Energy Identities of ADI-FDTD Method with Periodic Structure

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Applied Matheatics 5 6 65-73 Published Olie Februar 5 i SciRes. http://www.scirp.org/oural/a http://d.doi.org/.436/a.5.65 erg Idetities of ADI-FDTD Method with Periodic Structure Regag Shi aitia Yag 3 Departet of Applied Matheatics Dalia Uiversit of Techolog Dalia Chia College of Sciece Chia Uiversit of Petroleu Qigdao Chia 3 Departet of gieerig Mechaics Dalia Uiversit of Techolog Dalia Chia ail: srg83@63.co haitia@dlut.edu.c Received Jauar 5; accepted 8 Jauar 5; published 4 Februar 5 Copright 5 b authors ad Scietific Research Publishig Ic. This work is licesed uder the Creative Coos Attributio Iteratioal Licese (CC BY). http://creativecoos.org/liceses/b/4./ Abstract I this paper a ew kid of eerg idetities for the Mawell equatios with periodic boudar coditios is proposed ad the proved rigorousl b the eerg ethods. B these idetities several odified eerg idetities of the ADI-FDTD schee for the two diesioal (D) Mawell equatios with the periodic boudar coditios are derived. Also b these idetities it is proved that D-ADI-FDTD is approiatel eerg coserved ad ucoditioall stable i the discrete L ad ors. periets are provided ad the uerical results cofir the theoretical aalsis o stabilit ad eerg coservatio. Kewords Stabilit erg Coservatio ADI-FDTD Mawell quatios. Itroductio The alterative directio iplicit fiite differece tie doai (ADI-FDTD) ethods proposed i [] [] are iterestig ad efficiet ethods for uerical solutios of Mawell equatios i tie doai ad cause a researchers work sice ADI-FDTD overcoes the stabilit costrait of the FDTD schee [3]. For eaple it was proved b Fourier ethods i [4]-[8] that the ADI-FDTD ethods are ucoditioall stable ad have reasoable uerical dispersio error; Referece [9] studied the divergece propert; Referece [] studied ADI-FDTD i a perfectl atched ediu; Referece [] gave a efficiet PML ipleetatio for the ADI-FDTD ethod. B Potig s theore erg coservatio is a iportat propert for Mawell equatios ad good uerical ethod should cofor it. I Gao [] proposed several ew eerg idetities of the two diesioal (D) Mawell equatios with the perfectl electric coductig (PC) boudar codi- ow to cite this paper: Shi R.G. ad Yag.T. (5) erg Idetities of ADI-FDTD Method with Periodic Structure. Applied Matheatics 6 65-73. http://d.doi.org/.436/a.5.65

R. G. Shi. T. Yag tios ad proved that ADI-FDTD is approiatel eerg coserved ad ucoditioall i the discrete L ad ors. Is there a other structure which ca keep eerg coservatio for Mawell equatios? Is there a other eerg idetit for ADI-FDTD ethod? This two iterestig questios proote us to fid other eergcoservatio structure. I this paper we focus our attetio o structure with periodic boudar coditios ad propose eerg idetities i L ad ors of the D Mawell equatios with periodic boudar coditios. We derive the eerg idetities of ADI-FDTD for the D Mawell equatios (D-ADI-FDTD) with periodic boudar coditios b a ew eerg ethod. Several odified eerg idetities of D-ADI-FDTD i ters of the discrete L ad ors are preseted. B these idetities it is proved that D-ADI-FDTD with the periodic boudar coditios is ucoditioall stable ad approiatel eerg coserved uder the discrete L ad ors. To test the aalsis eperiets to solve a siple proble with eact solutio are provided. Coputatioal results of the eerg ad error i ters of the discrete L ad ors cofir the aalsis o the eerg coservatio ad the ucoditioal stabilit. The reaiig parts of the paper are orgaied as follows. I Sectio eerg idetities of the D Mawell equatios with periodic coditios i L ad ors are first derived. I Sectio 3 several odified eerg idetities of the D-ADI-FDTD ethod are derived the ucoditioal stabilit ad the approiate eerg coservatio i the discrete L ad ors are the proved. I Sectio 4 the uerical eperiets are preseted.. erg Coservatio of Mawell quatios ad D-ADI-FDTD Cosider the two-diesioal (D) Mawell equatios: ε ε ad µ t t t (.) i a rectagular doai with electric perittivit ε ad agetic pereabilit μ where ε ad μ are positive costats; ( ( t ) ( t )) ad ( t ) deote the electric ad agetic fields t ( T] ( ) Ω [ a] [ b]. We assue that the rectagular regio Ω is surrouded b periodic boudaries so the boudar coditios ca be writte as ( ) ( ) ( ) ( ) ( ) ( ) t at t bt t at (.) ( ) ( ) ( ) ( ) ( ) ( ) t b t t a t t b t. (.3) We also assue the iitial coditios ( ) ( ) ( ) ( ) ( ) ( ) ( ). (.4) It ca be derived b itegratio b parts ad the periodic boudar coditios (.)-(.3) that the above Mawell equatios have the eerg idetities: Lea. Let ( ( t ) ( t )) ad ( t ) ( t ) be the solutio of the Mawellsstes (.)-(.4). The ( ) ( ) (.5) where ad i what follows deotes the L or with the weights ε (correspodig electric field) or µ (agetic field). For eaple ( ) ( ) ( ) ( ) a b ε ( ) t t t t t dd. (.6) Idetit (.5) is called the Potig Theore ad ca be see i a classical phsics books. Besides the above eerg idetities we foud ew oes below. Theore. Let ad ( t ) be the solutio of the Mawell sstes (.)-(.4) the sae as those i Lea.. The the followig eerg idetities hold 66

