phase error per wavelength using Yee’s and the traditional (2,4) scheme.

2) Chebyshev Filters: In this case, we choose { } as the expansion basis in (31), where is the th corresponds to order first-kind Chebyshev polynomial, and as an example, the frequency band of interest. Again using (31) and (32) now become (36)–(37), shown at the bottom of the page. Following the same procedure as Butterworth case, we solve and of an approximate for the coefficients and obtain second-order Chebyshev filter. Here

where , and denote the coefficients of the corresponding Chebyshev polynomials. Since the Chebyshev filter is set to be of second order, nonmonotonic behavior is expected to occur if a larger in the passband (extra degree of freedom) is used [19]. In our problem, this can be explored to reduce the maximum accumulated phase error, as illustrated in the next section.

III. NUMERICAL COMPARISONS We compare the nonfiltered, Butterworth, and Chebyshev DRP (4,4) schemes [3], [4] against Fang’s (4,4) and Deveze’s (4,4) schemes. The phase velocity in free space is solved from (maxthe transcendental dispersion relation for imum CFL number from the stability bound). In the following, we consider uniform 3-D FDTD grids with cell size , and we drop the coordinate subscript of the coefficients. Tables I and II give the coefficients used in the various DRP (4,4) schemes and Deveze’s (4,4) scheme.2 The Butterworth . The filter and the Chebyshev filter are optimized at , scheme labeled as “Chebyshev1” is optimized with . while “Chebyshev2” is optimized with Figs. 1–3 depicts the maximum (for all angles) phase error , where denotes per wavelength, defined as the discrete wavenumber obtained by solving the transcendental dispersion relation and refers to the continuum wavenumber. Fig. 1 illustrates the typical dispersion error levels from lower order FDTD schemes (Yee’s and (2,4) schemes) as a 2The

dispersion relation of Fang’s scheme is taken directly from [12].

(36)

(37)

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Fig. 2. Comparison of the maximum (for all angles) phase error per wavelength using a nonfiltered DRP (4,4) scheme, Fang’s (4,4) scheme and the Deveze (4,4) scheme.

Fig. 3.

Comparison of the maximum (for all angles) phase error per wavelength using different DRP (4,4) schemes.

reference.3 Fig. 2 shows that the nonfiltered DRP scheme has a better performance than both Deveze’s (4,4) and Fang’s (4,4) schemes. Moreover, Fig. 3 shows that the DRP schemes with filters are able to further reduce the dispersion error around the central frequency. Figs. 4 and 5 show the dispersion error produced by specifying different in DRP schemes with Butterworth and Chebyshev 3These curves are generated using the largest CFL number from the stability bound of each algorithm, viz.  1 for Yee’s scheme and  = 6=7 for the traditional (2,4) scheme.

=

filters, respectively. It is observed that the Chebyshev filters tend to produce slightly smaller errors than the Butterworth filters, particularly at higher frequencies. Note that for the schemes with filters, the dispersion error at high frequencies can be actually made smaller than the error at low frequencies (contrary to traditional schemes). As mentioned in the Introduction, this is a desirable characteristic since, for a given FDTD grid size, high frequencies correspond to an electrically larger domain and hence their accumulated phase error is larger. Unless the local dispersion error decreases faster than lin-

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Fig. 4. Comparison of the maximum (for all angles) phase error per wavelength of DRP schemes with Butterworth filters using different q .

Fig. 5. Comparison of the maximum (for all angles) phase error per wavelength of DRP schemes with Chebyshev filters using different q for a fixed

early with frequency, the largest accumulated phase error in the computational domain is determined by the highest frequency components. On this respect, consider a 3-D FDTD simulation in a (typical) frequency range such that the lowest frequency and the highest frequency corresponds corresponds to . Fig. 6 shows the largest (for all angles) phase error to accumulated in a distance equal to the largest wavelength considered by employing the different DRP (4,4) schemes. As it can be seen from this figure, although the local dispersion error from the DRP schemes with Butterworth and Chebyshev filters

1 = 0 02. q

:

of Fig. 3 is larger at low frequencies than at high frequencies, the largest accumulated phase error is still dominated by the high frequency components. The DRP scheme denoted as “Chebyshev2” is the one that comes closer to producing an uniform accumulated phase error at all frequencies (and also the smallest overall). Fig. 7 shows the accumulated phase error at two angles with the largest relative error difference, and for different . From this Figure, has no discernible effect on we observe that increasing

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Fig. 6.

