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Development of Higher Order FDTD Schemes With Controllable Dispersion Error Theodoros T. Zygiridis and Theodoros D. Tsiboukis, Senior Member, IEEE
Abstract—A new methodology that facilitates the control of the inherent dispersion error in the case of higher order finite-difference time-domain (FDTD) schemes is presented in this paper. The basic idea is to define suitable algebraic expressions that reflect numerical inaccuracies reliably. Then, finite-difference operators are determined via the minimization of the error estimators at selected frequencies. In order to apply this procedure, an error expansion in terms of cylindrical harmonic functions is performed, which also enables accuracy enhancement for all propagation angles. The design process produces a set of two-dimensional (2-D) FDTD algorithms with optimized frequency response. Contrary to conventional methodologies, the proposed techniques adjust their reliability range according to the requirements of the examined problem and can be, therefore, more efficient in computationally demanding simulations. Index Terms—Finite-difference time-domain (FDTD) method, higher order methods, numerical dispersion.
I. INTRODUCTION
A
CHALLENGING issue in computational electromagnetics remains the reliable solution of several contemporary problems, such as scattering from large or multiple targets, antenna array modeling, propagation prediction in indoor/outdoor environments, broadband characterization of microwave components, that usually entail electrically-large domains and/or long-term evolution of wave interactions. The discrete models provided by Yee’s finite-difference time-domain (FDTD) method [1], [2] for this type of simulations are contaminated by nontrivial numerical dispersion errors, whose sufficient reduction postulates excessive computational resources and execution times. Among other methods with enhanced dispersion characteristics [3], higher order finite-difference and time-integration schemes [4]–[21] constitute more appropriate choices, capable of accurately reproducing electromagnetic-field properties on a lattice, even under demanding conditions. Although more computationally intensive, such approaches overwhelm the second-order technique, because they require coarser meshes for a specific error level. Due to this attractive aspect, exploiting the full potential of higher order methodologies is deemed a desirable objective. Conventional FDTD implementations are dominated by finitedifference formulas obtained from truncated Taylor series. However, in several instances [5], [6], [9], [11], [17], [20], [21] the Manuscript received June 10, 2004; revised March 7, 2005. This work was supported in part by the Greek General Secretariat of Research and Technology, under Grant PENED 01ED27. The authors are with the Applied and Computational Electromagnetics Laboratory, Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki GR-54124, Greece (e-mail:
[email protected]). Digital Object Identifier 10.1109/TAP.2005.854559
advantages of FDTD schemes with a degree of “problem-dependency” have been demonstrated. These approaches optimize their performance by exploiting certain information regarding the investigated problem (e.g., frequency band of interest), or details related to the corresponding discrete model, such as grid resolution or time-step. The designframework recently presented in [17] and [20] aims at the construction of such FDTD methods, which allow sufficient control of the dispersion error. The basic idea is to introduce suitable expressions for the inherent discretization error, and derive modified finite-difference approximations via the minimization of these formulas. The definition of a reliable error indicator is necessary, since the nonlinear grid-dispersion relation prohibits the explicit calculation of the exact numerical inaccuracies, which would simplify the design approach. In this paper, we utilize analogous concepts to develop a methodology for the design of two-dimensional (2-D) FDTD schemes, whose performance can be optimized according to problem-related requirements. The proposed techniques, which accomplish significant amelioration of the dispersion error, resemble Fang’s fourth-order FDTD method [4] (also denoted as (4, 4) scheme, indicating the order of accuracy in time and space). Although their only differences with the (4, 4) method lie on multiplying coefficients, the new schemes exhibit a completely different spectral behavior. For their construction, we initially introduce parametric expressions for the differential operators. Contrary to conventional trends, their determination is pursued through a dispersion-error minimization procedure. The latter is made feasible with the definition of proper error estimators, which originate from Maxwell’s equations and incorporate effects of both space and time discretization. An expansion of these expressions in terms of basis angular functions is subsequently conducted, thus facilitating the accuracy improvement regardless of the direction of propagation. Depending on the manner the parametric operators are finally determined, various FDTD algorithms can be obtained, whose diverse, yet controllable, wide-band characteristics render them suitable for the solution of a broad range of demanding electromagnetic applications. The quality of the proposed techniques is estimated by numerically computing the exact dispersion error, while several simulations reveal their practical merits and superiority over standard approaches. II. ESTIMATION OF DISPERSION ERROR IN HIGHER ORDER FDTD SCHEMES A. Parameter-Dependent Finite-Difference Operators We examine the 2-D case of waves , assuming a lossless homogeneous medium with
