Wireless Egieerig ad Techolog, 0,, 30-36 doi:0.436/we.0.005 Published Olie Jauar 0 (hp://www.scirp.org/joural/we) A Efficie Mehod o Reduce he umerical Dispersio i he IE- Scheme Jua Che, Aue Zhag School of Elecroic ad Iformaio Egieerig, Xi a Jiaoog Uiversi, Xi a, Chia. Email: chejua0306@ahoo.com.c Received Sepember d, 00; revised ovember d, 00; acceped ovember 8 h, 00. ABSTRACT A parameer opimied approach for reducig he umerical dispersio of he 3-D hbrid implici-eplici fiie-differece ime-domai (IE-) is preseed i his leer. B addig a parameer io he IE- formulas, he error of he umerical phase veloci ca be corolled, causig he umerical dispersio o decrease sigifical. The umerical sabili ad dispersio relaio are preseed aalicall, ad umerical eperimes are give o subsaiae he proposed mehod. Kewords: IE-, umerical Dispersio, Weakl Codiioall Sabili. Iroducio The fiie-differece ime-domai () mehod [] has bee prove o be a effecive meas ha provides accurae predicios of field behaviors for varieies of elecromageic ieracio problems. owever, as i is based o a eplici fiie-differece algorihm, he Coura Friedrich Lev (CFL) codiio [] mus be saisfied whe his mehod is used. Therefore, a maimum ime-sep sie is limied b miimum cell sie i a compuaioal domai, which makes his mehod iefficiec for he problems where fie scale dimesios are used. To rela he Coura limi o he ime sep sie of he mehod, a hree dimesioal (3-D) hbrid implici-eplici fiie-differece ime-domai (IE-) mehod has bee developed recel [3]. I his mehod, he CFL codiio is o removed oall, bu beig weaker ha ha of he coveioal mehod. The ime sep i his scheme is ol deermied b wo space discreiaios, which is eremel useful for problems where a ver fie mesh is eeded i oe direcio. ow- ever, he umerical dispersio error of he IE- scheme is larger ha ha of he coveioal mehod. I his leer, a simple ad efficie approach for reducig he umerical dispersio of he 3-D IE- mehod is proposed. umerical resuls idicae ha he umerical dispersio of he mehod ca be oabl reduced whe a proper parameer is iroduced [4]. As a resul, he usefuless ad effeciveess of he IE- mehod ca be sigifical ehaced. The umerical dispersio of he ew algorihm is sudied aalicall ad validaed b a umerical simulaio, ad he resuls are compared wih boh he sadard IE- mehod ad he coveioal mehod.. Formulaios To reduce he umerical dispersio of he 3-D IE- mehod, parameer is iroduced io he IE- discreiaio. The modified algorihm is described as follows: E i, j, ke i, j, k i, j, k i, j, k i, j, k i, j, k () Coprigh 0 SciRes.
A Efficie Mehod o Reduce he umerical Dispersio i he IE- Scheme 3 i, j, k i, j, k,,,, E i j k E i j k E i, j, k E i, j, k,, E i jk E i jk i jk i, jk, i, j, k i, j, k i, j, k i, j, k,, E i jk E i jk i jk i, jk, i, j, k i, j, k i, j, k i, j, k,, i j k i j k E i j k E i, j, k E i, j, k E i, j, k E i, j, k E i, j, k,, i j k i j k E i j k E i, j, k E i, j, k E i, j, k E i, j, k E i, j, k () (3) (4) (5) (6) ad are he ide ad sie of ime-sep,, ad are he spaial icremes respecivel i, ad direcios, i, j, ad k deoe he idices of spaial icremes respecivel i,, ad direcios, ad are he permiivi ad permeabili of he surroudig media, respecivel. Whe he value of parameer is equal o, Equaios ()-(6) is he formulaios of sadard IE- mehod [3]. Obviousl, updaig of E compoe, as show i Equaio (3), eeds he ukow compoe a he same ime, hus he E compoe has o be updaed implicil. Subsiuig Equaio (5) io Equaio (3), he equaio for E field is give as: E ( i, j, k) E,,,, i j k E i j k 4 E i, j, k,,,, i j k i j k E i, j, k,, E i j ke i j k E i j k,,,,,, 4 E i j k E i j k E i j k i, jk, i, jk, (7) Coprigh 0 SciRes.
