Operator Splitting Method for Coupled Problems: Transport and Maxwell Equations

Transcription

1 America Joural of Compuaioal Mahemaics,,, 6-75 doi:46/acm9 Published Olie Sepember (hp://wwwscirporg/oural/acm) Operaor Spliig Mehod for Coupled Problems: Traspor ad Mawell Equaios Absrac Jürge Geiser Deparme of Mahemaics, Humbold-Uiversiä u Berli, Uer de Lide, Berli, Germa geiser@mahemaikhu-berlide Received March, ; revised April, ; acceped Ma, I his aricle a ew approach is cosidered for implemeig operaor spliig mehods for raspor problems, iflueced b elecric fields Our moivaio came o model PE-CVD (plasma-ehaced chemical vapor deposiio) processes, meas he flow of species o a gas-phase, which are iflueced b a elecric field Such a field we ca model b wave equaios The mai coribuios are o improve he sadard discreiaio schemes of each par of he couplig equaio So we discuss a improveme wih implici Ruge- Kua mehods isead of he Yee s algorihm Furher we balace he solver mehod bewee he Mawell ad Traspor equaio Kewords: Operaor Spliig Mehod, Iiial Value Problems, Ieraive Solver Mehod, Sabili Aalsis, Beam Propagaio Mehods, Traspor ad Mawell Equaios Iroducio We moivae our sud b simulaig hi film deposiio processes ha ca be realied b PE-CVD (plasma ehaced chemical vapor deposiio) processes, see [,] For he deposiio process, he ifluece of he elecric fields o he raspored gases i a plasma reacor is ver impora, see [] Therefore we deal wih a simplified model of a coupled raspor ad Mawell equaios While he raspor equaios modeled he raspor of gaseous species ad he Mawell equaio he ifluece of he uderlig flow field We deal wih he followig equaios u u u u ve, v D u D, () u,, u,, () H, E,,,, T, H, E,,,, T, () (4) E, H H J source,,,,, T where u is he coceraio of he gaseous species, E is he elecric field ad H, H is he correspodig mageic field i wo dimesios Furher v v,v is he iflueced veloci of he raspor equaio We cocerae o he umerical modelig ad simulaio of elecrical fields, which are coupled wih raspor equaios Several mehods eis o solve elecric field ad are of ieres Oe mehod for a saioar case of he elecric field is a propagaio mehod (BPM) This is a powerful ool o aale liear ad oliear ligh propagaio i aiall varig waveguides like direcioal couplers, apered waveguides, S-shaped be waveguides, ad opical fibers [4-7] The mehod has is origi i he field of propagaio of elecromageic beams i amosphere, where he muli-phsics modelig was doe o he assumpio ha he coiuous gai medium ma be approimaed b a series of gai shees wih free propagaio bewee he shees [8,9] As i will be show laer o, his mehod is i fac a Srag-Marchuk operaor spliig mehod [,] Here we firs describe he BPM [] We iroduce he ieraive spliig idea o couple (5) Coprigh SciRes

2 64 J GEISER Mawell ad Traspor equaios Furher a spliig aalsis is preseed Numerical eperimes are preseed wih respec o decoupled ad coupled differeial equaios The paper is orgaied as follows The discreiaio mehods are described i Secio I Secio, he applied operaor spliig mehods are preseed The error aalsis of he coupled mehods is sudied i Secio 4 The eperimes of he ew discreiaio mehods ad spliig mehods are performed i Secio 5 A he ed of his paper we iroduce fuure works Discreiaio Mehod of he Mawell Equaio I he followig we discuss he discreiaio mehods for he Mawell equaio FDTD Mehod: Yee s Scheme Yee s scheme is he sadard fiie differece ime-domai (FDTD) discreiaio of he followig ime depede Mawell curl equaios H r E, (6) E r H, (7) where E E, E, E,, is he elecric field, H H, H, H,, is he mageic field, r, is he relaive permiivi (give daa), r (omageic maerial) is he mageic permeabili Here, are cosas I ca be show ha if he divergece free codiios r E ad H are saisfied a, he he are saisfied for all ime This is he case for our seig Therefore i is eough o cosider ol he above curl equaio Rewriig hem compoe-wise, we ge i our case H E E (8) H E E (9) E H H r () Le, are spaial discreiaios, ad ime sep We use he followig oaio,,, F i F i is a () Le represes a spaial coordiae such as, The goal of Yee s scheme is o compue he approimaios for he various compoes E of E ad H of H a he followig spaial locaios ad emporal isas: spaial coordiae : half ieger E oher spaial coordiae : ieger () ime : ieger spaial coordiae : ieger H oher spaial coordiae : half ieger () ime : half ieger Thus he disribuios/grid of various compoes are saggered i space ad i ime This is oe of he wo uique characerisics of he Yee s scheme The secod uique characerisic is ha he various spaial derivaives i Equaios (8) - () are compued across he oe spaial cell, ie he differece ceer for he ceral differece approimaio of he spaial derivaive is he mid poi of oe cell legh i he correspodig direcio of he derivaive Thus he Yee s scheme approimaes Equaios (8) - () a he followig pois: Equaio (8) i,, (4) Equaio (9) i,, (5) Equaio () i,, (6) Such a saggered ucollocaed arrageme gives he Yee s scheme several ice umerical ad phsical properies, see [] The we ge fiie-differece approimaios as: H i, E, H i, E i, i, H i,,, E H i, E i i, E i, E i, H i, H i, r, H i H i, r (7) (8) (9) Coprigh SciRes