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ZYGIRIDIS AND TSIBOUKIS: DEVELOPMENT OF HIGHER ORDER FDTD SCHEMES
stitutive parameters (the conclusions, however, are valid for the case as well). The schemes developed hereafter comply with common FDTD realizations, i.e., the continuous domain is discretized with a dual cartesian grid and truncated by some boundary condition (e.g., absorbing). Regarding the approximation of spatial derivatives in Maxwell’s equations, parametric finite differences are introduced, according to
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which is discretized according to (1) and (2). We also obtain
(5)
from (3). A finite-difference version of (4) is now derived as
(1) at point ( is the cell size along -direcand are appropriate constants). It is reminded tion, that, based on Taylor analysis, second-order approximations truncation error) can be ensured if is ( truncation enforced, whereas fourth-order operators ( and . Along error) are produced when similar lines, a parametric expression for temporal derivatives is adopted, at time instant
(6)
where the third-order spatial derivatives are approximated by
(2) with being the selected time-step. One may recall that the fourth-order approximation is retrieved from (2) by setting . In order to obtain explicit FDTD update equations, the thirdorder time derivatives are converted to a combination of spatial coones, discretized with second-order accuracy due to the efficient. This alteration is realized through
(7)
(3) where is the speed of light in free space. Appar, ently, (3) is valid for homogeneous media only at least in the range of the spatial stencil. The main advantage of this modification is that the resulting update equations preserve the structure and, consequently, much of the simplicity of the to classic FDTD method, since the update from remains a two-stage process. In the present approach, we treat , and parameters as unknowns, whose determination will be based on spectral, problem-related, accuracy requirements. B. Dispersion-Error Estimation As mentioned in the Introduction, the design of FDTD schemes with low levels of numerically-induced phase inaccuracies calls for an appropriate error estimator. It is desirable that its dependence on frequency and angle of propagation is similar to that of the true dispersion error. Since we can not find an analytical formula for the latter, the improvement of accuracy is alternatively pursued through the minimization of the error indicator. Bearing this idea in mind, the first equation from the 2-D Maxwell’s system is considered (4)
(8) can be expressed Additionally, (6) is further simplified, as . Assuming plane-wave propagation along the in terms of direction denoted by angle in the polar coordinate system, one may obtain (9) with the electric-field and magnetic-field intensities satisfying . According to (9), we apply
(10)
in (6). This step is necessary, because an equation involving only one field component ( in this case) is desired. The error estiwith plane-wave mator can be constructed, by first replacing solutions , where and is the discrete (numerical) wavenumber. Next, the additional constraint , i.e., the equality between the
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numerical and exact wavenumber, is directly enforced. We finally measure the FDTD error related to (4) as the difference between the two sides of the resulting equation
(11) Apparently, an exact (for all angles and frequencies) FDTD method would produce . It is readily understood that the aforementioned measure of the FDTD-related error is a function of both frequency and angle of propagation. By setting a specific value for , the error along the specific direction is estimated. However, we aim at the construction of schemes with increased reliability range irrespective of . For this reason, a fundamental step at this point is the analbasis ysis of (11) in terms of , functions. Taking into account that the following formula is utilized:
(17) if if
(18)
The ideal case for and a specific frefor In pracquency is equivalent to tice, we always seek a small (but nonzero) value for the error estimator, which can be ensured from the vanishing of the leading terms in the corresponding expansion series, due to the quite with respect to . In the next section, “regular” behavior of it becomes clear how the above analysis is practically exploited for the construction of FDTD schemes. For completeness, we also examine the expressions of the FDTD error, if the other two Maxwell’s equations are used. For instance, if we consider (19) as the starting point, the previous procedure will produce the following error formula
