IEEE TRANSACTIONS ON, VOL. XX, NO. Y, MONTH 000 100 Moeling Dielectric Interfaces in te FDTD-Meto: A Comparative Stuy C. H. Teng, A. Ditkowski, J. S. Hestaven Abstract In tis paper, we present special finite ifference scemes embee in te Cartesian Yee cell algoritm for moeling ielectric materials wit curve interfaces. Te accuracy an efficiency of te new meto is emonstrate by numerical computations. We also compare te performance of te new approac an more traitional staircase formulations wit an witout accuracy improving tecniques. Tese tecniques are basically using average material properties at staircase interfaces. Te numerical results sow tat te traitional tecniques can improve te accuracy of te solution only for electromagnetic fiels being continuous at te interface. On te oter an, numerical results obtaine by te new metos are more accurate tan tose obtaine by te oters. Most importantly, te solutions are secon orer accurate even for electromagnetic fiels being iscontinuous at te interface. Keywors Finite ifference scemes, Maxwells equations, Complex geometries, ielectric interfaces, Staircasing U I. Introuction SING te classical finite ifference time omain (FDTD) meto to simulate te interaction of electromagnetic (EM) waves wit geometrically complex objects, one very often encounters tat a simple Cartesian gri mes fails to properly escribe te bounaries of te object, e.g., material interfaces an metallic bounaries. A stanar approac to overcoming tis problem is to use staircase bounaries wic rougly approximate actual bounaries. Hence te problem is force to conform to te simple gri structure. It is well known, owever, tat tis approac as a severe impact on te overall sceme. It reuces te sceme to at best first-orer accuracy, an allows te possibility of localize non-convergent beavior. Several tecniques ave been propose in te past to ameliorate tis unfortunate situation. For eiter te wole omain or close to te object, te straigtforwar approac is to use irregular nonortogonal gris or curvilinear coorinates; instea of te simple Cartesian Yee cell. Clearly, tis is muc more complicate as compare to te simple FDTD approac. Empasizing simplicity, an alternative approac, centere aroun te iea of efining average material properties at te staircase material interfaces, as been use extensively in te past. It yiels superior results wen compare to utilizing only te staircase formulation. However, in [6], [7], it as been sown tat using te staircase approximation may ave a serious consequence: C. H. Teng is at te Division of Applie Matematics, Brown University, Provience, RI 091 E-mail: ct@cfm.brown.eu A. Ditkowski is at te Division of Applie Matematics, Brown University, Provience, RI 091 E-mail: ait@cfm.brown.eu J. S. Hestaven is at te Division of Applie Matematics, Brown University, Provience, RI 091 E-mail: JAN HESTHAVEN@BROWN.EDU. te non-convergence of te sceme as been proven by rigorous matematical analysis an numerical computations. Moreover, tey also evelope a general framework for esigning new scemes to use at te gri points near te bounaries in te Cartesian Yee cell. Tese new scemes are constructe by combining bot te one-sie ifference operator an te bounary conitions. Applying te new meto, one oes not nee to use te staircase approximation an ence overcomes te problems inuce by it. Most importantly, tenumerical solutions converge ata seconorer rate in L. In tis paper, following te same ieas as in [6], [7], we present novel scemes for simulating two imensional EM wave penomenon involving ielectric material interfaces, were EM fiels can be eiter continuous or iscontinuous. Te performance of te new meto is emonstrate by numerically solving scattering of EM waves by a ielectric cyliner, for bot transverse electric (TE) an transverse magnetic (TM) moes. Our results sow tat tese scemes are capable of anling suc tasks