Numerical solution of the Maxwell equations as stated above is now a highly developed modeling technique as exemplified by Prof. He s lectures.
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1 Mh KH DN74 Compol lecromgecs Fll ' Lecres o Mwell Hperbolc Ssems p (9 Lecre Mhemcs of he Mwell eqos Refereces: flove Jckso J.C.Mwell 867: B J m ; B D H J e ; D ρ D B H lecrc Mgec Feld V/m H A/m Fl des D C/m B Wb/m Merl Permv Permebl Crre J e A/m J m V/m des For smple merls ρj e d ρ J m H wh ρ elecrc (Ohmc ressv σ /ρ wh σ codcv d ρ mgec ressv. No-relvsc No-qzed Smple merl behvor ssmed: Cf. ferro-mgecs: hseress ec. Sc (elecroscs mgeoscs d qs-sc (dco behvor ws sded before Mwell who rodced he dsplceme crre erm dd/d whch gves rse o he elecromgec wves. he Mwell eqos re he PD formlo. he eqos re hperbolc d hve wve-lke solos. For cos d o crres or spce chrges we ob b kg he crl d og dv D dv B h for feld compoe.e. he wve eqo wh wve speed c speed of lgh. Nmercl solo of he Mwell eqos s sed bove s ow hghl developed modelg echqe s eemplfed b Prof. He s lecres. Whe merl properes re depede of feld mpldes oe ofe sdes mehrmoc solos he freqec dom lss. For cses wh pecewse cos d crres ol o erfce srfces s possble o formle srfce egrl eqos for he crres. Nmercl solo of properl chose feld egrl eqos reqres dscrezo of ol he srfces order of mgde fewer degrees of freedom h reqred for FDD. Whe he sze of he dom mesred wveleghs s lrge mercl resolo of he wvelegh becomes ver cosl. Dffere pprome mehods for shor-wvelegh compos hve bee devsed. I he zero wvelegh lm wve solos m be obed b rcg wvefro ormls or rs he geomercl opcs ppromo. B m smlos ms clde dffrco pheome d correcos o he geomercl
2 Mh KH DN74 Compol lecromgecs Fll ' Lecres o Mwell Hperbolc Ssems p (9 opcs sch s he Uform heor of dffrco hve bee developed srg wh Keller s work he ses o dffrco coeffces. Nmercs oom D me dom FD Freqec dom FD Fe Dfferece FV Fe Volme F Fe leme MoM Mehod of Momes me dom Freqec dom PD FDD (Yee 966 Lb FFD (m FVD e.g. Shg FD Bodesso Iegrl Chew Illos (998? MoM (Hrrgo 968 Ro lsso Wlso Lb Nedelec Mok Rs -?? Keller ol overvew Ierfce codos see homework. Frs order ler ssems of prl dfferel eqos wh me-lke vrble Frs order ssem of N PD wh s me-lke vrble spce dmesos: N Ak ( R k k Wve-lke solo S( ( ( e : mplde S :phse. A so-srfce locs of pos of eql phse S cos. s wvefro. Cos coeffces A k llow cos vecor. he S ( S ; I AS ( I A ( S S cosφ ( S No rvl solos : ms be egevecor d - he correspodg egevle. Hperbolc f A hs rel egevles d complee se of egevecors for rel{ }. Norml o srfce s (... he fro moves s orml dreco wh phsespeed. For ds Sd ds. ke d δ (smll dsplceme orml dreco. δ he Sd δ ( S d d
3 Mh KH DN74 Compol lecromgecs Fll ' Lecres o Mwell Hperbolc Ssems p (9 Noe m deped o he wve orml dreco. If o soropc. N dffere wves Le s look he Mwell eqos hs frmework pre l vle problem for ll of emp spce R o smplf mers ke (o resrco. he mrces become (cf. he crl forml A hs shows wh he sgs of crl he wo eqos dffer: h mkes he ssem smmerc becse crl self s -selfdjo operor. So. wh A d he egessem ssfes. o vecors orhogol ll he se of whose vr sbspce s d wo egevles wh egevecor hs egevle ; or hs for he 66 A k k mr here re wo sc felds (he zero egevles d for wves movg lef d rgh wh veloc remember. hese re rsversl wves: he egevecors re orhogol o he wvefro orml. mple Wves movg he (-ple H-feld H (H drecosoropc depede of lgh Speed of ; / / / / / / / / : ; ; c H H H H A A A A hree dffere kds of wves c orml.wo wves movg o wvefro H orml - feld ( elecrosc. o mgec feld orml o wvefro - feld prllel ( ±
4 Mh KH DN74 Compol lecromgecs Fll ' Lecres o Mwell Hperbolc Ssems p 4 (9 erg esmes. he wve solos bove eher grow or dec wh me. h hs s re for solo c be esl show g sg he smmer of he crl eqos. H H H H ( ( H H B D R ( H H dv ( H ( ( H dv R *** or : H(. (. cos. H(. (. mple: Show h he egrl *** vshes wheever d H vsh f. H: Iegro b prs. hs proves h he eerg s coserved. hs s more geerll re e.g. lso he presece of perfecl reflecg srfces. However oher mesres of he solo m grow: A mrror c focs he wves so h he mml feld (he L orm becomes essell boded.