(12) (20) as well as the expansion series
The expansion series of comprises only terms. In particular, it can be shown that (13)
(21)
where , and is the -th order Bessel function of the first kind. By applying (12) and (13) in (11), we have
where the expression for is derived from (15)–(18) by simply replacing with and vice versa. Finally, in the case of the third equation
(14)
(22)
where
the corresponding indicator
can be written as (23)
as long as the following formula for the time derivative of is adopted
(24) (15) (16)
It is found that the angular dependence of is expressed via functions. After some algebraic manipulations, we conclude that if the first terms of both and
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are cancelled, an equal number of terms of vanishes as well. and are necessary in practice. Therefore, only
III. OPTIMIZED HIGHER ORDER FDTD SCHEMES With the availability of the frequency dependent error indicators, some enhanced higher order FDTD algorithms can be derived. The characterization “enhanced” indicates that the new schemes exhibit smaller phase errors, compared to traditional counterparts, at specific points or parts of the frequency spectrum. Improving the performance of an FDTD method for all frequencies and without increasing the entailed computational burden is an extremely difficult task, but probably an unnecessary one, since the majority of the real-life problems exhibit a finite spectral content. This fact is not taken into account by traditional, based on Taylor series, FDTD methods, which improve their accuracy at lower frequencies. In our case, a simple of and direct way to determine the coefficients the differential operators is to request the minimization of the dispersion error at one or more frequencies. The “higher order” designation now simply implies the increased number of grid nodes taken into account, compared to Yee’s method. The expansion series of the error estimators enable their reduction regardless of the propagation angle, through the cancellation of their first terms. Theoretically, different levels of error compensation are possible by increasing the number of vanishing terms; however, the number of undetermined coefficients sets an upper limit on the correction that can be practically pursued. We subsequently demonstrate the usefulness of the described procedure by deriving three different, conditionally stable, FDTD algorithms. An expression involving the maximum time-step can be acquired from the corresponding amplification matrix, and the requirement that its eigenvalues should lie inside the unit circle on the complex plane. For the general case, the stability criterion is formulated as
Fig. 1. Error e versus mesh resolution for various FDTD methods. Scheme I is optimized at 10, 20, or 40 cells per wavelength.
e
TABLE I ERRORS VERSUS CELL SIZE FOR VARIOUS FDTD SCHEMES
general case (one, if we enforce
). For instance, in the case of
from which , and are determined (see the Appendix for more details). In order to theoretically evaluate the performance of the produced algorithm, we calculate the overall error associated with the numerical phase velocity (26)
(25) The above expression does not calculate the maximum time-step directly, due to the dependence of the differential operators on . However, (25) can be easily solved with a simple iterative procedure. It has been verified that (25) yields the known stability limits for traditional FDTD techniques (Yee’s method, Fang’s (2, 4) and (4, 4) schemes [4]). The novel FDTD approaches are constructed in a rather straightforward manner. The first scheme examined (labeled scheme I) is probably the simplest one, since it minimizes the dispersion error at one specific frequency . This can be achieved by imposing the zeroing of the first three terms of the error indicators at the selected design frequency for all directions. Since the series coefficients incorporate the unknown parameters, two 3 3 systems of equations are produced in the
and depict it in Fig. 1, for mesh resolutions ranging from 5 to 80 cells per wavelength (the same time-step is selected for all cases). Three schemes, optimized at frequencies corresponding , and cell sizes are considered. As seen, to the error is suppressed at the design frequencies, which in fact confirms the validity of the proposed estimators. Compared to traditional counterparts, this optimized single-frequency behavior unavoidably deteriorates at other frequencies. Consequently, such algorithms are suitable for certain (time-harmonic or very narrowband) types of simulations (e.g., calculation of radiation patterns). The attained single-frequency improvement can be better appreciated from Table I, where the values of are analytically presented. For comparison, the cases of Yee’s and Fang’s (4, 4) schemes are also shown. Evidently, the effective treatment of broadband simulations requires different characteristics from those of the previous method. Aiming at improving the FDTD behavior in a more wide-band sense, we propose two other approaches. Their
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Fig. 2. Error e versus mesh resolution for various FDTD methods. Scheme II is optimized at 10, 20, or 40 cells per wavelength.
