Optimized Finite-Difference Time-Domain Methods Based on the (2;4 ...

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 3, MARCH 2005

Optimized Finite-Difference Time-Domain Methods Based on the (2; 4) Stencil Guilin Sun, Student Member, IEEE, and Christopher W. Trueman, Senior Member, IEEE

Abstract—The higher order (2 4) scheme optimized in terms of Taylor series in the finite-difference time-domain method is often used to reduce numerical dispersion and anisotropy. This paper investigates optimization of the numerical dispersion behavior for a square Yee mesh based on the (2 4) computational stencil. It is shown that, for one designated frequency, numerical dispersion can be eliminated for some directions of travel, such as the coordinate axes or the diagonals, or numerical anisotropy can be eliminated entirely, resulting in a constant “residual” numerical dispersion. Using a coefficient-modification technique, the residual numerical dispersion can then be completely eliminated at that frequency, or for a wide-band signal, the numerical dispersion error and the averaged-accumulated phase error can be minimized. The stability of the method is analyzed, the numerical dispersion relation is given and validated using numerical experiments, and the relative rms errors are compared to the standard (2 4) scheme for the proposed methods. The optimized methods are second-order accurate in space and have higher accuracy than the standard (2 4) scheme. It has been found that the dispersion error of the (2 4) scheme is like that of a second-order accurate method, though it behaves like a fourth-order accurate method in terms of anisotropy. Index Terms—Computational electromagnetics, finite-difference time-domain (FDTD) method, higher order method, numerical anisotropy, numerical dispersion.

I. INTRODUCTION

T

HE finite-difference time-domain (FDTD) method originally proposed by Yee [1] has been widely used to solve Maxwell’s equations in numerous electromagnetic applications [2]. However, the numerical dispersion and anisotropy inherent in Yee’s FDTD method prohibits its application to electrically large objects because of the phase-error accumulation. In addition, the anisotropic velocity makes compensation for the phase error in the near-to-far transformation complex [3]. To overcome this limitation, numerous methods have been proposed. Cole [4], [5] described a high-accuracy algorithm with a nonstandard finite-difference method and adjacent nodes. Forgy and Chew [6] presented a nearly isotropic method on an overlapped lattice. Nehrbass et al. [7] demonstrated a reduced dispersion method without increasing the order. Juntunen and Tsiboukis [8] presented a method using artificial anisotropy. Rewienski and Mrozowski [9] introduced an iterative algorithm in cylindrical coordinates. Wang et al. Manuscript received March 8, 2004; revised May 25, 2004. The authors are with the Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, Canada H4B 1R6 (e-mail: Trueman@ ece.concordia.ca; [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842507

[10] suggested a parameter-optimized method for the alternating-direction-implicit finite-difference time-domain (ADI FDTD) method. Taflove and Hagness [2] analyzed several scheme methods, including the standard higher order [11]. Wang and Teixeira [12]–[15] proposed several methods such as angle-optimized and dispersion-relation-preserving schemes using complicated filters. Zygiridis and Tsiboukis [16] presented a method based on Taylor-series analysis of the numerical dispersion relation. Xie et al. [17] developed a scheme with numerical dissipation. Shao et al. [18] proposed a generalized higher order method. Zingg [19] compared some high-accuracy methods. Shlager and Schneider [20] compared the performance of several low-dispersion methods. Other methods may be found in the references of the above-mentioned papers. For a square mesh, Yee’s FDTD method can have no numerical dispersion along the diagonals and at the Courant limit time step size [2]. The standard higher order method at small Courant numbers can have no numerical dispersion along the diagonals or along the axes. The fact that a higher order method can reduce the numerical error in the FDTD implies that the larger computational stencils used in higher order methods can cancel some of the errors in Yee’s second-order formulation. With the optimization presented in this paper, better results than the scheme are obtained. The proposed methods are more flexible: by choosing special optimal parameters at certain mesh densities and time-step sizes, the numerical dispersion can be eliminated along a pre-assigned direction of travel such as along the axes or along the diagonals of a square mesh, or the anisotropy can be eliminated altogether at one frequency, with “residual” dispersion error, which can then be removed completely with a simple coefficient-modification (CM) technique, or, for a broadband signal, the numerical dispersion error and averaged-accumulated phase error can be minimized. Another feature of the proposed method is that it can be incorporated into the current algorithm directly including the ADI FDTD method [21] without additional computational cost. Since the proposed methods are second-order accurate in space, the material interface and boundary conditions can be easily treated using existing second-order methods. In contrast, the scheme must use higher order methods to treat the material interfaces in order to maintain the overall accuracy. Notice that [11]–[16] provide methods to reduce the numerical dispersion based on the same stencil with approximate dispersion relation analysis and using coefficient modifi-