R. G. Shi. T. Yag ( ) ( ) u u u u (.7) ( ) ( ) ( ) ( ) t t (.8) where u or ad is the or (the or of f is defied b f f f where f f f f f f f. f is also called the -sei or of f). Proof. First we prove quatio (.7) with u. Differetiatig each of the quatios i (.) with respect to leads to ε ε ad µ. t t t B the itegratio b parts ad the periodic boudar coditios (.)-(.3) we have where dd a b a b dd rt dd a b a b dd ( ) (.9) (.) b r ( at ) ( at ) ( t ) ( t ) d. (.) Multiplig the quatios (.9) b ε ε ad µ respectivel itegratig both sides a b ad usig (.) we have over [ ] [ ] d r. dt Fro (.) ad the boudar coditios (.)-(.3) we ote that ( t ) li ( t ) li ε µ ( t ) t t li ε µ li a t t a ( at ) ( at ). So r( t ). The b itegratig (.) with respect to tie over [ ] (.) (.3) T we get equatio (.7) with u. Siilarl the idetit (.7) with u ca be proved. Cobiig (.5) ad (.7) leads to (.8). The D-ADI-FDTD Schee The alteratig directio iplicit FDTD ethod for the D Mawell equatios (deoted b D-ADI-FDTD) was proposed b (Naiki 999). For coveiece i aalsis of this schee et we give soe otatios. Let i t tt t i i i i I a J b N N t T. I where Δ ad Δ are the esh sies alog ad directios is the tie step I J ad N are positive itegers. fαβ f α β t defie For a grid fuctio ( ) J 67

R. G. Shi. T. Yag f f f f α β α β αβ αβ fαβ fαβ where u or t. For ( V V i i ) o the Yee staggered grids (Yee 966) f αβ fαβ t αβ u v αβ u v αβ ( ) ( ) f f f u u V W i v defie soe discrete eerg ors based I J I J I J ε ε µ i i i i i i V V v V V v W W v ( i ) I J I J V V V V ε V v V ε V i i Other ors: V ad V are siilarl defied. Deote b the approiatios of ( u α β t ) (u ) ad ( t ) schee for (.) is writte as Stage : i u αβ v. ad αβ α β respectivel. The the D-ADI-FDTD i i i ε (.4) i i i ε (.5) i i µ i i (.6) Stage : i i i ε (.7) i i i ε (.8) i i µ i i (.9) For siplicit i otatios we soeties oit the subscripts of these field values without causig a abiguit. B the defiitio of cross product of vectors the boudar coditios for (.)-(.3) becoe 68

R. G. Shi. T. Yag i i J I I I i ij i ij I I i i J i i J where or. Fiall the iitial values obtaied b the iitial coditio (.4). ad αβ αβ (.) of thed-adi-fdtd schee are 3. Modified erg Idetities ad Stabilit of D-ADI-FDTD i Nor I this Sectio we derive odified eerg idetities of D-ADI-FDTD ad prove its eerg coservatio ad ucoditioal stabilit i the discrete or. Theore 3. Let > ( ) i i ad be the solutio of the ADI-FDTD schee i (.4)-(.9). The the followig odified eerg idetities hold ( ) ( ) (3.) (3.) where for u ad or Proof. First we prove (3.). Applig tie levels we have u u u u u u. to the quatios (.4)-(.9) ad rearragig the ters b the i i i (3.3) ε i ε i i (3.4) i µ i i µ i (3.5) t i i i (3.6) ε t i i ε i (3.7) t t. i µ i i µ i (3.8) Multiplig both sides of the equatios (3.3)-(3.4) b ε respectivel ad those of (3.5) b µ ad takig the square of the updated equatios lead to 69