common feature is that they both require the vanishing of the first two terms in the error expressions at a specific frequency . Their difference lies on the manner their wide-band characteristics are established. In one case (designated as scheme II), second-order formal accuracy . In is enforced to the spatial operators the other case (scheme III), the vanishing of the first term of the FDTD error is required at a second design frequency . Since the dispersion error at higher frequencies has a more severe impact than at lower ones, we of scheme III is selected lower than (in suggest that any case, the opposite choice will lead to another optimized FDTD approach, with improved accuracy at lower frequencies). Figs. 2 and 3 indicate that desirable wide-band features are attained with these algorithms. Both methods exhibit an error minimum at the design frequency points. Moreover, scheme II accomplishes a constantly reducing error toward lower frequencies (an effect due to second-order formal accuracy), whereas the range of improvement is specified primarily by the two design frequencies, in the case of scheme III. Interestingly, the optimized methods remain more accurate than the traditional (4, 4) scheme in the high frequency band. It can be argued that scheme II is a limit case of scheme III, obtained by selecting a very low value for . The dispersion error of scheme II at various optimization frequencies is also referenced in Table I (scheme III obtains a similar performance). Regarding the stability of the optimized algorithms, it should be mentioned that schemes II and III retain practically the same with the Yee method, irrespective of the cell’s maximum shape. On the other hand, scheme I is characterized by a slightly , which decreases as larger upper limit, in the case of the cells become more irregular (e.g., when ). IV. DISCUSSION Before proceeding to the numerical results, we would like to briefly comment on two matters. The first one is related to the fundamental differences of the presented methodology from
Fig. 3. Error e versus mesh resolution for various FDTD methods. Scheme III is optimized at (=20; =40); (=20; =60) and (=20; =80) cells per wavelength.
other relevant approaches presented in the literature. For instance, apart from the different type of operators, the developed (4, 4) techniques are characterized by an increased reliability range, compared to the modified (2, 4) schemes of [21]. Furthermore, their optimization is carried out through a totally different approach. The basic element of this paper’s methods (i.e., the construction of estimators based on Maxwell’s equations) may be considered conceptually similar to the dispersion-relation-preserving (DRP) techniques of [17] and [20]. It must be noted, though, that the error formula of [20] is based on a (2, 4) scheme and the optimized coefficients are then implemented in a frequency-dependent (and not constant) form. For a time-domain realization, this approach inevitably increases the computational stencil, since additional grid points are introduced, and the resulting FDTD technique resembles a (4, 6) one. Our approach seems to be more direct, because it does not alter the initial (4, 4) discretization scheme. In addition, we avoid using the filtering processes of [20] for controlling the wide-band behavior, as dispersion errors are now directly reduced at preselected frequencies. Other approaches, such as Tam and Webb’s DRP methods for acoustic problems [5], or most of the optimized schemes of [11], deal with the spatial operators and time integration processes separately. In these cases, spatial finite differences are usually optimized for one-dimensional propagation, without consideration for anisotropy errors in higher dimensions. By following different concepts, we have treated space and time inaccuracies simultaneously and special attention has been given to all propagation directions. Next, in the aforementioned analysis we have assumed the homogeneity of the background media. Apparently, if different materials are present in the computational domain, a set of optimized operators should be derived for the areas occupied by each material. Although the important issue of material discontinuities has not been dealt with in this paper, some approaches presented in the literature can be used in combination with the proposed schemes (such as the smoothing of material parameters [8], one-sided finite differences [13], derivative matching
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TABLE II MAXIMUM L (t) ERRORS IN THE SQUARE CAVITY PROBLEM
Fig. 4.
L (t) errors in the simulation of the parallel-plate waveguide.
[22]), so that the solution is not contaminated by improper treatment of material interfaces. Subject to an appropriate application of boundary conditions, the optimized schemes are expected to ensure improved performance over the conventional methods. V. NUMERICAL RESULTS In the following paragraphs, we apply the new FDTD techniques in 2-D simulations and compare their performance to that of traditional ones. It is noted that the implementation of the finite-difference operators in the vicinity of infinite PEC walls is facilitated by the computation of image fields, according to symmetric or antisymmetric formulas. Furthermore, in case of timeharmonic problems, the optimization of the FDTD schemes at the corresponding excitation frequency is implied. When wideband simulations are considered, the design frequencies are explicitly stated. In the first numerical test, wave propagation within a highly elongated guiding structure is examined. Specifically, a