3 J GEISER 65 I Equaio (9) he relaive permiivi r is compued a he correspodig differece ceer as give b Equaio (6) A he ierface bewee wo media, r is approimaed b he average value Codiios for he Yee s algorihm: The CFL sabili codiio for he Yee s FDTD mehod is c () where c is he speed of ligh i vacuum, see [] To resric he ubouded domai o fiie domai, oe uses absorbig boudar codiio like he perfecl mached laers, see [4,5] Remark Ofe for more accurae problems a Yee s algorihm which is secod order i ime ad secod order i space is ofe o low For higher order mehods i ime ad space ca be cosruced bu are ofe o delicae ad epesive o impleme, see [6,7] We propose o improve wih higher order implici Ruge-Kua mehods wih a idea o sparse marices schemes, which saves addiioal memor Improved Time Discreiaio Mehods for Mawell Equaio Based o he problem of recosrucig a higher order Yee s algorihm, we deal wih separae improveme of he discreiaio schemes While he spaial discreiaio of he Yee s algorihm is a secod order differece scheme, he ime discreiaio is also ol a secod order scheme Here we see he deficis of ol improvig he spaial scheme wih higher order schemes ad leave he imediscreiaio wih a secod order scheme We propose a improved ime-discreiaio scheme of higher order ad appl fie spaial grids, while he ime error is a leas larger, see [8] We deal wih higher order ime-discreiaio mehods Therefore we propose he Ruge-Kua as adaped imediscreiaio mehods o reach higher order resuls For he ime-discreiaio we use he followig higher order discreiaio mehods We deal wih he followig semi-discreied parial differeial equaios, such equaios are used i each ieraive spliig sep: u Au f, () u u, () where A is he operaor ha we implici solve i he equaio ad f Bu is he eplici operaor, wih a previous soluio u, eg las ieraive soluio Higher Order Time-Discreiaio Mehods wih Ruge-Kua Mehods We deal wih he followig Mawell equaio, give as: E H H J () I H, H, J BH BH J H E II E CE (4) H E III E CE (5) For he boudar codiios we assume periodic boudar codiios Tha meas we use he ideificaio E N, i E, i (6) E i, N E i, i,, N (7) Remark For he saioar field, we appl a periodic boudar codiio, which is sufficie The Mur absorbig boudar codiio, see [5], is used for he isaioar field, while respecig he ifluece of he chages a he boudaries To ge a firs realiaio of a ope boudar i he case of he lie-source we use smmer ad a combiaio of PBC ad Mur s firs order ABC For he boudars orhogoal o he propagaio direcio of he field (lef-righ) i is useful o work wih Mur s ABC Mur s ABC We ca ierpre he elecromageical field as a wave ha has o fulfill he homogeeous wave equaio r r c c (8) (9) c D D D c D D c D D c D c D D c D () () Coprigh SciRes