an proucing solutions wit secon-orer accuracy in L. We also test several traitional approaces, te staircase approximation an te averaging tecniques, by te same experiments. From te results, we observe tat te traitional metos can improve te accuracy only for EM fiels being continuous at te ielectric interface. Te paper is organize as follows. In te next section, we start wit a general pysical problem formulate by Maxwell's equations subjecte to te ielectric material interface bounary conitions. We ten iscuss several traitional approaces, an te formulation of te new scemes in greater etail. Te numerical results from all tese metos are presente in Section 3. Implementation issues an te efficiency of te new metos are also iscusse. In te last section, we aress some concluing remarks an ieas for future work. II. Formulation A. Maxwell's equations an Yee scemes Consier two meia meeting at an interface as sown in Fig.(1). For simplicity, we assume tat eac meium is caracterize by a constant permitivity ffl an a constant permeability μ. In te meia, te EM wave penomenon is governe by Maxwell's equations an te bounary conitions at te interface. Witout losing generality, let us consier te Maxwell's equations for te TE waves in two imension of te form, ffl ff @ t E z;ff = @ x H y;ff ; (1)
IEEE TRANSACTIONS ON, VOL. XX, NO. Y, MONTH 000 101 ε,µ x Replacing te spatial erivatives in Maxwell's equation by te central ifference approximations yiels ε 1,µ 1 n z ffl i+ 1 ;j t E z i+ 1 = H y i+ 1 ;j+ 1 H y i+ 1 ;j 1 ;j x ; (7) bounary ffl i;j+ 1 t E xi;j+ 1 = H y i+ 1 ;j+ 1 H y i 1 ;j+ 1 z (8) Fig. 1. ffl ff @ t E x;ff = @ z H y;ff ; () μ ff @ t H y;ff = @ x E z;ff @ z E x;ff ; (3) were ff = 1; enote ifferent materials. Te variables, E z,e x an H y, are te electric an magnetic fiel components. Te bounary conitions at te interface are n z E x;1 n x E z;1 = n z E x; n x E z; ; (4) (n z E z;1 + n x E x;1) = ffl (n z E z; + n x E x;) ; (5) H y;1 = H y; ; (6) were n z an n z are te z an x components of a unit normal vector n, pointing outwar from material 1 to. Fig.. To solve te set of equations by te FDTD meto, a two imensional staggere gri mes is constructe by te tensor prouct of te gri points efine as z i = z(i); z i+ 1 = z(i + 1 ); x j = x(j); x j+ 1 = x(j + 1 ); were z an x enote te gri sizes in z an x irections. An illustration of te mes is given in Fig.(). Let E z i+ 1, E ;j x i;j+ 1 an H y i+ 1 ;j+ 1 be te values of te EM fiel components evaluate at te gri points: E z i+ 1 ;j = E z (z i+ 1 ;x j) E x i;j+ 1 = E x(z i ;x j+ 1 ) H y i+ 1 ;j+ 1 = H y (z i+ 1 ;x j+ 1 ): X Hy Ex Z μ i+ 1 ;j+ 1 t H y i+ 1 ;j+ 1 = E z i+ 1 ;j+1 E z i+ 1 ;j x E x i+1;j+ 1 E xi;j+ 1 : (9) z In te Eq.(7), (8) an (9), ffl i+ 1 ;j, ffl i;j+ 1 an μ i+ 1 ;j+ 1 are te values of te material properties of te meium in eac region. It is well known tat solving te system numerically oes not give promising results ue to te staircase formulation. We woul like to iscuss some tecniques, using te average constitution parameters at te staircase interface, to improve te accuracy. To istinguis wit te oter approaces given later, te staircase approximation witout using moifie material properties will be refere as meto one. B. Classical tecniques: averaging metos Te averaging meto is very simple to apply. For gri points at te staircase interface, it basically reassigns te values of ffl i+ 1, ffl ;j i;j+ 1 an μ i+ 1 ;j+ 1 by some formulas. Te formulas are taking mean values of te material properties uner some specific consierations. Hence, applying te metos in a pre-stage, one can use te same Yee algoritm as usual. We woul like to iscuss tree commonly use averaging tecniques. Te tree formulas referre as meto,3 an 4 are: Meto ffl ef f = + ffl ; μ ef f = μ 1 + μ : (10) Meto 3 p ffl1 + p p ffl ffl ef f =( ) μ1 + p μ ; μ ef f =( ) : (11) Meto 4 (armonic mean) ffl ef f = ffl + ffl ; μ ef f = μ 1μ μ 1 + μ (1) In meto, te aritmetic mean of te material parameters is use, wic results in a smoot cange of te parameters at te interface instea of a step jump. One can also use te mean of te pase spees of te electromagnetic waves traveling in ifferent meia. In meto 3, we use te formulas for eiter ffl or μ being te same in ifferent meia, e.g., a non-magnetic meium place in free space. All tese tecniques sare te avantage of te simplicity of te Yee sceme. However, tey all suffer some rawbacks ue to te staircasing approximation from te following points of view. Because of te failure of escribing
IEEE TRANSACTIONS ON, VOL. XX, NO. Y, MONTH 000 10 te curve interface, te information suc as te location of te curve interface in te mes is cange. Consequently, te normal vector at te interface, wic is neee in te bounary conitions, are incorrectly set to irections parallel to gri axes. Furtermore, te flux at eac interface gri point, were te erivatives of te fiels may not be continuous in general, are evaluate by using values from eac region. As a result, one soul not expect tat suc an approac will always give accurate solutions. C. Special scemes at gri points near te interface After introucing te classical approaces for improving te computations, we woul like to apply te ieas propose in [6] to overcoming te improper situations. Our purpose is to esign new scemes for gris near te interface. To complete tis task, it is necessary to classify ifferent configurations of ow te bounary intersects te staggere gri mes. Consier a magnetic fiel point an te nearest four electric fiel points as a group. Wen an interface passes troug te group of points, it separates some electric fiel points an te magnetic fiel point into ifferent meia. We may use tis caracter, te number of electric fiel points separate from te group, to classify te configurations into two ifferent classes. Two examples from eac class are presente in Fig.(3), (4). We will use tese configurations as examples to emonstrate te ieas for constructing te new scemes. Ex i,j+1/ i+1/,j+1 γ z z Hy i+1/,j+1/ Fig. 3. i+1/,j A n Ex i+1,j+1/ Type 1 configuration Te caracter of te first class is tat te bounary separates one electric fiel point to te rest of te four. As sown in Fig.(3), te interface intersects te line connecting H y i+ 1 ;j+ 1 an E x i+1;j+ 1 at point Aantus E xi+1;j+ 1 is separate from te group. Note tat fl z z enotes te istance from H y i+ 1 ;j+ 1 to point A. In anoter wor, fl z is te ratio of te istance to te gri size in z irection. Due to te separation, we nee to construct scemes for upating te values at H y i+ 1 ;j+ 1 an E x i+1;j+ 1. Te algoritms for upating H y i+ 1 ;j+ 1 an E x i+1;j+ 1 in tis kin of configuration ave been introuce in [6]. In ere, we woul like tosow a simplifie version. Hy Ex Te sceme for upating H y i+ 1 ;j+ 1 μ i+ 1 ;j+ 1 t H y i+ 1 ;j+ 1 is = E z i+ 1 ;j+1 E z i+ 1 ;j x E x;1 E xi;j+ 1 : fl z +1 z (13) were E x;1 is te fiel value at te bounary point A in meium one. Comparing te two scemes,(9)(13), one see tat a one-sie ifference operator is use in (13), instea of a central ifference operator as in (9). As a consequence, Eq.(13) contains aitional information, te istance of te interface to te H y i+ 1 ;j+ 1 in z irection caracterize by fl z. To use Eq.(13), one nees to provie te value of E x;1 wic is, unfortunately, not given from te pysical problem. However, we can utilize te bounary conitions an te nearby fielvalues to estimate te value of E x;1. From te bounary conitions Eq.(4) an (5) at point A, we can write E x;1 as ffl E x;1 = n x + ffl n E x; + (ffl )n z n x z n x + ffl n E z;1 : (14) z In Eq.(14), it implies tat one can approximate te values of E x; an E z;1 first, an ten substitute te two values into Eq.