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cation. This paper treats the weight parameter and the CM parameter separately. Methods presented in this paper are given that eliminate or minimize dispersion, or eliminate anisotropy, using the exact numerical dispersion relation. This paper is organized as follows. Section II gives the formulation, amplification factor, and numerical dispersion relation for all the proposed methods in terms of a weight parameter. Sections III–V describe how to obtain the optimal values of the weight parameter in various senses. Section VI discusses the stability, time-step size limit, accuracy, and the CM technique to reduce the residual numerical dispersion. Section VII proposes methods to minimize the dispersion error over a broad bandwidth. (2) II. WEIGHTED TWO-DIMENSIONAL (2-D) FDTD FORMULATION wave in a linear isotropic nondisFor simplicity, a 2-D persive medium is assumed. Based on the Taylor-series analysis [2], [11], the first-order spatial derivatives in the scheme are approximated by the use of the conventional Yee’s elements plus “one-cell-away” elements. Both elements use a second-order finite-difference formula to eliminate third-order and higher odd-order terms. The fourth-order accuracy is obtained by the cancellation of the two second-order error terms from the Yee’s elements and the “one-cell-away” elements. In this paper, we do not pursue fourth-order accuracy based on the Taylor series. Instead, the methods are based on the optimization of the numerical dispersion error. To achieve this, is introduced to optimize the relative a weight parameter contributions of Yee’s elements and the “one-cell-away” elements. The new update equations are given by (1)–(3) as follows:

(3) where , , is the time step size, and are the permittivity and permeability of the material, reand are the spatial meshing sizes, and spectively, are the indices of the computational cells, and is the index of time step. The weight parameter is to be determined. If , (1)–(3) becomes the conventional Yee’s method [1], , the above formulation is exactly the same as [2]. If the scheme [2], [11]. In Sections III-V, the weight parameter will be chosen to optimize the dispersion behavior in various ways. With the Fourier analysis method [22], the amplification factor for this formulation can be obtained as

(4)

where (1)

,

, ,

, and

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The phase constant of the numerical wave is . The angle is the direction of travel with respect to the -axis, and and . The amplification factor can be expressed in terms of the and the angular frequency as . time-step size Thus, with some manipulation, the numerical dispersion equation can be obtained as

Fig. 1. Optimal value of weight parameter as a function of the mesh density for various methods.

(5)

Since the numerical dispersion depends on the weight parameter, the dispersion behavior can be optimized in various ways. III. AXES-OPTIMIZED METHOD (AOM) The AOM has no numerical dispersion along the axes for any time-step size within the stability limit at a designated frequency. To eliminate the numerical dispersion error along the ), use (5) to examine the nuaxes for a square mesh ( merical dispersion along the -axis to obtain Fig. 2. Numerical dispersion of the AOM optimized at 10 CPW and the (2; 4) method.

(6)

where is the numerical phase constant along the axes. Solving (6) obtains the “axes-optimized” value of the weight as parameter

(7)

To have a solution of (7), set equal to the theoretical phase at the designated frequency. Since the mesh constant size can be expressed in terms of mesh density as , is a function of mesh density (7) shows that the optimal value , the signal frequency, and time-step size. Fig. 1 shows the optimal values of the weight parameter as a function of mesh density for four Courant numbers , , , , where is the Courant limit in the 2-D and case. As the Courant number increases, i.e., as the time-step size at a fixed mesh density increases, the optimal value of the weight parameter decreases. The optimal value also decreases monotonically as the mesh density increases. Fig. 2 graphs the numerical dispersion (quantified as the relwhere is the numerical velocity) versus the ative velocity direction of travel , at a mesh density of 10 cells per wave, , , length (CPW) for Courant number . For comparison, the numerical dispersion of the and scheme is also shown. For the AOM, Fig. 2 shows that, indeed, there is no numerical dispersion along the axes. Both

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Fig. 3. PNDE for the AOM along 0 and 10 , the DOM along 45 and 55 , optimized at 10 CPW, and the minimum dispersion error of the (2; 4) scheme along the axes.