4 66 J GEISER D D V D D V c () ( ) c () Waves ha saisf ol propagae i -direcio ad hose ha saisf ol propagae i -direcio A aalogous formulaio ca be give for he ad direcio To hadle i is comforable o do a Talor epasio aroud V V V V 4 V OV V V V 5 4 V O V OV V (4) (5) (6) Cosiderig (6) equaio () urs o, c c (7) which is Mur s ABC wih firs order accurac As a firs aemp o model a ope boudar we will use his Lef boudar ( ) For he lef boudar we have do discreie he followig equaio: (8) c This ca be doe wih a FDM-scheme as follows wih ad,, ; (9),,,,, ;,,,,,,,,, ; (4) (4) his leads o c c,,,, (4) I is eas o see ha his ool does o saisf compleel because i ol has firs order accurac ad eve more impora i ol absorbs he par of he wave ha propagaes orhogoal o he boudar Bu here are also a few advaages Mur s ABC has o be applied ol o he E field because H ad H are deal wih auomaicall hrough he ordiar updaesep The secod advaage of Mur's ABC is he low umeric epese For he boudaries parallel o he propagaio direcio (op ad boom) we use he PBC The s mmer of our seig garaies ha he iflow ad he ouflow of he field equalie each oher Bu wih he ee o he e simulaios wih less smmer i seams o be ecessar o use perfecl mached laers These equaios above mark he sarig poi The spaial par of each equaio is discreised ad is calculaed wih he help of he mari-operaors B, B, C, C (ceered differeces correspodig o he dimesioal Yee-larice) I he followig we are usig he geeral Bucher-able for (-sage) Ruge-Kua-mehods o ge a clear oaio (4) Le deoe he seppig ime ad c i i The se p from o i () - ( 5) ca ow be wrie i he followig wa i i i I I I i i II II II i i III III III k bk bk E E bk b k b k i (44) H H bk b k b k (45) H H b i i i i where M,,, (46) k M H H E J for M I, II, III ad,, Wih i I i i E E E alkl l II a kl (47) i i i H H H l (48 ) l H H H a k i i i III l l l (49) Coprigh SciRes

5 i i J J which is kow (i our ca se) (5) For a beer legibili ad because he focused poi of ime does o chage, we wrie E, H, H, J isead of E, H, H, J i i i i Combiig () - (5) ad (47) - (5) give 9 equaios l l l E E al BH BH J l (5) l H H al CE (5) l l H H al CE,, l Remark ad are also realied as marices, such ha, ad, Remark 4 The scheme above is ol correc for isoropic media because i he o isoropic case i is ecessar o cosider, Takig he 6 equaios of (5) ad (5) ad puig hem ogeher w ih he equaios of (5) lead s o he followig liear equaio ssem which eeds o be solved J GEISER 67 (5) Q E Q E Q E A A A A A A R, C S I R, C S R, C S A A A A A A R, C S R, C S I R, C S A A A A A A R, C S R, C S R, C S I E E E (54) Q E as I as as E Q E as as I as E (55) Q E as as as I E Q E S I E Q E S I E (56) Q E S I E A A Q : R, BH BH R, J (57) R : -h rowof A (58) A A C : -h colum of A (59) :,, (6) J : J, J, J (6) S : BC i i i B C (6) A A a : A R, C (6) a a a a : a a a a a a i ai m imes a i mimes (64) (65) I : Idei m (66) wher e a i, bi ad c are he Ruge-Kua coefficies To be more precisel: If we have m pois i our regio, here are m equaios o solve Wih his resul we are able o calculae (5) ad (5) So ha i is fiall possible o do he sep ((44) - (46)) Remark 5 For a opimiaio of he ime-discreiaio scheme, we ca eglec some of he ouerdiagoals of he RK mehods, which leads o S DIRK mehods We have he beefi i faser compuaios, wihou reducig he accurac For higher ime-discreiaios we have o ake io accou also higher spaial discreiaio scheme Sabili Aalsis of he Implici Discreiaios We deal wih he followig discreied equaio ssems: E I A E C H C H J u J u i, i (67) where i is he ieraio ide of he couplig scheme Defiiio We have a posiive defiie mari M T ( real smmeric mari), if A > for all o-ero vecors wih real eries, where T deoes he raspose of Eample For fiie differece discreiaio, eg, i is sufficie o show, ha he sum of he ouer-diagoals are equal or less ha he diagoal a i, aii for i,, ad is he umber of discreiaio pois We have he followig assumpios: Assumpio ) We assume A is posiive defiie, ad herefore we have Coprigh SciRes