(14) to estimate te value of E x;1. To approximate te values of E x; an E z;1, we can use extrapolation an interpolation of te nearby fiel values. Te formulas are: E x; =( 3 fl z)e xi+1;j+ 1 (1 fl z)e xi+;j+ 1 i+ E z;1 = (1 + fl z ) 1 ;j+1 + E z i+ 1 ;j1 fl z i 1 ;j+1 + E z i 1 ;j1 ; (15) : (16) Terefore, te value of E x;1 at point A is obtaine byusing Eq. (14), (15) an (16), an one can use Eq.(13) to upate H y i+ 1 ;j+ 1. To upate E x i+1;j+ 1, one simply use te following sceme, ffl i+1;j+ 1 t E xi+1;j+ 1 = H y i+ 3 ;j+ 1 H y; ; ( ~fl z +1) z (17) were ~fl z = 1 fl z. Te variable H y; is te magnetic fiel value at point A in meium two. Here, we encounter a similar problem, not knowing te value of H y;. However, we can follow te same proceure to estimate te value of E x;1. Te resulting formula is given as H y; = H y;1 =(1+ ~fl z )H y i+ 1 ;j+ 1 ~fl z H y i 1 ;j+ 1 : (18) Finally, we ave presente te scemes for upating H y i+ 1 ;j+ 1 an E xi+1;j+ 1 an te auxiliary scemes to use tem. To anle all te configurations in te first class,
IEEE TRANSACTIONS ON, VOL. XX, NO. Y, MONTH 000 103 tree oter sets of algoritms are neee. Eac one is use in te configuration wen E z i+ 1, E ;j+1; x i;j+ 1 or E z i+ 1 is separate from ;j H y i+ 1 ;j+ 1 byteinterface, an all te scemes can be constructe by te same proceure presente ere. ε 1,µ 1 Ex i,j+1/ γ x x Hy i+1/,j+1/ i+1/,j+1 i+1/,j B γ z z A n Ex i+1,j+1/ ε,µ Hy Ex vali. Te reason is tat te average of E z can not be use ue to ifferent meia. However, one may try anoter approac toapproximate E x;1. From te bounary conitions (4),(5), E x;1 can be written as E x;1 = n z + ffl n x E x; + ffl n z n x E z;: (0) To estimate te value of E x;1, we can substitute two approximate values of E x; an E z; into Eq.(0), an tus two auxiliary equations are neee. Fortunately, since we can approximate E x; by Eq.(15), we only nee to erive an auxiliary equation for E z;, an te proceure is te following. We evaluate one interpolate value of E z fiel at te coorinate (z i+1= + fl z z; x j+1) using E z i+ 1 ;j+1 an E z i+ 3 ;j+1, an anoter one at te coorinate (z i+1= + fl z z; x j+) usinge z i+ 1 ;j+ an E z i+ 3 ;j+. We ten use te interpolate values to compute a extrapolate value, E z; at point A. Hence, te expression of te auxiliary equation for E z; is Fig. 4. Type configuration Te caracter of te secon class is tat two electric fiel points are separate from te group of points. Atypi- cal configuration in te secon class is sown in Fig.(4). In Fig.(4), te interface intersects one line, connecting H y i+ 1 ;j+ 1 an E x i+1;j+ 1, at point A, an anoter line, connecting H y i+ 1 ;j+ 1 an E z i+ 1 ;j+1,atpointb.wecan now caracterize tis configuration by fl z an fl x, were fl z z an fl x x are te istances from H y i+ 1 ;j+ 1 to point A an B. In tis configuration, we nee to esign scemes for upating te fiel values, H y i+ 1 ;j+ 1, E x i+1;j+ 1 an E z i+ 1 ;j+1. Comparing te two configurations, one observes tat te situations of E xi+1;j+ 1 an E z i+ 1 ;j+1 in Fig.(4) are similar to te situation of E xi+1;j+ 1 in Fig.(3). Hence, we can use Eq.(17) an (18) to upate E xi+1;j+ 1 irectly, an E z i+ 1 ;j+1 after canging te variables an te corresponing gri inices. Terefore, we only ave to consier ow to construct te set of scemes for upating H y i+ 1 ;j+ 1. Following te same proceure evelope previously, we ave asceme for upating H y i+ 1 ;j+ 1 as μ i+ 1 ;j+ 1 t H y i+ 1 ;j+ 1 = E z;1 E z i+ 1 ;j fl x +1 x E x;1 E x i;j+ 1 ;(19) fl z +1 z were te E x;1 an E z;1 are fiel values at point A an B in meium one. As before, since te values of E x;1 an E z;1 are not known, we neetofinaway toapproximate tem. Let us first consier ow tofine x;1. If we try to estimate E x;1 using Eq.(14),(16) an (15) evelope in te previous case, we immeiately know tat Eq.(16) is no longer E z; = 3 (E z i+ 1 ;j+1 + fl z (E z i+ 3 ;j+1 E z i+ 1 ;j+1 )) 1 (E z i+ 1 ;j+ + fl z (E z i+ 3 ;j+ E z i+ 1 ;j+ )): (1) Terefore, using Eq.(0), (15) an (1), we ave E x;1 for Eq.