parallelplate waveguide with 5 cm width is considered. In order to excite the first TM mode, the exact field solution is used to prescribe initial values throughout the entire domain, as well as field values to the two ports of the guide at every time-step. The excitation frequency is 6 GHz, which produces a propagation constant of 108.9 rad/m. The performance of FDTD algorithms error. The strucis evaluated by computing the respective ture, which extends approximately 53.4 free-space wavelengths in the longitudinal direction, is discretized with 800 15 cells, and the total simulation time is 10 000 time steps. Fig. 4 exerrors, where the improvement hibits the evolution of the ensured by the optimized schemes is revealed. For instance, the for scheme I, and for maximum error is scheme II, which denote a superior performance compared to , and Fang’s (4, 4) methods . Yee’s In a similar framework, a square cavity with PEC walls and 1 m sides is modeled. The exact field distribution of the mode (which appears at 540.47 MHz) is assigned to all field components as initial values. The cavity is discretized with three different resolutions, using 20, 40, and 80 cells per axis. The
evolution of the field is monitored for 5000 time-steps in the case error are of the coarse mesh. The maximum values of the reported in Table II for various methods and grids. It can be seen that the enhanced quality of scheme I, which concentrates the error reduction on the resonant frequency, ensures an extremely low error value, even when compared to that of the conventional fourth-order method (for the 20 20 mesh, the error is almost times smaller). Scheme II, although developed primarily for broadband simulations, improves the performance of the (4, 4) method about 6 times. Compared to the second-order approach, the error mitigation is even more substantial. Although the computational savings of the optimized schemes are quite problem-dependent, it is instructive to demontrate their efficiency in the previously simulated configuration. For example, we estimate the increased node density required by the conventional methods, in order to reach the level of accuracy of the new techniques in the 20 20 grid. According to the formal convergence rates, Yee’s method requires a lattice that is 12.25 and 1500 times finer, in order to perform as reliably as schemes II and I, respectively. In the case of the standard (4, 4) method, we predict that the used cell size should be reduced 1.55 and 17.35 times, respectively. Since single-frequency simulations correspond to a specific class of practical problems, we proceed with some tests for the wide-band performance of the optimized schemes. Initially, the first 25 resonant frequencies of a square cavity with 1 m sides and perfectly conducting boundaries are calculated. Using pulsed excitation, the cutoff frequencies, which refer to the frequency range from 149.9 MHz to 1.08 GHz, are extracted by applying a Fourier transform to recorded field samples. The simulations are carried out within a 15 15 mesh and for time steps, yielding a resolution of 13 kHz in the frequency domain. Although the computational space is not electrically large, numerical dispersion artifacts are expected to emerge, due to the extended time-integration process. In Fig. 5, the relative error (in megahertz) is given for 27 frequency points, because different modes exhibit the same cutoff frequency in two cases, but appear as different ones in the spectrum due to numerical anisotropy. More specifically, the results concerning the conventional second-order approach, (2, 4) method, (4, 4) method, scheme II, optimized at 900 MHz, and scheme III, optimized at 900 and 500 MHz are shown. Evidently, the new higher order approaches provide the most reliable simulations. In fact, the succesful reduction of the dispersion error in the high frequency band improves the overall performance, despite the slight deterioration at low frequencies. This superiority is also supported by calculating the mean absolute error , which is indicative of the attained improvement, as we obtain: 1.16% for the Yee method, 1.218% for the (2, 4) scheme, 0.149% for the (4, 4)
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Fig. 5. Error (in megahertz) in the calculation of cutoff frequencies in the case of a square cavity.
Fig. 6. Maximum L (t) errors as a function of the simulation time, for multimode excitation in the square cavity problem.
method, 0.054% for scheme II, and 0.047% for scheme III. It is noted that since the design frequencies have been selected empirically, other choices may lead to even better results. Considering a similar configuration (a 10-cm cavity), three and ) are simultaneously exdistinct modes ( cited, by initially prescribing their exact wave patterns. The corGHz, responding cutoff frequencies are GHz and GHz. Simulations are carried out for different mesh densities (the number of time steps is determined so that the same time period is examined in all cases). error, as well as the overall computaThen, the maximum tional time, is recorded. As expected, simulations with Yee’s scheme are not as time-consuming as those employing higher order approaches, for a specified cell size. However, the higher order methods are proven more efficient (Fig. 6), by guaranteeing significantly lower error levels under the same computational effort (or, equivalently, the total simulation time). In addition, it can be seen that the optimized FDTD schemes always
Fig. 7. Absolute % errors in the calculation of the cutoff frequencies in the case of the cavity with a metallic ridge.