6 68 J GEISER see [9] ) We assume I A (68) E C H C H J u J u E (69) i The sabili is give wih i he followig Theorem: Theorem Give is he umerical scheme (7) ad we have he assumpios The scheme is sable for all ieraive seps i Proof Based o he assumpios we ca boud he iverse mari, also he previous soluio is bouded The scheme is sable We have he followig proof idea: Based o he assumpio, A is posiive defiie ad he esimaio of he remaiig erm, we have : E (7) i, E So we have a upper boud of he ieraive resuls, give b he previous soluio a ime Discreiaio Mehods of he Covecio-Diffusio Equaio For he dimesioal covecio-diffusio equaio we appl a secod order fiie differece scheme i space ad a higher order discreiaio scheme i ime u vudu, u u u u v v v D u u D D, u, u, We appl dimesioal spliig o our problem u A u A u Au where u u A v D We use a s order upwid scheme for ad a d order ceral differece scheme for B iroducig he arificial diffusio cosa D D v we achieve a d order fii e differece scheme uu Lu v uuu D because he ew diffusio cosa elimiaes he firs order error (ie he umerical viscosi) of he Talor epasio of he upwid scheme Lu ad Lu are derived i he same wa For he discreiaio i ime we use several eplici Ruge-Kua ad Adam-Bashforh mehods, his leads o resricios of he sep-sie i ime bu o he oher had he cos of implici mehods is much o high i his -dimesioal case Adam-Bashforh Mehods s h b f, b (7) s u id u,,, s (7)! s! i, i We cosider here s (firs order) h f, f, (7) ad s (secod order) 6 h f, f, (74) 5 f, Eplici Ruge-Kua Mehods I geeral a s-sage Ruge-Kua mehod ca be wrie i he followig wa: where s k h b (75) s k f hc, h alkl (76) l We will ake io accou he followig wo: Heu s hird-order ad Kua s classical fourh-order (77) Coprigh SciRes

7 J GEISER 69 (78) Spliig Mehods o Couple Mawell ad Covecio Diffusio Equaio We cocerae o he spliig mehods, which ca be classified as classical ad ieraive spliig mehods We propose ieraive spliig mehods b discussig he addiive ieraive spliig mehods, see [,] We cosider he followig he liear problem c Ac Bc, where he iiial codiios are (79) c c The opera- A ad B are spaiall discreied operaors, eg he ors correspod i space o he discreied covecio ad diffusio operaors (marices) Hece, he ca be cosidered as bouded operaors Ieraive Spliig Mehods The followig algorihm is based o he ieraio wih fied spliig discreiaio sep sie O he ime ierval, we solve he followig subproblems cosecuivel for i,,,m, cf [,] c c Ac Bc, wih c c, i i i i i (8) Aci Bci, wih ci c,(8) where c ad c is he kow spli approimaio a ime level The spli approimaio a ime level efied as is d c c m (Clearl, he fucio ci depeds o h e ierval,, oo, bu for he sake of simplici, i our oaio we omi he depedece o ) I he followig we aale he covergece ad he rae of he covergece of he mehod (8) - (8) for m edig o ifii for he liear operaors A, BX : X, where we assume ha hese ope raors ad heir sum are geera ors of he C semigroups We emphasie ha hese operaors are' ecessaril bouded, hus he covergece is eamied i a geeral Baach space seig Theorem Le us cosider he absrac Cauch prob- lem i a Baach space X c c c Ac Bc, < T,, (8) where A, BA, BX : X are give liear operaors beig geeraors of he C semigroup ad c X is a give eleme The he ieraio process (8) - (8) is coverge ad he rae of he covergece is of higher order The proof ca be foud i [] Remark 6 Whe A ad B are marices (ie (8) - (8) is a ssem of ordiar differeial equaios), for he growh esimaio we ca use he cocep of he logaclasses of marices we ca prove he validi rihmic orm, see eg [] Hece, for ma impora Remark 7 We oe ha a huge class of impora differeial operaors geerae a coracive semigroup This meas ha for such problems-assumig he eac solvabili of he spli subproblems-he ieraive spliig mehod coverges i higher order o he eac soluio I he e subsecio we prese he used ime-discreiaio mehods 4 Error Aalsis: Couplig Mehods For he couplig mehods we deal wih oliear differeial equaios of he followig pe: d c, wih A c c B c c c c, (8) d where c H, H, E, u, wih H, H is he mageic field, E is he elecric field ad u is he coce- of he species raio The mai idea is o boud he operaors Ac ad Bc i he discreied equaio o saisf a sable mehod A firs idea is he fi-poi scheme, ha is discussed i he followig subsecio Ieraive Operaor-Spliig Mehod as a Fi-Poi Scheme The ieraive operaor-spliig mehod is used as a fi-poi scheme o liearie he oliear operaors, see [,4] We resric our aeio o ime-depede parial differeial equaios of he form: d u, A u u B u u wih u c, (84) d where Au, Bu : X X are liear ad desel defied i he real Baach space X, ivolvig ol spaial derivaives of c, see [5] I he followig we discuss he sadard ieraive operaor-spliig mehods as a fipoi ieraio mehod o liearie he operaors Coprigh SciRes