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Comparison of the maximum (for all angles) accumulated phase error over the largest wavelength using different (4,4) optimized schemes.

Fig. 7. Comparison of the anisotropy of the accumulated phase error over one wavelength corresponding to the minimum frequency of interest and employing different q for the Chebyshev filters in DRP schemes with fixed q : .

1

=01

the dispersion anisotropy. By choosing an appropriate , the magnitude of the maximum accumulated phase error can be further minimized by having near symmetric curves with respect to the zero error level. As in this case, we find that exhibit more closely this characteristic, with the . largest accumulated phase error lower than 0.2 at Fig. 8 illustrates the anisotropy of the accumulated phase error for such a Chebyshev DRP scheme over one wavelength . Fig. 9 illustrates the frequency dependency of the at . This anisotropy of the accumulated phase error for

shows that despite the increase on the anisotropy at larger frequencies, the magnitude of the total accumulated phase error is no larger than 0.2 . IV. CONCLUSION We have described a dispersion-relation-preserving (DRP) methodology to optimize higher order FDTD schemes for largescale 3-D problems. By first expanding the numerical dispersion error in spherical harmonics and properly choosing the

WANG AND TEIXEIRA: DISPERSION-RELATION-PRESERVING FDTD ALGORITHMS FOR LARGE-SCALE 3-D PROBLEMS

Fig. 8. The anisotropy of the accumulated phase error over one wavelength corresponding to the minimum frequency of interest at FDTD scheme with Chebyshev filter designed with q : and q : .

=01

1 = 0 035

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q

= 0:1 for an DRP 3-D

Fig. 9. The anisotropy of the accumulated phase error over one wavelength corresponding to the minimum frequency of interest for an DRP 3-D FDTD scheme : and q : .  is fixed to be 0 . with Chebyshev filter designed at q

=01

1 = 0 035

FDTD coefficients to cancel the leading terms of the series, the dispersion error can be minimized in a minimax sense for all propagation angles. Time-domain DRP schemes are obtained by expanding the resulting frequency dependent coefficients in a Taylor series, and by employing either Butterworth or Chebyshev filters, the DRP schemes are fine-tuned to reduce the local dispersion error in a preassigned frequency range of interest. In particular, the dispersion characteristics can be adjusted to yield a smaller local dispersion error at high frequencies than at low frequencies so that less accumulated phase error (up to a preas-

signed frequency) results. A possible venue of future investigation is the extension of this methodology to irregular grids and a more detailed investigation of their (discrete) group velocity dispersion properties.

ACKNOWLEDGMENT The authors gratefully acknowledge R. Lee and J.-F. Lee for helpful discussions.