the AOM and methods have velocity larger than the physfor the scheme. Both ical speed, except at methods have larger velocity along the diagonals than along the axes, similar to Yee’s method. Fig. 2 shows the results from numerical experiments as small circles, with good agreement with the theory. All the numerical experiments in this paper are performed in a 2000 2000 . The excitation cell space with mesh density source is located at the center. As time advances, the field values are recorded every five degrees on two quarter-circles: an inner circle located at 50 wavelengths away; and an outer circle located 60 wavelengths away from the source. The numerical velocity is then calculated from the time delay for propagation between the two circles. To compare accuracy, the numerical dispersion error can be defined as . Fig. 3 shows the percentage numerical disfrom mesh persion error (PNDE) along the axis and for densities from 8 to 45 CPW for the AOM at the time step sizes and . The AOM method is optimized of at 10 CPW, where the axial dispersion error is zero. This figure also shows the minimum dispersion error along the axis of the method. It can be seen that the numerical dispersion error for the AOM in the sector of 10 is much smaller than that of the scheme. For example, at , the maximum error for the AOM at approximately 14 CPW is 0.016% along , whereas the minimum disthe axis, and 0.0218% at method is persion error at the same mesh density for the much larger at 0.8624%. Note that the AOM is intended for use in problems where most waves travel within a sector around the axis, such as the laser cavity and other examples indicated in [14]. For problems where the prominent direction of wave is along the diagonal, the method in Section IV can be used. IV. DIAGONALLY OPTIMIZED METHOD (DOM) A DOM has no numerical dispersion error along for a square mesh. To derive a formula for calculating the optimal

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Fig. 4. Numerical dispersion of the DOM optimized at 10 CPW and the (2; 4) method.

weight parameter, use (5) to examine the numerical dispersion along 45 to obtain

(8)

Set , and solve (8) for the optimal value of the to obtain weight parameter

(9)

The optimal value as a function of the mesh density is graphed in Fig. 1 and has similar behavior to that of the AOM method. Fig. 4 graphs the numerical dispersion at four different Courant numbers for the DOM. It can be seen that, as expected, there is no numerical dispersion along the diagonal. Different from the scheme, shown in Fig. 4, all the velocities AOM and the in the DOM method are below or equal to the physical speed. The small circles are the results of numerical experiments and agree well with the theory. Fig. 3 graphs the PNDE for the DOM and at the time step sizes of along the diagonal and optimized at 10 CPW. It can be seen that the PNDE is much smaller than that of , and smaller , the maxthan that of the AOM. For example, at imum dispersion error for the DOM at approximately 14 CPW , is 0.00336% along the diagonal, and 0.00937% at

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which is approximately 25.7 and 9.2 times smaller than that of scheme, respectively, the minimum dispersion error of the at the same mesh density. Note that the AOM and DOM are special cases which eliminate numerical dispersion in a designated direction of travel. The formulation of (1)–(3) allows the elimination of the nu. The merical dispersion at any specific angle of interest weight parameter can be found from

Fig. 5. Numerical dispersion for the IOM optimized at 10 CPW.

(10) (11a) (11b) The solution is straightforward, but messy and will not be given here. V. ISOTROPIC OPTIMIZATION METHOD It has been shown that the weight parameter can be chosen to make the numerical phase constant exactly equal to the theoretical value in one direction of travel. The value of the weight parameter depends on the mesh density and time step size, as shown in Fig. 1. If the AOM and DOM are combined together, the numerical velocity can be made independent of the direction of travel. After eliminating the time-step size term, (6) and (8) can be solved to obtain the “isotropic” optimal value of the weight parameter, given by (12), shown at the bottom of this page. This optimal parameter is only a function of mesh density, and is independent of both the time step size and the signal frequency. The numerical dispersion using is graphed in Fig. 5 with solid lines for a mesh density of 10, at four different Courant numbers. The “curves” are, in fact, horizontal lines, demonstrating that the numerical velocity is indeed independent of the direction of travel. There is no anisotropy, as expected, but there is numerical dispersion. This method is termed