8 7 J GEISER We spli our oliear differeial equaio (84) b applig: du i d i i i i A u u B u u, wih u c, du i d i i A u u B u u, wih u c, i i i i (85) (86) where he ime sep is The ieraios are i,,, m u c is he sarig soluio, where we assume he soluio c is ear c, or u So we have o solve he local fi-poi probime lem c is he kow spli approimaio a he level The spli approimaio a ime level is defied as c u m We assume he operaors Aui, Bui : X X o be liear ad desel de- fied o he real Baach space X, for i,,, m Here he lieariaio is doe wih respec o he ieraios, such ha Au, i Bui are a leas o-de- equaios, ad we ca pede operaors i he ieraive appl he liear heor The lieariaio is a leas i he firs equaio Aui Aui, ad i he secod equaio Bui Bui We have i i Au uu Au u wih sufficie ieraios i,,,m Remark 8 The lieariaio wih he fi-poi scheme ca be used for smooh or weak oliear operaors, oherwise we loose he covergece behavior, while we did o coverge o he local fi-poi, see [] 5 Eperimes I he followig eperimes, firs we deal wih he decoupled equaios, meas Mawell ad raspor equaios, o verif our mehods I he hird eperime, we cosider a sim ple PE-CVD process ad cocerae o he coupled raspor ad Mawell equaio 5 Tes Eperime : Mawell Equaio The ime-depede Mawell equaios i D is give as: H, E,,,, T, (87) H, E,,,, T, E, H H J source,,,, T, where J, si source (88) (89) We have o impleme he ouflow codiio, via he uderlig discreiaio mehod (we assume fiie dif o he cell wih he spaial ferece mehods), meas how ma coceraio is flowig via he ime-sep sep : The relaive spaial sep is give as relaiv The perceage of he ouflow is give as: relaiv rel E, ou rele,, The same is also give for he H, H Here w e appl he FDTD mehod of Yee s algorihm For spaial ad ime discreiaio i is impora o balace such schemes We assume o have fiie differece schemes i ime ad space Therefore he CFL (Coura Friedrichs Lev) codiio is impora o balace he schemes: While we are dealig wih wave-equaios: where, are he spaial ad ime seps To corol he elecric field E,, we have he followig lie source: J, si source where,, The corol of he paricle raspor is give b he elecric field i Figure The elecric ad raspor siuaio is give wih cu of he hree dimesioal model i Figure I he followig we have he lie sources wih he resuls give i Figure : Remark 9 We cosider he Mawell equaio, ha models a periodic elecric field i he reacor We appl Yee s algorihm o obai a leas a secod order scheme i ime ad space Based o he slower ime-scales of he Mawell equaios, which is less siff ha he raspor equaios, we have sufficie accurac i he full coupled ssem A higher order discreiaio scheme is ecessar for he raspor par Coprigh SciRes