3 A Efficie Mehod o Reduce he umerical Dispersio i he IE- Scheme Similarl, updaig of E compoe eeds he ukow compoe a he same ime-sep. Subsiu- ig Equaio (6) io Equaio (4), we obai he discree equaio for E field, E ( i, j, k ) E,,,, i j k E i j k 4 E i, j, k,,,, i j k i j k E i, j, k,,,,,, E i j k E i j k E i j k (8),,,,,, 4 E i j k E i j k E i j k i, j, k i, j, k Therefore, he field compoes are updaed b usig Equaios (), (), ad (5)-(8). Compoes E ad are eplicil updaed firs b usig Equaios () ad (). The, E ad E compoes are updaed implicil b solvig he ridiagoal mari equaios b usig Equaios (7) ad (8). Afer E ad E are obaied, compoes ad are eplicil updaed sraighforward b usig Equaios (5) ad (6). 3. Weakl Codiioall Sabili The relaios bewee field compoes of Equaios ()- (6) ca be represeed i a mari form as: a 0 D 0 D 0,, E 0 a D 0 0,, D E 0 0 0 0 0,, b,, 0 0 0 D b D,, 0 0 0,, D b D E 0 0 0 0 0 a a 0 0 0 D 0 E,, 0 a 0 D 0 0 E,, 0 0 0 D D b,,,, 0 0 0 0 D b,, 0 0 0 0 D,, b E 0 0 0 D D a (9) Coprigh 0 SciRes.
A Efficie Mehod o Reduce he umerical Dispersio i he IE- Scheme 33 where a, b, D w w ( w,, ) represes he firs derivaive operaor wih respec o w. Wih o loss of geerali, he field compoes ca be wrie as follows: p,, p,, f (0) f,, ep( jk jk jk ) () where E,, p,,, j. k k k are wave umbers. idicaes growh facor. p are he ampliude of he field compoes, respecivel. I a discree space, f,, ca be deoed as: f m, l, p ep( jk m jk l jk p ) The: () D f D f f m, l, p,, f m l p f m, l, p (3) f m, l, p,, f m l p f m, l, p (4) D f f m, l, p,, f m l p f m, l, p (5) where, k j si, k j si, k j si Subsiuig hese represes io Equaio (9), he mari becomes: a a 0 0 0 E f 0 0 0 a a E f b b 0 0 0 f 0 0 0 0 f b b 0 0 0 f b b 0 0 0 E f a a (6) For a orivial soluio of (6), he deermia of he coefficie mari i (6) should be ero. I ca be obaied: r 4r 4r 0 where: si r c k, (7) si r c k, si r c k, c is he speed of ligh i he medium. B solvig Equaio (7), he growh facor is obaied (8) Coprigh 0 SciRes.
34 A Efficie Mehod o Reduce he umerical Dispersio i he IE- Scheme r r r r r r r 34 56 r (9) To saisf he sabili codiio durig field advaceme, he module of growh facor ca be larger ha. I is evide ha he module of is ui. For he values of 34 ad 56, whe he codiio r r is saisfied, 34 56 ca be obaied. The limiaio for ime-sep sie ca be calculaed as follows: c si k c si k c c c (0) This scheme is weakl codiioall sable. The ime sep is ol deermied b wo space discreiaios. The parameer does affec he weakl codiioall sabili of he IE- mehod. 4. umerical Dispersio Aalsis We ow sud he umerical dispersio i he modified j IE- algorihm. Subsiue e io Equaio (6), i ca be obaied: k k k si si si si cos c () For compariso, we ake a look a he umerical dis- persio relaio of he sadard IE- mehod. k k k si si si si cos c () Compared o he dispersio Equaio () of he sadard IE- mehod, i ca be obaied ha here is a facor added o he las erm i he righ-had side of he umerical dispersio relaios of Equaio (). Whe a proper value of parameer is seleced, he umerical dispersio of he IE- mehod ca be corolled, causig he umerical dispersio o decrease sigifical, which is validaed i e secio. 5. umerical Validaio Suppose ha a wave propagaig a agle ad is i he k k si cos, spherical coordiae ssem. The, k k si si, ad cos k k. B subsiuig hem io dispersio relaios (), umerical phase veloci v p of modified IE- mehod ca be k solved umericall. To make he discussio simple ad eas, ol he uiform cell is cosidered here. is se o be /0, wih he operaig frequec. O he k k plae 0, k ksi si 0. I ca be easil see ha he umerical dispersio of modified IE- mehod is he same as ha of sadard IE- mehod. So we ol cosider he dispersio performace compariso bewee he modified IE- ad sadard IE- mehod o oher plaes. Figures -4 show he ormalied phase velociies wih respec o agle for differe CFL values. is se as 45 ad 90 respecivel. The CFL is defied as he raio of he ime-sep sie ad he maimum ime-sep sie saisfied wih he 3-D CFL codiio of coveioal mehod. Parameer equal o represes he ormalied phase veloci of sadard Coprigh 0 SciRes.