(19). To estimate te value of E z;1 at point B,we can perform a similar proceure as for te value of E z;1. Hence, we irectly present te auxiliary equations: E z;1 = n x + ffl n z E z; + ffl n z n x E x;: () E z; =( 3 fl x)e z i+ 1 ;j+1 (1 fl x)e z i+ 1 ;j+ ; (3) E x; = 3 (E xi+1;j+ 1 + fl x(e xi+1;j+ 3 E xi+1;j+ 1 )) 1 (E xi+;j+ 1 + fl x(e x i+;j+ 3 E xi+;j+ 1 )): (4) After giving te equations for estimating E x;1 an E z;1, we can use tem wit Eq.(19) to upate H y i+ 1 ;j+ 1. We ave iscusse te scemes for upating te tree fiel values. As in te first case, we also nee tree oter sets of algoritms to anle all te configurations in te secon class. Te scemes are constructe in a similar fasion for computing fiel values, wen E z i+ 1 ;j+ an E z i+ 1 ;j+, E z i+ 1 ;j+ an E z i+ 1 ;j+,ore z i+ 1 ;j+ an E z i+ 1 ;j+ are not place in te same meium wit H y i+ 1 ;j+ 1. III. Numerical Computation an Discussion To compare te performance of all te metos mentione in te previous section, we use tem to solve a funamental pysical problem, scattering by a ielectric cyliner in free space wit TE an TM wave excitations. Te reasons of coosing tis simple problem are te following. First, te problem as an exact solution. Hence, we can
IEEE TRANSACTIONS ON, VOL. XX, NO. Y, MONTH 000 104 measure te errors in te numerical solutions compute by all te metos. Te secon reason is tat te scatterer as a curve bounary. Hence, we will know if te new scemes ave te capability of anling complex geometries. Te last one is to emonstrate ow well te metos perform wen EM waves are eiter continuous or iscontinuous at te interface. All te computations are marce to t=0 were te numerical solutions ave reac a perioic pattern. We ten measure te iscrete L errors at tat time. Te resolution of te numerical solution is caracterize by N,te number of points per wavelengt (ppw) measure from te incient wave. Te error versus of te EM fiels are sown in in Fig.(6),(7) an (8) for te TE moe, an Fig.(9),(10) an (11) for te TM moe. PML ε 0,µ 0 r ε 1,µ 0 δ H y 10 Fig. 5. Te computation is setup by using total/scattering fiel formulation were te range of te total fiel is [1; 1] [1; 1]. Insie te total fiel region, a ielectric cyliner is place at te center. To simulate te outgoing waves in te finite computational omain, a perfect matce layer (PML) type absorbing bounary conition [5] is use. An illustration is sown in Fig.(5). In te numerical test, we use te following parameters. Te permittivity an permeability in free space are given as ffl 0 = 1 an μ 0 = 1. Te material properties an te raius of te cyliner are ffl =ffl 0 ;μ = μ 0,anr =(ß=6). Te incient waves are time-armonic fiels. Te angular frequency is equal to ß an te amplitue as well as te wavelengt 0 measure in free space are set to unity. All te scemes presente in Section are in semiiscrete form. To upate te fiel values in time, we use te fourt-orer Runge-Kutta algoritm in all five metos. Te time step use in te computations is t = min( z; x) p c (5) were c is te spee of ligt after normalization. Terefore, c is equal to one in tis particular case. In all te computations, we let z = x = uring mes refinement an compute te time step aaptively by Eq.(5). To compare te performances of ifferent metos, we measure te iscrete L error, jj ffiu jj, efine as jj ffiu jj = s z x X i;j (U i;j U exact ) : (6) Te variable U i;j is te fiel value compute at te gri point, an U exact refers to exact one. Te analytic solutions to tis particular problem can be foun in [1]. 10 1 10 Fig. 6. Te iscrete L errors of H y by te ifferent metos: 4: staircase, Π :( + ffl )=, Λ :(( p + p ffl )=) : ffl =( + ffl ), ffi: new meto. δ E x 10 10 1 10 Fig. 7. Te iscrete L errors of E x by te ifferent metos: 4: staircase, Π :( + ffl )=, Λ :(( p + p ffl )=) : ffl =( + ffl ), ffi: new meto. Let us examine te errors of te H y fiel from te TE waves in Fig.(6). It sows tat te staircase approximation as te largest error among all te metos for te same N. Moreover, te numerical solutions compute p by te meto converge slowly, from O() to O( ) as N increases. Te results obtaine by te averaging metos sow tatte tecniques o improve teaccuracy of te