outperform the standard (4, 4) approach, by mitigating dispersion errors at the high frequency band (the new techniques are GHz). optimized at The following simulation involves the investigation of a cavity with a metallic ridge, depicted as an inlet sketch in Fig. 7 m, m, m, m). In this ex( ample, 20 cutoff frequencies are detected, which belong to the frequency band 138.23–953.36 MHz. The cavity is discretized with a rather coarse lattice (10 5 cells), and the simulations are performed for 65 536 time steps, yielding a frequency resolution of approximately 64.7 kHz. As reference values, we consider the result calculated with a finite-element simulation and an extremely refined mesh. Fig. 7 illustrates the absolute (percentage) errors that characterize the determination of the cutoff frequencies. The potential of the optimized approaches is evident here as well, even when compared to the standard (4, 4) scheme. The mean error values are also indicative: 3.579% for the Yee method, 1.4% for the (4, 4) method, 0.802% for scheme II, and 0.798% for scheme III. In the last numerical example, a pulse propagating in the presence of an elongated, perfectly conducting scatterer with size 4.794 0.06 m is examined. Most of the pulse’s energy is confined in the frequency band from 4 to 8 GHz. The corresponding computational domain comprises 871 102 cells, using a cell size of 6 mm and a 16-cell perfectly matched layer [23] for mesh truncation. Assuming that the origin of the coordinate system coincides with the lower left vertex of the scatterer, a -oriented m, and the magsource is inserted at m. The outnetic field is recorded at come of the (4, 4) method, when applied within a three-times finer mesh, is considered as a reference solution. The recorded fields are shown in Fig. 8, where it is clear that the secondorder scheme fails to remain accurate, due to the relatively extended domain and the rather low mesh resolution (approxiat 8 GHz). On the other hand, the higher order mately schemes accomplish more reliable calculation of the magnetic field. The new techniques, optimized according to the pulse’s
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mula of (15) (the analytical expressions are quite lengthy and , and , consider impractical). For the calculation of a system of the form (27) We also define
as (28)
(29) Fig. 8. Time evolution of the recorded magnetic field, in the problem of pulse scattering from an elongated object.
frequency content, produce the most accurate results (further demonstrated by the inlet figure, which focuses on the early time of the field evolution).
are obtained from (16), (17). If is the optiwhere and the frequencies for mization frequency for scheme I and are the correschemes II and III ( sponding wavenumbers), the system (27) for scheme I is determined by
VI. CONCLUSION The examination of computationally demanding, contemporary electromagnetic applications calls for highly accurate solvers of Maxwell’s equations, with better performance than traditional approaches. In this paper we have discussed the development of FDTD methods that are suitable for this type of simulations. The proposed schemes are as computationally expensive as Fang’s (4, 4) method. However, in contrast with the Taylor-based derivation of conventional techniques, they can be more efficient as they allow satisfactory control of their reliability range, depending on the simulation’s requirements. To achieve this, a suitable estimator for the inherent disperion error has been proposed and analyzed in harmonic functions of the propagation angle. Different FDTD schemes with adjustable wide-band behavior have been derived, through the minimization of the dispersion error in one or two preselected frequencies. A number of numerical experiments have verified the validity of the error formulas and, consequently, the success of the corresponding FDTD approaches. Although derived for homogeneous media, the proposed methods can be combined with other specialized techniques, proposed in the literature, for the treatment of material boundaries. Regarding a potential extension of the optimization procedure to three-dimensional cases, proper estimators may be defined in a similar manner. To enable the calculation of the unknown coefficients, the expansion of the error in surface harmonic functions [20] appears to be a suitable choice, as will be analyzed further in a future paper. APPENDIX As mentioned above, the unknown coefficients can be determined by solving simple equation systems, based on the for-
The corresponding values for scheme II are
Finally, for scheme III we have