R. G. Shi. T. Yag i i i i i t ( ) ( ) ε ε µ t i i i i i ε µ ε i i i i µ ε t i i i i µ ε Applig suatio b parts we see that I J I I J i i ij i i i i i I I J i i i i i (3.9) (3.) (3.) (3.) I J i i where we have used that ad that i ij which ca be obtaied fro the periodic boudar coditios. Siilarl we get that ij i I J I J i i i i. (3.3) So if suig each of the qualities (3.9)-(3.) over their subscripts addig the updated equatios ultiplig both sides b ΔΔ ad usig the two idetities (3.) ad (3.3) together with the ors defied i Subsectio. we arrive at ( ) Siilar arguet is applied to the secod Stage (3.6)-(3.8) we have ( ).. (3.4) (3.5) Cobiatio of (3.4) ad (3.5) leads to the idetit (3.). Idetit (3.) is siilarl derived b repeatig the above arguet fro the operated quatios (.4)-(.9) b. This copletes the proof of Theore 3.. I the above proof if takig as the idetit operator we obtai that Theore 3. Let > ad be the solutio of D-ADI-FDTD. The the followig eerg idetities hold 7

( ) ( ) R. G. Shi. T. Yag (3.6) Cobiig the results i Theores 3. ad 3. we have Theore 3.3 If the discrete sei-or ad or of the solutio of D-ADI-FDTD are deoted respectivel b the the followig eerg idetities for D-ADI-FDTD hold ( ) ( ). 4 4 (3.7) µε µε ( ) ( ). 4 4 (3.8) µε µε Reark 3.4 It is eas to see that the idetities i Theores 3. 3. ad 3.3coverge to those i Lea. ad Theore. as the discrete step sies approach ero. This eas thatd-adi-fdtd is approiatel eerg-coserved ad ucoditioall stable i the odified discrete for of the L ad ors. 4. Nuerical periets I this sectio we solve a odel proble b D-ADI-FDTD ad the test the aalsis of the stabilit ad eerg coservatio i Sectio 3 b coparig the uerical solutio with the eact solutio of the odel. The odel cosidered is the Mawell equatios (.) with ε µ t T ad its eact so- Ω [ ] [ ] ( ] lutio is: cos π( ) π t. ( ) It is eas to copute the ors of this solutio are ( ) ( ) 4.. Siulatio of the rror ad Stabilit ( ) ( π ) 8. To show the accurac of D-ADI-FDTD we defie the errors: ( ) ( ) ( ) where ( t ) ( t ) ( ) t t ad t i i i i i i i i i t are the true values of the eact solutio. Deote the error ad relative error i the ors defied i Sectio 3 b rl R-rL r ad R-r i.e. rl - R rl ( ) rl 7

R. G. Shi. T. Yag r ( rl ) ( ) ( h) ( h ) r rror R- r Rate log rror where log is the logarithic fuctio. Table gives the error ad relative error of the uerical solutio of the odel proble coputed b D- ADI-FDTD i the ors ad the covergece rates with differet tie step sies Δt 4h h ad h whe Δ Δ h. is fied ad T. Fro these results we see that the covergece rate of D-ADI-FDTD with respect to tie is approiatel ad that D-ADI-FDTD is ucoditioall stable (whe Δt Δ Δ h the CFL uber c t > ). Table lists the siilar results to Table whe Δt.h is fied Δ Δ varies fro h h ad.5h ad the tie legth T. Fro the colus Rate we see that D-ADI-FDTD is of secod order i space uder the discrete L ad or. 4.. Siulatio of the erg Coservatio of D-ADI-FDTD I this subsectio we check the eerg coservatio of D-ADI-FDTD b coputig the odified eerg ors derived i Sectio 3 for the solutio to the schee. Deote these odified eerg ors b ( ) ( ) u u Iu u u u u u u ( ) ( ) I ( ) ( ) ( ) ( ) I I I I. I Table 3 are preseted the eerg ors ( I ) ( u ) u of the solutio of the D-ADI-FDTD schee at the tie levels ad 4 (the third to fifth rows) ad the absolute values of their differece (the last two rows) where the sies of the spatial ad tie steps are Δ Δ. Δt.4. The secod row shows the four kid of eergies of the eact solutio coputed b usig the defiitios of u ( ) I u. Fro these value we see that D-ADI-FDTD is approiatel eerg-coserved. Table. rror of ( ) i L ad with Δ Δ h ad differet Δt. Δt R-rL rl Rate R-r r Rate 4h 6.84e 8.554e 6.87e 7.6675e h.664e.3e.89.665e.595e.89 h 5.57e 3 7.93e 3.657 5.57e 3 6.59e.38 Table. rror of ( ) i L ad with Δt.h ad differet spatial step sies. Δ Δ R-rL rl Rate R-r r Rate h 5.9e 3 8.38e 3 5.9e 3 7.4333e 3 h.498e 3.86e 3.7393.498e 3.894e 3.7393.5h 4.e 4 5.685e 4.8979 4.e 4 5.834e 4.8978 7