A Efficie Mehod o Reduce he umerical Dispersio i he IE- Scheme 35.00 ormalied phase veloci 0.999 IE- = 0.997 IE- =.000 IE- =.00 IE- =.004 IE- =.006 0 0 40 60 80 Thea(degree) ormalied phase veloci 0.994 0.99 0.99 IE- = IE- =.000 IE- =.00 IE- =.004 IE- =.006 0 0 40 60 80 Thea(degree) Figure. Whe CFL =, he ormalied phase velociies wih respec o agle for differe parameer ( 45 ). ormalied phase veloci.0005 0.9995 0.999 5 0.9975 IE- = 0.997 IE- =.000 IE- =.00 5 IE- =.004 IE- =.006 0 0 40 60 80 Thea(degree) Figure. Whe CFL =., he ormalied phase velociies wih respec o agle for differe parameer ( 45 ). ormalied phase veloci 0.999 0.997 0.995 0.994 IE- = IE- =.000 0.993 IE- =.00 IE- =.004 0.99 IE- =.006 0.99 0 0 40 60 80 Thea(degree) Figure 3. Whe CFL =, he ormalied phase velociies wih respec o agle for differe parameer ( 90 ). Figure 4. Whe CFL =., he ormalied phase velociies wih respec o agle for differe parameer ( 90 ). IE- mehod. For compariso, he ormalied phase veloci of coveioal mehod is also ploed i hese figures. I ca be see from hese figures ha he umerical dispersio error of he sadard IE- (=.000) scheme is larger ha ha of he coveioal mehod, especiall whe is close o 90. Whe CFL =., he dispersio error alog he ais ( 90, = 90 ) of sadard IE- mehod is almos 4 imes as ha of coveioal mehod. For he modified IE- mehod, he dispersio error is reduced as he value of parameer icrease. Uder he CFL =, wih =.004, he ormalied phase velociies of he modified IE- is almos he same as ha of he coveioal mehod for boh 45 ad 90 plaes. Apparel, he dispersio performace of IE- mehod ca be corolled b selecig parameer. owever, whe he value of eceeds he value.004, he ormalied phase velociies will eceed, which is o he performace we epec. So, selec a proper value for parameer is he ke facor for reducig he dispersio error of IE- mehod. I ca be easil decided b Equaio () umericall. 6. Coclusios A parameer opimied IE- mehod is preseed i his leer. The parameer is iroduced o miimie he dispersio error. The sabili aalsis shows ha his algorihm is also weakl codiioall sable. umerical eperimes show ha his algorihm ca dramaicall reduce he dispersio error wihou iroducig addiioal compuaioal cos. Coprigh 0 SciRes.
36 A Efficie Mehod o Reduce he umerical Dispersio i he IE- Scheme 7. Ackowledgemes This work was suppored b aioal aural Sciece Foudaios of Chia (o. 600039 ad 6050004), ad also suppored b he Research Fud for he Docoral Program of igher Educaio of Chia (009000030). REFERECES [] K. S. Yee, umerical Soluio of Iiial Boudar Value Problems Ivolvig Mawell s Equaios i Isoropic Media, IEEE Trasacios o Aeas ad Propagaios, Vol. 4, o. 5, Ma 966, pp. 30-307. [] A. Taflove, Compuaioal Elecrodamics, Arech ouse, orwood, 995. [3] J. Che ad J. Wag, A 3-D brid Implici-Eplici Scheme wih Weakl Codiioal Sabili, Microwave ad Opical Techolog Leers, Vol. 48, o. 3, March 006, pp. 9-94. doi:0.00/mop.898 [4] M. Wag, Z. Wag ad J. Che, A Parameer Opimied ADI- Mehod, IEEE Aeas ad Wireless Propagaio Leers, Vol., o., Februar 003, pp. 8-. doi:0.09/lawp.003.8583 Coprigh 0 SciRes.