9 J GEISER 7 5 Tes eperime : Covecio-Diffusio Equaio Figure Elecric field i he apparaus Algorihm 44 ) Iiialie Covecio-Diffusio equaio, ill sar ) Solve Elecric Field equaio wih s, we obai ar, E for sar ) Solve Covecio Diffusio equaio wih ad use for sar for he ukow 4) Do sar sar ad go o ) ill sar ed sar Figure Elecric field i he apparaus E, Figure Lie source of he Elecric field i he apparaus We deal wih he -dimesioal advecio-diffusio equaio ad periodic boudar codiios u v udu, u u u u v v D D, u, u, wih he parameers v v D 5 The give advecio-diffusio problem has a aalical soluio v ua, ep 4D which we will use as a coveie iiial fucio:,, u u a We appl dimesioal spliig o our problem where u A u A u u u A v D We use a s order upwid scheme for ad a d order ceral differece scheme for B iroducig he arificial diffusio cosa D D v we achieve a d order fiie differece scheme u u Lu v u uu D because he ew diffusio cosa elimiaes he firs order error (ie he umerical viscosi) of he Talor epasio of he upwid scheme Lu is derived i he same wa We appl a BDF5 mehod o gai 5h order accurac i ime For simplificaios, we o e ha he depedecies of u, are suppressed as u 7 Lu u 5u 5u 6 (9) 5 u u u Coprigh SciRes

10 7 J GEISER To compare he four mehods we have he followig geeral seig Le,,,, he ui cube There we se up he iiial coceraio a u ep wih a 5,5,5 which is us he aalical soluio u a, ep T v 4D (9) (9) (9) wih v ad D a 5 o Durig he followig eperimes we will se v ad cosider a equidisa laice of N pois ( N ) The resul is show wihi he followig Figures 4 ad 5: Remark We cosider he raspor equaio, ha models he mass raspor of he ioied species from he lower-lef o he middle of he reacor We use higher order ime ad spaial discreiaio schemes o obai higher order soluios Such mehods, we ca appl wih larger ime ad spaial seps ad obai sufficie accurae resuls Based o he fas ime-scales of he raspor equaios, which is siffer ha he Mawell equaio, we ca balace he larger ime-seps wih sufficie accurae soluio of he raspor regime i he coupled ssem H, E,,,, T, T H, E,,,, T, E, H H J source,,,,, The advecio-diffusio problem has a aalical so luio a he begiig for u a, ep, sar v 4D which we will use as a coveie iiial fucio:,, u u a Furher he fucio: (, ) for, (, ) (, ) for v E sar v E E where, sar sar 5 Tes Eperime : Couplig Covecio-Diffusio ad Elecric Field Equaios (Weak Couplig) Here, we cosider a simple PE-CVD process, ha a uderlig mass raspor of a gaseous species is iflueced b a elecric field, see [,] For raspor i a plasma evirome, we assume a homogeeous medium ad ha he ifluece of he elecric field ca be simulaed b a coupled raspor ad Mawell equaio, see cielieb5 For simplificaios, we deal wih he -dimesioal advecio-diffusio equaio ad elecric field equaio:, u u u v E v u u D D, u,, u,, Figure 4 Iiial gaseous coceraio a 5 Figure 5 Gaseous coceraio a 5 Coprigh SciRes

11 J GEISER 7 Boh equaios have he same domai,, Numericall we solve he equaio, as i he followig algorihm 5: The followig figures show he developig of he coceraio uder he ifluece of he elecric field, we deal wih a ormalied ime scale i sec Furher we have 7, sar 5 ad v for sar The resuls are give i Figure 6 - Remark Based o he raspor of he ioied species from he lower-lef o he middle of he reacor, we see ad ifluece of he species The former circular coceraio is spread ou o a diffusive ellipse Here, we ca corol he species i he reacor wih a elecric field Numericall, i is impora o deal wih he differ- e ime- ad spaial scales of he uderlig raspor ad Mawell equaio Via ieraive spliig, we could couple he wo equaios ssems ogeher ad reduce he umerical errors wih addiioal ieraive seps Figur e 8 Gaseous coceraio afer 6 wih a firs ifluece of he elecric field Figure 6 Gaseous coceraio afer 8 Figure 9 Elecric field afer 6 Figure 7 Elecric field afer 8 Figure Gaseous coceraio afer 48 wih a firs ifluece of he elecric field Coprigh SciRes