the isotropically optimized method (IOM). The IOM obtains a circular numerical wavefront since the numerical velocity is uniform in all directions. The residual numerical dispersion in the IOM can be corrected with the technique that will be briefly discussed in Section VI-G. This isotropic method makes the phaseerror compensation easier for the near-to-far transformation. VI. DISCUSSION A. Optimal Parameter The above analysis shows that, by suitable choice of the weight parameter in (1)–(3), numerical dispersion and anisotropy can be made to exhibit different properties: zero velocity error along a designated direction of travel, such as the coordinate axes (AOM) or along the diagonals (DOM) or zero anisotropy (IOM) for a square mesh. The update (1)–(3) for the AOM, DOM, and IOM are the same, except for the value of the weight parameter . Fig. 6 shows that the numerical velocity along the diagonals (solid curves) and along the axes (dashed curves) increases monotonically with the value of the weight parameter, for mesh density of 10 CPW at four different Courant numbers. Where each curve intersects with the horizontal line for relative velocity 1.000, there is no numerical dispersion. Where two of the curves for the same time-step size intersect one another, there is no anisotropy for that value of the weight parameter. These “special” optimal values correspond to the values given in (7), (9), and (12). Other optimal values of the weight parameter can be found in different ways, which will be shown in Section VII.

(12)

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Fig. 6. Relative numerical velocity along the axes and along the diagonals as a function of the weight parameter.

Fig. 7. Anisotropy at different optimized wavelength with the IOM and the (2; 4) method.

B. Stability and the Time-Step Size Limit

the anisotropy is less than a pre-set maximum. The mesh density in Fig. 7 can be transformed into wavelength by

Numerical stability relies on the magnitude of the amplification factor in (4) [22]. Equation (4) implicitly assumes that the following inequality holds: (13) Following the analysis for the Yee’s method [2], set the sine term equal to unity. The time step size is then bounded by (14a) where (14b)

is the same Courant limit as the Yee’s method. For where scheme, and the time-step size limit is 6/7 the of the Courant limit, which is the expected result. Since the optimal value of weight parameter of the IOM at mesh density 10 is 1.133733, its corresponding time-step size limit is 0.84867 times of the Courant limit, which is a little smaller than 6/7. is used in this This is why the Courant number paper. In the limit of zero mesh size, the optimal value of is 9/8, thus, the time-step size limit is the same as the standard scheme.

s

= 0:848s for

(15) Hence, the bandwidth can be estimated. For optimization at cells per wavelength, the anisotropy increases sharply as increases above , reaches the maximum , and then decreases. If the of 1.4 10 at upper bound on the anisotropy is, e.g., 1.4 10 , then the . The shortest mesh density must be larger than wavelength for which the anisotropy is less than 1.4 10 is . For comparison, Yee’s FDTD requires and scheme requires for the same anisotropy. Note that the anisotropy for Yee’s is nearly independent of the time-step size, hence, it is about the same at . For optimization of the IOM the Courant limit and , the anisotropy is very large at coarse mesh at sizes. For anisotropy less than 1.4 10 , we must choose ; hence, the shortest wavelength is . , the anisotropy is less than Note also that, for 9.8 10 , much smaller than the chosen bound, whereas the scheme requires for anisotropy 1.4 10 . increases, the residual numerical dispersion deNote that as has the largest discreases. Optimization at , the dispersion is about persion error and, at scheme. In all the cases, the the same as the standard dispersion error decreases monotonically with increasing mesh density.

C. Anisotropy for Other Frequencies The IOM is designed for zero anisotropy at one frequency of operation. Fig. 7 shows the anisotropy as a function of mesh density for three different optimizations: optimized for zero , , and , respectively, anisotropy at with fixed Courant number , which results in the largest numerical dispersion. Suppose at the frequency of and the mesh density . optimization, the wavelength is At higher and lower frequencies, there will be some numerical anisotropy, and it is useful to know the bandwidth over which

D. Relative RMS Error Comparison For the three methods presented in this paper, their numerical dispersion errors are quite different because the numerical velocity can be either larger or smaller than the theoretical velocity. Thus, it is difficult to compare them. The rms truncation (or discretization) error of (1)–(3) based on the Fourier method [23] is a useful way to compare methods, and provides more information than the numerical dispersion error. Fig. 8 gives the maximum relative rms truncation errors for the AOM, DOM,