R. G. Shi. T. Yag Table 3. erg of ( ) ad its error whe Δ Δ h. Δt 4h ad 4. Fields\Nors I ( ) I ( ) I ( ) I ( ) ( ) 8.9367 8.9367.46.783 ( ) 8.9367 8.9367.46.783 ( ) 8.9367 8.9367.46.783 ( 4 4 ) 8.9367 8.9367.46.783 ( ) ( ) 3.685e 3 3.685e 3 5.43e 4 4.678e 3 4 4 ( ) ( ) 3.685e 3 3.685e 3 5.43e 4 4.678e 3 5. Coclusio I this paper the odified eerg idetities of the D-ADI-FDTD schee with the periodic boudar coditios i the discrete L ad ors are established which show that this schee is approiatel eerg coserved i ters of the two eerg ors. B the derivig ethods for the eerg idetities ew kid of eerg idetities of the Mawell equatios are proposed ad proved b the ew eerg ethod. Nuerical eperiets are provided ad cofir the aalsis of D-ADI-FDTD. Refereces [] Naiki T. (999) A New FDTD Algorith Based o Alteratig-Directio Iplicit Method. I Trasactios o Microwave Theor ad Techiques 47 3-7. http://d.doi.org/.9/.79575 [] Zheg F. Che Z. ad Zhag J. () Toward the Developet of a Three-Diesioal Ucoditioall Stable Fiite- Differece Tie-Doai Method. I Trasactios o Microwave Theor ad Techiques 48 55-558. http://d.doi.org/.9/.8697 [3] Yee K. (966) Nuerical Solutio of Iitial Boudar Value Probles Ivolvig Mawell s quatios i Isotropic Media. I Trasactios o Ateas ad Propagatio 4 3-37. http://d.doi.org/.9/tap.966.38693 [4] Naiki T. ad Ito K. () Ivestigatio of Nuerical rrors of the Two-Diesioal ADI-FDTD Method [for Mawell s quatios Solutio]. I Trasactios o Microwave Theor ad Techiques 48 95-956. http://d.doi.org/.9/.883876 [5] Zheg F. ad Che Z. () Nuerical Dispersio Aalsis of the Ucoditioall Stable 3D ADI-FDTD Method. I Trasactios o Microwave Theor ad Techiques 49 6-9. http://d.doi.org/.9/.965 [6] Zhao A.P. (4) Cosistec of Nuerical Dispersio Relatio pressed i Differet Fors for the ADI-FDTD Method. Microwave ad Optical Techolog Letters 4-3. http://d.doi.org/./op.7 [7] Garcia S.G. Rubio R.G. Bretoes A.B. ad Marti R.G. (6) O the Dispersio Relatio of ADI-FDTD. I Microwave ad Wireless Copoets Letters 6 354-356. http://d.doi.org/.9/lmwc.6.87569 [8] Fu W. ad Ta.L. (7) Stabilit ad Dispersio Aalsis for ADI-FDTD Method i Loss Media. I Trasactios o Ateas ad Propagatio 55 95-. http://d.doi.org/.9/tap.7.893378 [9] Sithe D.N. Car J.R. ad Carlsso J.A. (9) Divergece Preservatio i the ADI Algoriths for lectroagetics. Joural of Coputatioal Phsics 8 789-799. http://d.doi.org/.6/.cp.9.6.5 [] Gede S.D. Liu G. Rode J.A. ad Zhu A. () Perfectl Matched Laer Media with CFS for a Ucoditioal Stable ADI-FDTD Method. I Trasactios o Ateas ad Propagatio 49 554-559. http://d.doi.org/.9/8.9649 [] Wag S. ad Teieira F.L. (3) A fficiet PML Ipleetatio for the ADI-FDTD Method. I Microwave ad Wireless Copoets Letters 3 7-74. http://d.doi.org/.9/lmwc.3.8875 [] Gao L. () Stabilit ad Super Covergece Aalsis of ADI-FDTD for the D Mawell quatios i a Loss Mediu. Acta Matheatica Scietia 3 34-368. http://d.doi.org/.6/s5-96()684-73

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