IEEE TRANSACTIONS ON, VOL. XX, NO. Y, MONTH 000 105 δ E z 10 δ E y 10 10 1 10 10 1 10 Fig. 8. Te iscrete L errors of E z by te ifferent metos: 4: staircase, Π :( + ffl )=, Λ :(( p + p ffl )=) : ffl =( + ffl ), ffi: new meto. Fig. 9. Te iscrete L errors of E y by ifferent metos: 4: staircase, : ffl =( + ffl ), ffi: new meto. numerical solutions. We also observe tat te errors ecay to secon orer for N being large. Te numerical solutions obtaine by te new meto are te most accurate wen EM waves are well resolve. Moreover, te errors ecay by a factor of four as N increases twice. Tis inicates tat te new metos is secon orer accurate except for N = 160 ppw, an te exception is ue to te reflection from te PML. In Fig.(7) an (8), we present te results from te E x an E z fiel computation. As sown in bot figures, te performance of te staircase formulation is very poor an te numerical solutions are less tan first orer accurate. Unfortunately, te averaging metos can only reuce te errors approximately by alf but fail to improve te orer of accuracy of te metos as in te H y fiel computations. Te reason for te convergence rates being so slow istat te staircase formulation fails to correctly moel te iscontinuities of E x an E z fiels at te interface. On te oter an, te new meto oes not ave suc problems an te results obtaine by tat are secon orer accurate. Due to ifferent convergence rates, for bot E z an E x fiels, te new meto as te L errors approximately one orer of magnitue lower tan te oters for N =80 ppw. Let us turn our attention to te TM moe, a case were all te EM fiels are continuous at te bounary. In Fig.(9),(10) an (11), te results sow tat te solutions are only first orer accurate for staircase approximation. Using te armonic mean of te material parameters gives less error an te orer of accuracy is recover for large N ppw. Wen using te new meto, one observes tat te errors are of secon orer an te smallest among all oter metos. Te new meto is also more efficient tan te oters, wen one emans te error in te numerical solution to be lower tan a certain level. For instance, let us consier δ H x 10 10 1 10 Fig. 10. Te iscrete L errors of H x by ifferent metos: 4: staircase, : ffl =( + ffl ), ffi: new meto. te global L error to be aroun 0:01, wic can be viewe as 1% relative error efine as te ratio of te L error to te amplitue of te incient wave. Uner suc request for te TE moe, one sees tat 40 ppw will be enoug for te new meto but 160 ppw for te oters ue to te iscontinuous fiels. Hence, te computation memory an te total time for computing erivatives are reuce by 16 times wen te new meto is use. Moreover, since te gri sizes in te new meto are 4 times larger tan tose in te oter metos, te time step is also 4 times larger. Terefore, te new meto gives results 64 time faster tan te oters, ue to te saving combine from te temporal an spatial part teoretically. Similarly, for TM moe, we nee to use 40 ppw for te new meto but 80 ppw for te traitional ones to satisfy te same accuracy request. Hence, te new meto is still 8 times faster tan te traitional metos. However, to gain all tese avantages,