5 Mh KH DN74 Compol lecromgecs Fll ' Lecres o Mwell Hperbolc Ssems p 5 (9 Lecre Bscs of Dfferece Schemes for Wve Problems Refereces: H-O.Kress B.sfsso J.Olger: me-depede Problems he scheme o be preseed del s he Yee or sggered grd Lep-frog scheme whch hs mber of dvges. I order o beer pprece Yee we wll rodce lss ools whch dspl mpor properes of schemes d ppl hem o oher (ver smple schemes. Cosder frs dfferece schemes for pre l vle problems D pls me. Dervves replced b dfferece qoes for emple f f ( f ( p O(. p order of ccrc : Forwrd dfferece. f rd : f ( f ( O(.Cerl dfferece. {( }. j. ( j j... j Coceps: Covergece cossec well-posedess growh of solos sbl j A ecessr proper s covergece whch mes h for fed d s d -> (sch h here lws s j ecl he rgh spo ( j( (. Cossec mes h whe j re chose s smples of smooh fco ( he dfferece scheme formll coverges o he dfferel eqo. he lss s doe b sbsg lor epsos d ccellg erms. hs opero s mechcl d dfferece operor clcls c be sed o smplf he cl mplos. Well-posedess s mhemcl chrcerzo of problem whch reqres esece of "locll qe" solo h he solo be coos s fco of he problem prmeers sch s l d For l vle problems well-posedess s eqvle o boded growh: he homogeeos problem wh l vle ( f( s well-posed f d ol f here es coss K d α depede of f sch h (. Ke α f he mercl coerpr o well-posedess s sbl whch roghl spekg for l vle problems mes boded growh of perrbos he dscree scheme. Le he mercl solo me be ( j- j j he he scheme ppled o he eqo s clled sble f d ol f here ess b depede of f d sch h / b As es problem we ke hperbolc cos coeffce ler ssems j
6 Mh KH DN74 Compol lecromgecs Fll ' Lecres o Mwell Hperbolc Ssems p 6 (9 A (. mple he secod order wve eqo sded flove (hdo c be wre s frs order ssem c c v v c A c c (eerl heorecl lss of Bodr codos s osde he scope of hs corse. he ssem s ssmed hperbolc d c be dgolzed b echgg he prmve vrbles for he chrcersc vrbles w reled b Vw where V s he mr of (rgh egevecors of A. he ssem becomes w Λ w Λ dg(... N or w k w k k k... N For ler scheme he operos of dscrezo d dgolzo comme so s cler h we c resrc or sd o he sgle eqo o whch we ppl he smples frs order ccre me d spce scheme: j j j j or j ( σ j σ j where σ / s clled he Cor (somemes he Cor-Fredrchs-Lew mber whch ells how m cells he wve rvels mesep. Forer (or Vo Nem lss he dfferece formls js s he dfferel eqos llow epoels s solos d we look for solo wh wve mber k ( π/ wvelegh: k j j e s clled he growh fcor; f > he wve grows d < mes dmped wve. If s rel he phse depeds o ol: sdg wve o propgo. Noe: Oe eed o dgolze he ssem for he sbl lss. For ssem of s eqos becomes s s mr. B sbsg he sz we ob k ( σ σe σ ( cosθ sθ where θ k phse shf per cell. he relev rge for θ s π o π; wve wh shorer wvelegh s dsgshble o he grd from oe wh wvelegh hs rge. hk of he Sho smplg heorem. he growh fcor for he ec solo s H ep(-k ep(-σθ. Is phse speed s of corse -(rg H/(k. Smlrl he phse speed of he mercl solo s -(rg /(k he frher lss reqres ssmpo of how d ed o ; he sl ssmpo s h σ s kep cos. We hve 4 σ ( θ / ( θ θ / 6 O( θ H σθ σ θ /...