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Fig. 8. Largest rms errors for the AOM, DOM, IOM, (2; 4), and Yee methods. TABLE I LARGEST RELATIVE rms ERRORS FOR DIFFERENT METHODS IN PERCENTAGE

accuracy increases are calculated to be 4 for Yee, 2.83 and 3.91 scheme, respectively, and 4 for the optimized for the methods. These results are quite close to those in Table II. The larger increase in Table II for the optimized methods may imply that higher order spatial terms have more effects in canceling errors at small Courant numbers. Further computations show scheme, the anisotropy error obeys the that, for the fourth-order accuracy, but the numerical dispersion error is second order. For example, from Fig. 4, the anisotropy error scheme, decreases 15.5 (approximately 2 ) times for the and 4.13 (approximately 2 ) times for Yee’s FDTD method, AOM, and DOM when the mesh density increases from 10 to . On the other 20 CPW at the Courant number hand, the numerical dispersion error decreases 3.92 times for scheme, 4.13 times for the AOM and DOM, and 4.21 the for the IOM from Figs. 2, 4, and 5. Thus, the scheme functions as a second-order method in terms of dispersion error. Note that the numerical dispersion error is defined as the difference between unity and the relative numerical velocity. F. Three-Dimensional (3-D) Case

TABLE II ACCURACY INCREASE FROM MESH DENSITY 10–20 CPW FROM TABLE I

IOM, Yee’s FDTD method, and methods at Courant numand . The maximum error occurs along the bers diagonals for the AOM, and occurs along the axes for DOM. Their smallest dispersion errors along their designated direction were given previously. As the time-step size increases, Yee’s FDTD method has a smaller error and the other methods have larger errors. Table I gives some typical values for the largest percentage rms error, and will be used to demonstrate the properties of different methods in the following. E. Accuracy By Taylor-series analysis, the optimized methods in this paper are second-order accurate. Due to the partial cancellation of the higher order terms, the truncation error is compensated to a certain extent. By extracting information from Table I, the accuracy increase at the mesh density of 20 CPW compared to that at 10 CPW is listed in Table II. scheme has the lowest increase It seems strange that the in accuracy in Table II. This is because the truncation error is a function of both the time-step size and the spatial increment. Analysis based on the Taylor series with the fundamental plane-wave solution of Maxwell’s equations shows that the for Yee’s FDTD accuracy is on the order of method, and for the scheme, where is from the fourth-order term of the Taylor-series expansion, and for the optimized methods. Thus, for the same Courant number, from mesh density 10–20 and at , according to the above analysis, the at

The optimized methods are readily extended to 3-D. The update equations for the 3-D case are similar to (1)–(3), but will not be given here for brevity. The amplification factor is given by (16) as

(16) which has unity magnitude within the time-step size limit given term. The numerical in (14a). Equation (14b) needs a similar dispersion relation is

(17)

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Based on (17), the optimal values of the weight parameter for the 3-D AOM and DOM cases are similar to the 2-D case. For the IOM, the optimal value is given by (18), shown at the bottom , the optimal weight parameter of this page. If is 1.1327, thus, it has a time-step size limit of 0.84965 times the 3-D Courant limit. In contrast, the filter scheme [13] has a limit . 0.765 times the Courant limit with

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For example, to eliminate the axial numerical dispersion, the parameters are chosen to be

G. CM Techniques Many techniques mentioned in Section I modify the coeffiand in the update equations in order to reduce cients of the numerical dispersion, thus, they are refereed as CM techand , and niques. In general, can be replaced by can be replaced by and , respectively, where the parameters , , , and can be modified differently. , , and ; [10] [8] uses , , and ; [4] and [5] uses use and in conjunction with adjacent cells. Reference [16] uses in addition to modifying the spatial difference. References [6], [7], [12]–[14] also provide methods to modify these parameters. The parameter values are obtained according to different criteria such as low numerical dispersion. These methods all change the “speed”: in ; the standard FDTD method, the physical speed is in the CM techniques, the new speed is for a , , , , square mesh ( ). Note that, from (14b), it can be seen that such modifications change the time-step size limit, which has been pointed out in times the original [2] and [20]. The new stability limit is limit before modification. For the IOM, from Fig. 5, the numerical velocity is larger than the speed of light. Therefore, the coefficient modification will increase the time-step size limit. If the problem contains only one material, these techniques are the same. However, if the problem uses different materials, then unless all the materials are modified by the same factor, the reflection coefficient at a material boundary changes and there is artificial reflection. Thus, this paper uses coefficient modification for all the update equations. In addition to the methods mentioned above, we have derived several formulas to optimize the values of these “ ”-parameters in different senses. A simple, straightforward way to find the values of -parameters is to reduce or eliminate the numerical dispersion at one mesh density along one direction of travel.