IEEE TRANSACTIONS ON, VOL. XX, NO. Y, MONTH 000 106 δ H z 10 Meto CPU time jj ffie x jj jj ffie z jj (sec) New 40 878 0.0139 0.0093 Meto 4 40 877 0.0343 0.065 Meto 4 160 59871 0.015 0.0118 TABLE I A measurement of te total CPU time by ifferent metos for TE moe wit L error aroun 0:01 10 1 10 Fig. 11. Te iscrete L errors of H z by ifferent metos: 4: staircase, : ffl =( + ffl ), ffi: new meto. an aitional workloa, flux correction for te gris near te bounary, is require. Hence, it is necessary to know ow tis aitional part affect te computations. To explain te effect of te aitional workloa, implementation issues are iscusse in te next. To upate fiel values by special scemes, one nees te corresponing values of fl z, fl x an te te normal vectors. Fortunately, all tese values only nee to be compute once an store in a pre-stage. Wit all te values in an, one first computes te erivatives at all te gri points as in te traitional metos, an ten a an aitional stage to recompute te erivatives at te gri points using special scemes. Te proceure given ere is suitable wen te Runge-Kutta meto in time is use. If te Leap Frog sceme staggere in time is use, ten one may use te following proceure. Te first step is to upate te fiel values at te special gri points using te new meto, an store tem in an aitional array. Ten we upate te fiel values at every gri point as usual. Te last step is to replace new fiel values at te special gri points by te values obtaine in te first one. Altoug te erivatives are compute twice at te gri points near te bounary, te aitional work is muc less tan te work for computing erivatives for all te gri points. Tis is ue to te fact tat te total number of te special gri points an all te gri points being O(N) ano(n ) respectively. In table (I), a irect measurement of te total CPU time of tree computations terminate at time t = 0 is provie. Surprisingly, as sown in te table, te total CPU time by te new meto an te traitional meto are only one secon in ifference for computation using N = 40 ppw. In anoter wor, aing an aitional 0:1% workloa relative to te traitional meto gives all te avantages mentione above. In te table, it is sown tat using new metos wit 40 ppw is 68 times faster tan using traitional meto wit 160 ppw. IV. Conclusions In tis paper, we present a new approac to moel te ielectric interface in te FDTD meto. We compare te new meto to te classical staircase formulation wit an witout averaging tecniques. Te results sow tat te averaging tecniques can only improve te accuracy of numerical solutions wen te EM fiels at te interface are continuous. On te oter an, te results obtaine by te new metos are of secon orer accurate, an tus muc more accurate tan tose compute by te oter metos. Te numerical eviences inicate tat te new meto is inee capable of simulating te EM waves being continuous an iscontinuous at te curve interface. Applying te same iea presente ere, one can construct special scemes to improve te existing algoritms, Currently, special scemes for iger imension an ig orer algoritms are uner investigation. We ope to report suc applications in te future. Acknowlegments Tis work was partially supporte by AFOSR grant F4960-96-1-046, NSF grant DMS-007457, an DOE grant 98-ER-5346. JSH acknowleges partial support as a Sloan Researc Fellow from te Sloan Founation. References [1] R. F. Harrington, Time-Harmonic Electromagnetic Fiels, McGraw-Hill Book Company, Inc. 1961. [] K. S. Yee, Numerical Solution of Initial Bounary Value Problems Involving Maxwell's Equations in Isotropic Meia, IEEE Trans. Antennas Propag. 14, 1966. [3] K. S. Kunz an R.J. Luebbers, Te Finite Difference Time Domain Meto for Electromagnetics, CRC Press, Inc., 1993. [4] A. Taflove, Computational Electroynamics, Te Finite Difference Time Domain Meto, Artec House, Norwoo, MA, 1995. [5] J. S. Hestaven, P.G. Dinesen, an J.P. Lynov, Spectral Collocation Time-Domain Moeling of Diffractive Optical Elements, J. Comput. Pys. 1999 - to appear [6] A. Ditkowski, K. Drii an J.S. Hestaven Convergent Cartesian Gri Metos for Maxwells Equations in Complex Geometries, Journal of Computational Pysics, 000 - to appear. [7] K. H. Drii, J.S. Hestaven an A. Ditkowski, Staircase Free Finite-Difference Time-Domain Formulation for General Materials in Complex Geometries, IEEE Trans. Antennas Propagat. 1999 - submitte.