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[5] C. K. W. Tam and J. C. Webb, “Dispersion-relation-preserving finite difference schemes for computational acoustics,” J. Comput. Phys., vol. 107, pp. 262–281, 1993. [6] 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 Propag., vol. 45, no. 2, pp. 254–264, Feb. 1997. [7] J. L. Young, D. Gaitonde, and J. J. S. Shang, “Toward the construction of a fourth-order difference scheme for transient EM wave simulation: Staggered grid approach,” IEEE Trans. Antennas Propag., vol. 45, no. 11, pp. 1573–1580, Nov. 1997. [8] E. Turkel, “High-order methods,” in Advances in Computational Electromagnetics: The Finite-Difference Time-Domain Method, A. Taflove, Ed. Norwood, MA: Artech House, 1998, ch. 2, pp. 63–110. [9] N. V. Kantartzis and T. D. Tsiboukis, “A higher-order FDTD technique for the implementation of enhanced dispersionless perfectly matched layers combined with efficient absorbing boundary conditions,” IEEE Trans. Magn., vol. 34, no. 5, pp. 2736–2739, Sep. 1998. [10] K. Lan, Y. Liu, and W. Lin, “A higher order (2, 4) scheme for reducing dispersion in FDTD algorithms,” IEEE Trans. Electromagn. Compat., vol. 41, pp. 160–165, May 1999. [11] D. W. Zingg, “A review of high-order and optimized finite-difference methods for simulating linear wave phenomena,”, Technical Rep. 97-2088, June 1997. AIAA. [12] E. Turkel and A. Yefet, “On the construction of a high order difference scheme for complex domains in a Cartesian grid,” Appl. Numer. Math., vol. 33, pp. 113–124, 2000. [13] A. Yefet and P. G. Petropoulos, “A staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell’s equations,” J. Comput. Phys., vol. 168, pp. 286–315, 2001. [14] T. Hirono, W. Lui, S. Seki, and Y. Yoshikuni, “A three-dimensional fourth-order finite-difference time-domain scheme using a symplectic integrator propagator,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 9, pp. 1640–1648, Sep. 2001. [15] Z. Xie, C.-H. Chan, and B. Zhang, “An explicit fourth-order orthogonal curvilinear staggered-grid FDTD method for Maxwell’s equations,” J. Comput. Phys., vol. 175, pp. 739–763, 2002. [16] S. V. Georgakopoulos, C. R. Birtcher, C. A. Balanis, and R. A. Renaut, “Higher-order finite-difference schemes for electromagnetic radiation, scattering, and penetration: Part I—Theory,” IEEE Antennas Propagat. Mag., vol. 44, pp. 134–142, Feb. 2002. [17] S. Wang, F. L. Teixeira, R. Lee, and J.-F. Lee, “Dispersion-relationpreserving (DRP) 2-D finite-difference time-domain schemes,” in Proc. IEEE Antennas Propagat. Soc. Int. Symp., vol. 3, San Antonio, TX, 2002, pp. 264–267. [18] H. Spachmann, R. Schuhmann, and T. Weiland, “Higher order time integration schemes for Maxwell’s equations,” Int. J. Numerical Modeling: Electron. Networks, Devices and Fields, vol. 15, pp. 419–437, 2002. [19] Z. Shao, Z. Shen, Q. He, and G. Wei, “A generalized higher order finite-difference time-domain method and its application to guided-wave problems,” IEEE Microw. Theory Tech., vol. 51, no. 3, pp. 856–861, Mar. 2003.
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Theodoros T. Zygiridis was born in Thessaloniki, Greece, in 1978. He received the Diploma degree in electrical and computer engineering from Aristotle University of Thessaloniki (AUTH), Thessaloniki, Greece, in 2000, and is currently working toward the Ph.D. degree at the AUTH. His research interests include computational electromagnetics, with emphasis on the development and implementation of higher order FDTD and finite element algorithms.
Theodoros D. Tsiboukis (S’79–M’81–SM’99) received the Diploma degree in electrical and mechanical engineering from the National Technical University of Athens, Athens, Greece, in 1971, and the Dr.Eng. degree from the Aristotle University of Thessaloniki (AUTH), Thessaloniki, Greece, in 1981. From 1981 to 1982, he was with the Electrical Engineering Department, University of Southampton, U.K., as a Senior Research Fellow. Since 1982, he has been with the Department of Electrical and Computer Engineering (DECE), AUTH, where he is currently a Professor. He has served in many administrative positions, including Director of the Division of Telecommunications at the DECE (1993–1997) and Chairman of the DECE (1997–2001). He is also the Head of the Applied and Computational Electromagnetics Laboratory at the DECE. His main research interests include electromagnetic field analysis by energy methods, computational electromagnetics (FEM, BEM, vector finite elements, MoM, FDTD method, absorbing boundary conditions), inverse and EMC problems. He has authored or coauthored six books, over 100 refereed journal papers and over 90 international conference papers. He was the Guest Editor of a Special Issue of the International Journal of Theoretical Electrotechnics (1996). Dr. Tsiboukis is a Member of various societies, associations, chambers and institutions. He has been the recipient of several awards and distinctions. He was the Chairman of the local organizing committee of the 8th International Symposium on Theoretical Electrical Engineering (1995).