(19a)

(19b) Note that is the theoretical phase constant at a designated frequency to have zero numerical dispersion, and is not necessarily to be the same as , where zero anisotropy is desired. When they are the same, the combined IOM and the simple CM technique can have zero anisotropy and zero numerical dispersion at one designated frequency. Such a CM technique does not change the anisotropy. Note that different methods to put the -parameters into the update equations are equivalent in terms of numerical dispersion, but have slightly different rms errors. In practical applications, it is the accumulated phase error, not the numerical dispersion error, that affects the result. Thus, evaluation of the accumulated phase error is important. It is defined as (20) in degrees, where is the distance of wave travel, which is in this paper, is the wavelength that changes with the signal frequency. Anisotropy makes the accumulated phase error in (20) dependent on the direction of travel. Define the averagedaccumulated phase error for all the directions of travel as (21) where ( ) is the numerical velocity along . Fig. 9 shows the averaged-accumulated phase error for the scheme and IOM at time-step size , and for the IOM with

(18)

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Fig. 9. Accumulated phase errors of the (2; 4) and IOM methods, the IOM method with coefficient modification (IOM-CM), and the HAM FW method with coefficient modification, optimized at 11.5 CPW.

coefficient modification (IOM CM) using (19), optimized at 10 and 11.5 CPW, respectively. The phase error for at its step size limit is larger, which is not given. It can be seen that both the have large phase errors at coarse mesh density IOM and (high frequency), and the error monotonically decreases as the mesh density becomes large. With coefficient modification for the IOM, the phase error is zero at the optimized frequency, but increases to a maximum at a somewhat lower frequency, and then decreases at much lower frequencies. This is a phenomenon common to all the CM techniques. VII. HIGH-ACCURACY METHODS WITH LARGER TIME-STEP SIZES In previous sections, several methods have been proposed based on the optimization of the numerical dispersion in different senses, and the formulas for the optimal values of the weight parameter are given. Other “optimal” values for the scheme is weight parameter can also be found. The obtained from the Taylor series with a fixed value of the weight parameter for whatever the mesh density is based on the order of accuracy analysis. Next, we will find a similar constant value of the weight parameter that is “optimal” in terms of numerical dispersion error. , AOM, DOM, and IOM methods have larger disThe persion error as the time-step size increases, which is contrary to the Yee’s method. In addition, their numerical phase velocity is generally larger than the physical speed. Since these methods are based on the weighted contributions of Yee’s elements and the “one-cell-away” elements, the larger numerical velocity indicates that the methods are “over-weighted.” When the weight parameter becomes smaller, the numerical velocity becomes smaller, and approaches to the physical speed, as can be seen from Figs. 1–5. Expanding the sine terms in (12) with the Taylor’ series up to the third order, a constant value of the weight parameter is . Using (14a), the time-step size limit is obtained as scheme. 39/41 (0.9512195), which is larger than 6/7 of the The averaged-accumulated phase errors at the time step sizes

Fig. 10. Numerical dispersion of the HAM-FW method, HAM-S method, and the (2; 4) scheme at different time step sizes.

and are shown in Fig. 9 using a CM method. It can be seen that the phase error of this method is and IOM methods at coarse much smaller than that of the mesh sizes. This method is termed the “high-accuracy method with fixed-weight parameter” (HAM FW). The numerical dispersion is shown in Fig. 10, where the results from the numerical experiments are in small circles, with very good agreement to the theory. The numerical dispersion error is smaller than that and Yee’s methods. For example, for a mesh denof the sity of 10 CPW, at their corresponding time-step size limits, the largest dispersion error is 0.81%, 0.59%, and 0.43% for the Yee, , and HAM FW methods, respectively, without coefficient modification. As the time-step size decreases, the error increases for the HAM FW method, which is similar to the Yee’s FDTD method. The optimal value of the weight parameter can also be found by a search algorithm once the optimization criterion to is set. For a range of weight parameter from with a increment of 0.0001, and for Courant numto with a increment of 0.0001, a bers from where simple search algorithm was used to find values of is less the error than 10 . The search finds pairs, which meet the error criterion. However, the time-step size should not be larger than the upper bound given from (14a) . This eliminates many pairs, as leaving useful pairs with in the range from 1.0546 to 1.0743, which correspond to a time-step size limit larger than 0.9 times the Courant limit of Yee’s FDTD method. For the , , which pair is greater than , thus, the method is stable. For the pair , , thus, this method is also stable. With the increase of the weight parameter’s value, the limit of the time-step size decreases. Note that, at the time-step size limit, the dispersion error is not the least with the values of the weight parameter found by search. This method is termed the high-accuracy method by searching (HAM-S). and , the dispersion from Using theory and numerical experiment is graphed in Fig. 10 for