7 Mh KH DN74 Compol lecromgecs Fll ' Lecres o Mwell Hperbolc Ssems p 7 (9 So rg H rg o frs order θ.e. d d O( s -> for fed Cor mber. For fe sep-szes he phse speed depeds o he wvelegh. hs s clled dsperso. I s cler h H cf. he L orm lss for he Mwell eqos he eerg s coserved. B depeds o σ d θ. < s dsspo (dmpg > s growh. Here s plo of he locs of (θσ he comple ple for σ (sold.4.8 d. (dshed Oe-sded scheme.5.5 Img Rel -π θ π. he crcle s mrked b. For σ - he scheme hs for < σ < s dmped ( < : σ s eresg d for oher σ s sble. rg he spce dfferece he oher w we ge j j j j or j ( σ j σj whch s dmped for < σ < d sble for oher σ. I ppers h he dfferece scheme shold be chose o reflec he dreco of he chrcerscs: pwd or psrem dfferecg s sefl. Wh hs s so c be llsred b he Cor-Fredrchs-Lew sffce codo for o-covergece. Coceps: dom of depedece dom of flece. Cosder g l vle problem for ssem of PD me d oe spce dmeso l vles ( f(. he dom of depedece s he sb-se D( of he -s sch h f( for osde D hs o flece o (. For hperbolc ssems plo of he chrcerscs hrogh (** mmedel revels D: s he seco c off b he ereme chrcerscs -* m( j (-* d -* m( j (-*. h D s boded s referred o s fe speed propgo of formo. For he he eqo D s ll of spce (- d formo rvels fel fs lhogh ds pos flece he solo less h er-b pos. he dom of flece s he reverse: he se I(* for whch f(* fleces (. See he fgre. For he mercl scheme he defo s logos.
8 Mh KH DN74 Compol lecromgecs Fll ' Lecres o Mwell Hperbolc Ssems p 8 (9 I( o D ( Dom of depedece D d Iflece I for secod-order wve eqo c he slopes re d/d ± /c. I s ow cler h for covergece o be possble he mercl dom of depedece N ms clde he mhemcl dom of depedece D: hs s he CFL codo whch s ecessr for covergece. If does o hold we c chge he ec solo wll b mplg f( for osde N b D.e. who chgg he mercl solo. he L qvlece Prcple shows h wh hppes s mercl sbl: he prcple ss - g roghl b wh he proper prepros hs s heorem - covergece sbl & cossec So for cosse scheme (whch ll of he schemes cosdered here re d s es o check s ol sbl h c go wrog. Hece volo of he CFL-codo mples sbl. he CFL codo s es o pcre b overlg he compol grd o he dom of depedece. For he frs sgle sded scheme bove he pcre s below: shows cse wh (σ > sble becse he dfferece scheme looks he wrog w b slope d/d / N b s (possbl sble cse d c g s sble becse he mesep s oo lrge. he plos of cofrm h b s cll sble. he fl sbl codo for he sgle-sded scheme s - < σ < d wh he dfferece red he oher w < σ < he followg scheme (L-Fredrchs s smmerc d vods he ecess o swch drecos: j * j j where * ( j j c
9 Mh KH DN74 Compol lecromgecs Fll ' Lecres o Mwell Hperbolc Ssems p 9 (9 As s esl see he CFL codo s σ < d hs s lso sffce codo s m be show b he Forer lss. I s lef s homework o eperme wh he scheme d look s dsspo d dspersve properes. Noe: hs scheme s ever sed for he Mwell or smple wve eqos becse of s ecessve dmpg. I c be sed s srg po for more ccre schemes for oler problems sch s gs dmcs where dsspo s bsolel ecessr o corol shock formo ec.
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