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REFERENCES [1] A. Taflove, Ed., Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method. Boston: Artech, 1998. [2] A. C. Cangellaris and R. Lee, “On the accuracy of numerical wave simulations based on finite methods,” J. Electromagn. Waves Applicat., vol. 6, pp. 1635–1653, 1992. [3] J. Fang, “Time Domain Computation for Maxwell’s Equations,” Ph.D, Univ. California at Berkeley, 1989. [4] T. Deveze, L. Beaulieu, and W. Tabbara, “A fourth order scheme for the FDTD algorithm applied to Maxwell’s equations,” Proc. IEEE AP-S Int. Symp., vol. 1, pp. 346–349, July 1992. [5] E. Turkel, “High order methods,” in Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, A. Taflove, Ed. Boston, MA: Artech House, 1998, pp. 63–110. [6] C. K. W. Tam and J. C. Webb, “Dispersion-relation-preserving finite difference schemes for computational acoustics,” J. Comp. Phys., vol. 107, pp. 262–281, 1993. [7] M. F. Hadi and M. Piket-May, “A modified FDTD (2,4) scheme for modeling electrically large structures with high-phase accuracy,” IEEE Trans. Antennas Propagat., vol. 45, pp. 254–264, Feb. 1997. [8] J. B. Cole, “A high accuracy FDTD algorithm to solve microwave propagation and scattering problems on a coarse grid,” IEEE Trans. Microwave Theory Tech., vol. 43, pp. 2053–2058, Sept. 1995. , “A high-accuracy realization of the Yee algorithm using nonstan[9] dard finite difference,” IEEE Trans. Microw. Theory Tech., vol. 45, pp. 991–996, June 1997. [10] E. A. Forgy and W. C. Chew, “A new FDTD formulation with reduced dispersion for the simulation of wave propagation through inhomogeneous media,” Proc. IEEE AP-S Int. Symp., vol. 2, pp. 1316–1319, 1999. [11] J. W. Nehrbass, J. O. Jevic´, and R. Lee, “Reducing the phase error for finite-difference methods without increasing the order,” IEEE Trans. Antennas Propagat., vol. 46, pp. 1194–1201, Aug. 1998. [12] K. L. Shlager, J. G. Maloney, S. L. Ray, and A. F. Peterson, “Relative accuracy of several finite-difference time-domain methods in two and three dimensions,” IEEE Trans. Antennas Propagat., vol. 41, pp. 1732–1737, Dec. 1993. [13] K. L. Shlager and J. B. Schneider, “Relative accuracy of several lowdispersion finite-difference time-domain schemes,” in Proc. IEEE APS Int. Symp., vol. 1, 1999, pp. 168–171. [14] S. Wang and F. L. Teixeira, “DRP schemes for broadband and electrically large 2-D FDTD simulations of Maxwell’s equations,” Progress Electromagn. Res., submitted for publication. [15] F. L. Teixeira and W. C. Chew, “Lattice electromagnetic theory from a topological viewpoint,” J. Math. Phys., vol. 40, no. 1, pp. 169–187, 1999.

[16] T. Weiland, “On the unique solution of Maxwellian eigenvalue problems in three dimensions,” Particle Accelerators, vol. 17, pp. 227–242, 1985. [17] G. Cohen and P. Joly, “Fourth order schemes for the heterogeneous acoustics equation,” Comp. Meth. Appl. Mech. Eng., vol. 80, no. 1-3, pp. 397–407, 1990. [18] L. N. Trefethen, “Group velocity in finite difference schemes,” SIAM Rev., vol. 24, no. 2, pp. 113–136, 1982. [19] B. Porat, A Course in Digital Signal Processing. New York: Wiley, 1997.

Shumin Wang received the B.S. degree in physics from Qingdao University, P. R. China, and the M.S. degree in electronics from Beijing University, P. R. China, in 1995 and 1998, respectively. He is currently working toward the Ph.D. degree in electrical engineering at The Ohio State University, Columbus. Since 1999, he has been a Graduate Research Associate with the ElectroScience Laboratory (ESL), The Ohio State University. His research interests include electrostatic and magnetostatic lens design, time-domain differentialequation-based methods, high-frequency asymptotic methods and their applications to scattering, packaging, microwave circuit, and antenna analysis.

Fernando L. Teixeira (S’89–M’99) received the B.S. and M.S. degrees in electrical engineering from the Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Brazil, in 1991 and 1995, respectively, and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 1999. From 1999 to 2000, he was a Postdoctoral Research Associate with the Research Laboratory of Electronics at the Massachusetts Institute of Technology (MIT), Cambridge. Since 2000, he has been an Assistant Professor at the ElectroScience Laboratory (ESL) and the Department of Electrical Engineering, at The Ohio State University, Columbus, OH. His current research interests include analytical and numerical techniques for wave propagation and scattering problems in communication, sensing, and devices applications. He has edited one book, Geometric Methods for Computational Electromagnetics (PIER 32, EMW: Cambridge, Mass., 2001), and has published over 30 journal articles and 50 conference papers in those areas. Dr. Teixeira was awarded the Raj Mittra Outstanding Research Award from the University of Illinois, and a 1998 MTT-S Graduate Fellowship Award. He received paper awards at the 1999 USNC/URSI National Radio Science Meeting (Boulder, CO) and at the 1999 IEEE AP-S International Symposium (Orlando, FL), and received a Young Scientist Award at the 2002 URSI General Assembly. He is a Member of Phi Kappa Phi and was the Technical Program Coordinator of the Progress in Electromagnetics Research Symposium (PIERS), Cambridge, MA, in 2000.