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HAM-S. It can be seen that the absolute numerical dispersion error is 0.247% along the axis, and 0.246% along the diagonal. The averaged-accumulated phase error is 0.00886 at 10 CPW, with a minimum of 0.0001 at 10.7 CPW, and a maximum of 0.0091 at 13.8 CPW, too small to be shown in Fig. 9. and was also verified The case of with numerical experiments. The maximum dispersion error is approximately 0.184%. Since the two cases have very close dispersion, the result is not shown in Fig. 10. However, caution must be taken when using the searching method. The time-step size limit for the IOM can be found from (14a) directly since the value of the optimal weight parameter (12) is independent of the time-step size. However, in the searching method, the value of the optimal weight parameter depends on the time-step size. One must make sure that the time-step size used in practice is smaller than the bound given by (14a) according to the value of the weight parameter. The HAM method, particularly the HAM-S method, is a broad-band more efficient method, with a time-step size larger method. The IOM method is relatively than that of the narrow band, but can have no numerical error at one designated or obfrequency. Note that using (10) at , and eliminates error along the corresponding tains angles, which is close to the search method. The search method is more general. VIII. CONCLUSION Starting with the computational stencil, a weighted FDTD method was formulated, which can be optimized by choosing a proper value of weight parameter based on the exact numerical dispersion relation, which includes all higher order terms. Methods to find the specific values of the weight parameter were given to eliminate or minimize numerical errors in various senses. The CM technique was used to reduce the numerical dispersion error further. The methods presented in this paper can have better accuracy than the standard scheme even though they are second-order accurate in space. In particular, the HAM-S has much higher accuracy than the scheme. Numerical experiments were used in this paper to validate the numerical dispersion predicted by theory in 2-D and good agreement was found. The optimized FDTD methods are simple, and easy to be incorporated into existing scheme. No additional comFDTD codes employing the putational effort is involved because the value of the weight parameter is computed before the FDTD algorithm is run. scheme is It is shown that, in terms of anisotropy, the fourth-order accurate; however, in terms of dispersion error, it is only second-order accurate. This paper has shown that, with the same computational stencil, the numerical dispersion, anisotropy, and the averaged accumulated phase errors are quite different with different values of weight parameter, coefficient modification, and timestep size. The optimization can be in different senses. A method can be designed having suitable optimization for a specific practical problem. For a nonsquare mesh, one can first partially use the CM technique to lift up the velocity along the axis with the coarse cell size, and then use the methods in this paper.