12 74 J GEISER 6 Coclusios Figure Elecric field afer 48 We prese a coupled model based o Mawell ad Traspor equaios, ha ca be applied for simplified raspor model for a ioied gaseous species i a PECVD reaor Based he differe scale models, we have icluded he opimal discreiaio mehods for each separae equaio Spliig mehods are used o couple he separae equaios ogeher Furher, we dis- cussed he spliig aalsis Numerical eamples are preseed o discuss he ifluece of decoupled ad coupled ssems I fuure, we will aale he validi of he models wih phsical eperimes 7 Refereces [] M Ohrig, Maerials Sciece of Thi Films, d Ediio, Academic Press, Sa Diego, New York, Boso, Lodo, [] J Geiser ad M Arab, Simulaio of a Chemical Vapor Deposiio: Mobile ad Immobile Zoes ad Homogeeous Laers, Joural of Porous Media, Begell House Ic, Reddig, 9, Vol, No, pp -4 [] L Rudiak, Numerical Simulaio of Chemical Vapour Deposiio Process i Elecric Field, Compuers & Chemical Egieerig, Vol, Suppleme, 998, pp [4] J Va Roe, J va der Dok ad P E Lagasse, Beam-Propagaio Mehod: Aalsis ad Assessme, Joural of he Opical Socie of America, Vol 7, No 7, 98, pp 8-8 doi:64/josa78 [5] M D Fei ad J A Fleck Jr, Aalsis of Rib Waveguides ad Couplers b he Propagaig Beam Mehod, Joural of he Opical Socie of America A, Vol 7, No, 99, pp 7-79 doi:64/josaa77 [6] M D Fei ad J A Fleck Jr, Ligh Propagaio i Graded-Ide Opical Fibers, OSA Applied Opics, Vol 7, No 4, 978, pp doi:64/ao799 [7] L Thle, E M Wrigh, G I Segema, C T Seao ad J V Moloe, Beam-Propagaio Mehod Aalsis of a Noliear Direcioal Coupler, OSA Opics Leers, Vol, No, 986, pp doi:64/ol79 [8] J A Fleck, J R Morris Jr ad M D Fei, Time-depede Propagaio of High Eerg Laser Beams hrough he Amosphere, Applied Phsics, Vol, No, 976, pp 9-6 doi:7/bf896 [9] M La, J H Baeh ad G P Agrawai, Chaelig of Iese Elecromageic Beams, Joural of Applied Phsics, Vol 5, No, 98, pp 9-5 doi:6/ 844 [] G I Marchuk, Some Applicaios of Spliig-up Mehad Compario of Dif- ods o he Soluio of Mahemaical Phsics Problems, Aplikace Maemaik, Vol, 968, pp - [] G Srag, O he Cosracio ferece Schemes, SIAM J Numerical Aalsis, Vol 5, No, 968, pp doi:7/754 [] K Okamoo, Fudameals of Opical Waveguides, Academic Press, New York, 5 [] A Taflove, Compuaioal Elecrodamics: The Fiie Differece Time Domai Mehod, Arcech House Ic, 995 [4] J P Bereger, A Perfecl Mached Laer for he Absorpio of Elecromageic Waves, J Comp Phs, Vol, 5, pp 85- [5] S D Gede, A Aisoropic Perfecl Mached Laer-Absorbig Medium for he Trucaio of Fdd Laices, IEEE Tra A Prop, Vol 44, No, 996, pp 6-69 [6] W Sha, X Wu, M Che ad Z Huag, Applicaio of he High-Order Smplecic Fdd Scheme o he Curved Three-Dimesioal Perfecl Coducig Obecs, [7] T Hiroo, W Lui, S Seki ad Y Yoshikui, A Three- Dimesioal Fourh-Order Fiie-Differece Time-Domai Scheme Usig a Smplecic Iegraor Propagaor, [8] J Geiser, Numerical Simulaio of a Model for Traspor ad Reacio of Radiouclides,, Proceedigs of he Large Scale Scieific Compuaios of Egieerig ad Eviromeal Problems, Soopol, [9] W Hackbusch, Ieraive Losug Groser Schwachbeseer Gleichugssseme, Teuber-Verlag, Sugar, 99 [] I Farago ad J Geiser, Ieraive Operaor-Spliig Mehods for Liear Problems, Ieraioal Joural of Compuaioal Sciece ad Egieerig, Vol, No 4, 7, pp 55-6 [] J Kae, C Miller ad C Kelle, Covergece of Ieraive Spli-Operaor Approaches for Approimaig Coprigh SciRes

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