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REFERENCES [1] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. AP-14, no. 5, pp. 302–307, May 1966. [2] A. Taflove and S. C. Hagness, Computational Electrodynamics—The Finite-Difference Time-Domain Method, 2nd ed. Boston, MA: Artech House, 2000. [3] T. Martin, “An improved near-to-far-zone transformation for the finitedifference time-domain method,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 9, pp. 1263–1271, Sep. 1998. [4] J. B. Cole, “A high accuracy FDTD algorithm to solve microwave propagation and scattering problems on a coarse grid,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 9, pp. 2053–2058, Sep. 1995. [5] , “High-accuracy Yee algorithm based on nonstandard finite differences: New development and validations,” IEEE Trans. Antennas Propag., vol. 50, no. 9, pp. 1185–1191, Sep. 2002. [6] E. A. Forgy and W. C. Chew, “A time-domain method with isotropic dispersion and increased stability on an overlapped lattice,” IEEE Trans. Antennas Propag., vol. 50, no. 7, pp. 983–996, Jul. 2002. [7] J. W. Nehrbass, J. O. Jevtic, and R. Lee, “Reducing the phase error for finite-difference methods without increasing the order,” IEEE Trans. Antennas Propag., vol. 46, no. 8, pp. 1194–1201, Aug. 1998. [8] J. S. Juntunen and T. D. Tsiboukis, “Reduction of numerical dispersion in FDTD method through artificial anisotropy,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 4, pp. 582–588, Apr. 2000. [9] M. Rewienski and M. Mrozowski, “An iterative algorithm for reducing dispersion error on Yee’s mesh in cylindrical coordinates,” IEEE Microw. Guided Wave Lett., vol. 10, no. 9, pp. 353–355, Sep. 2000. [10] M. Wang, Z. Wang, and J. Chen, “A parameter optimized ADI-FDTD method,” IEEE Antennas Wireless Propag. Lett., vol. 2, no. 9, pp. 118–121, Sep. 2003. [11] K. Lan, Y. Liu, and W. Lin, “A higher order (2; 4) scheme for reducing dispersion in FDTD algorithm,” IEEE Trans. Electromagn. Compat., vol. 41, no. 2, pp. 160–165, May 1999. [12] S. Wang and F. L. Teixeira, “A three-dimensional angle-optimized finitedifference time-domain algorithm,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 811–817, Mar. 2003. [13] , “Dispersion-relation-preserving FDTD algorithms for large-scale three-dimensional problems,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 8, pp. 1818–1828, Aug. 2003. [14] , “A finite-difference time-domain algorithm optimized for arbitrary propagation angles,” IEEE Trans. Antennas Propag., vol. 51, no. 9, Sep. 2003. [15] , “Grid-dispersion error reduction for broadband FDTD electromagnetic simulations,” IEEE Trans. Magn., vol. 40, no. 2, pp. 1440–1443, Mar. 2004. [16] T. T. Zygiridis and T. D. Tsiboukis, “Low dispersion algorithms based on the higher order (2; 4) FDTD method,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1321–1327, Apr. 2004. [17] Z. Xie, C.-H. Chan, and B. Zhang, “An explicit fourth-order staged finite-difference time-domain method for Maxwell’s equations,” J. Comput. Appl. Math., vol. 147, pp. 75–98, 2002. [18] Z. Shao, Z. Shen, Q. He, and G. Wei, “A generalized higher order finitedifference time-domain method and its application in guided-wave problems,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 856–861, Mar. 2003. [19] D. W. Zingg, “Comparison of high-accuracy finite-difference methods for linear wave propagation,” SIAM J. Sci. Comput., vol. 22, no. 2, pp. 476–502, Jul. 2000. [20] K. L. Shlager and J. B. Schneider, “Comparison of the dispersion properties of several low-dispersion finite-difference time-domain algorithms,” IEEE Trans. Antennas Propag., vol. 51, no. 3, pp. 642–653, Mar. 2003. [21] J. Chen, Z. Wang, and Y. Chen, “higher order alternative direction implicit FDTD method,” Electron. Lett., vol. 38, no. 22, pp. 1321–1322, Oct. 2002. [22] J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations. Pacific Grove, CA: Brooks/Cole, 1989. [23] G. Sun and C. W. Trueman, “Quantification of the truncation errors in finite-difference time-domain methods,” in Can. Electrical and Computer Engineering and Humane Technology Conf., Montreal, QC, Canada, May 4–7, 2003.

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Guilin Sun (S’02) was born in Henan, China, in 1962. He received the B.Sc. degree from the Xi’An Institute of Technology, Xi’An, China, in 1982, the M.Sc. degree from the Beijing Institute of Technology, Beijing, China, in 1988, respectively, both in optical engineering, and is currently working toward the Ph.D. degree in electrical and computer engineering at Concordia University, Montreal, QC, Canada. From 1988 to 1994 he was an Assistant Professor with the Xi’an Institute of Technology. From 1994 to 2000, he was an Associate Professor with the Beijing Institute of Machinery. From 1998 to 1999, he was a Research Associate with the University of Southern California, Los Angeles, where he was involved with the characterization of electrooptical (E/O) polymers used in photonic devices. He has authored or coauthored approximately 70 journal and conference papers in optical engineering and several papers on FDTD methods. He appears in Marquis Who’s Who in the World, 16th edition, 1999. His current research interest is in computational electromagnetics, particularly in new methods of the FDTD method in microwave and optical frequencies. Mr. Sun was the recipient of several awards and honorable titles.

Christopher W. Trueman (S’75–M’75–SM’96) received the Ph.D. degree from McGill University, Montreal, QC, Canada, in 1979. His doctoral dissertation concerned wire-grid modeling aircraft and their high-frequency (HF) antennas. He is currently a Professor with the Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, Canada. His research concerns computational electromagnetics uses moment methods, the FDTD method, and geometrical optics and diffraction. He has been involved with electromagnetic compatibility (EMC) problems with standard broadcast antennas and high-voltage power lines, the radiation patterns of aircraft and ship antennas, EMC problems among the many antennas carried by aircraft, and on the calculation of the radar cross section (RCS) of aircraft and ships. He has studied the near and far fields of cellular telephones operating near the head and hand. He has recently been concerned with indoor propagation of RF signals and electromagnetic interference (EMI) with medical equipment in hospital environments.