Finite Differences. Centered vs. forward/backwards. ( x) Consider the Taylor expansion of f(x+h) around x:

Transcription

1 FD Bass C 66 Nueal Mehods Phoos Fe Dffeees Ceeed vs. fowad/bawads Cosde he Talo epaso of f(h) aoud : h f ( h) f h f f K Fo whh we ge he fowad fs dffeee ( h) f f f h f ( h) f f h h h O( h) Now epad boh f(h) ad f() aoud f(h/) ad suba h h f h 8 h h 8 h f 3 h h 48 h f f f f K To ge he eeed fs dffeee ( h) f O( h ) h f f h ( h) f ( h) O( h ) Robe R. MLeod Uves of Coloado Copuao Mehods fo M & Mowaves Booo 7 3 h h f K 48 f f Aleae odal veso h ( h) f f ( h) f f h No ve auae O h Ceeed d dffeee Moe auae

2 C 66 Nueal Mehods Phoos Robe R. MLeod Uves of Coloado 8 FDTD fo he D wave equao Devao of he e-advaee foula FDTD FD Bass L L L ± D sala wave equao Fao opeao o Fowad ad bawads popagao opeaos Subsue eeed d dffeee opeaos Solve fo o ge epl e-ahg equao Sel: Noe ha soehg eesg happes whe

3 C 66 Nueal Mehods Phoos Robe R. MLeod Uves of Coloado 9 FDTD FD Bass FDTD fo he D wave equao Mag e sep Le he advaee equao Subsue egesoluos o opeaos L ± f B B L f F F L o he advaee equao above [ ] [ ] [ ] [ ] F f f f f F F F F ad slal B B whh ae he ea soluos. Ol possble D hoogeeous spaes bu good fo udesadg.

4 C 66 Nueal Mehods Phoos Robe R. MLeod Uves of Coloado 3 FDTD fo he D wave equao Dspeso he sapled spae FDTD FD Bass Plug ouous plae wave o d ode wave equao e Phase ad goup velo follow fo dspeso elao v p v g Plug dsee plae wave o e-ahg equao e ~ ~ ~ e e e e Cael ou oo es o ge [ ] ~ os os whh elds dspeso elao fo dsee spae:

5 C 66 Nueal Mehods Phoos Robe R. MLeod Uves of Coloado 3 [ ] os os ~ v p π λ FDTD fo he D wave equao Dspeso he sapled spae (o) FDTD FD Bass ~ Che ha hs edues o ouous as ~ Wha abou ag e sep? [ ] ~ os os ~ I geeal (solve fo ) [ ] os os ~ Fo whh we a we phase velo of dsee spae ~

6 FD Bass C 66 Nueal Mehods Phoos FDTD fo he D wave equao Dspeso he sapled spae (eaple) PC (µ) ( Robe R. MLeod Uves of Coloado Jaques Lou NMP W assge 3

7 FD Bass C 66 Nueal Mehods Phoos FDTD fo he D wave equao Choosg ad Poedue: Reque ha eeg s bouded hough he odo. Fd hs odo b wg eseppg equao as a equao ad solvg fo egevalues. Resul: λ / v p /.75 / Lg dspeso (phase eo) eques D Coua sabl l I o 3D ad/o pobles wh a dele we a opeae gh a hs l whh leads o ueal dspeso: v p λ λ Nqus l Robe R. MLeod Uves of Coloado 33

8 Yee algoh C 66 Nueal Mehods Phoos 3D FDTD The Yee ehod We ou Mawell ul equaos Ad defe dsee gd ( ) ( ) le ad age feld gds ae offse b ½ ell all 3D o pe eeed fsdffeees of he ul equaos. K. S. Yee I Tas. o Aeas ad Popagao. Robe R. MLeod Uves of Coloado Vol. 4 pp Ma

9 Yee algoh C 66 Nueal Mehods Phoos 3D FDTD Te advaee equaos Apee s Law gves e advaee of ele felds a edges of ell o ege e seps: D J Faada s Law gves e advaee of age felds a faes of ell o half-ege e seps: B Robe R. MLeod Uves of Coloado 35

10 Yee algoh C 66 Nueal Mehods Phoos 3D FDTD Devao fo egal equaos S B ds () C dl [ BX BX ] [ ] Y Z Faada s law egal fo Taslae o dffeee equao w/ eeed dffeees: Y Z Dvde b ad eaage o ge sae updae equao Robe R. MLeod Uves of Coloado 36

11 Yee algoh C 66 Nueal Mehods Phoos Does Yee shee obe Gauss Laws? Ol used Apee s ad Faada s laws o deve. Does shee obe Gauss Laws? Sa wh egal fo soue fee ego: S D ds 6 D S S V ds 6 dl C D ds ρ dv? Gauss Law Dffeeae wh espe o e Appl o faes of Yee ell Use Apee s Law ah edge of ube s used we wh oppose sg so S D ds Q osa so f all hage fee (Q) sas hage fee ad Gauss Law s sasfed. Robe R. MLeod Uves of Coloado 37

12 Yee algoh C 66 Nueal Mehods Phoos Sabl ad dspeso 3D Coua odo ad 3D dspeso elao Usg sae poedue as fo D sabl l: ( ) 3 Iepeao (daw fo D): / 3D Coua sabl l Fo ubal ells Te sep us be shoe ha shoes popagao e bewee adae odes whh s o he dagoal of he ell. As we saw D dspeso s ed b opeag as lose as possble o hs l. Ufouael 3D hs s que fa fo he eo-dspeso odo Dspeso elao 3D: ~ ~ ~ s s s s Depeds o deo so dsee spae s asoop. Robe R. MLeod Uves of Coloado 38

13 Bouda odos C 66 Nueal Mehods Phoos Bouda odos PC ad PMC PC: Fo eaple se ad : B D PMC: Fo eaple se ad : B B defo (odd se) B Gauss Law (eve se) B Faada s Law (odd se) B Apee s Law (eve se) B defo (odd se) B Gauss Law (eve se) D B Apee s Law (odd se) B Faada s Law (eve se) Robe R. MLeod Uves of Coloado 39

14 Bouda odos C 66 Nueal Mehods Phoos Radao bouda odo gqus ad Mada -wa wave eq. I 3D he oe-wa wave opeaos fo a wave avelg deo (fo eaple) beoe: ± L ( ) ( ) Usg a bouda Good ews: gqus ad Mada showed ha hs wll absob a wave a popagag a a agle. Bad ews: The squae oo aes que dfful o plee. Soluo (Mu): Appoae squae oo as Talo sees. Robe R. MLeod Uves of Coloado B. gqus ad A. Mada Mah. Copu. vol. 3 pp

15 Bouda odos C 66 Nueal Mehods Phoos Radao bouda odo Fs-ode Mu odo The fs e he Talo epaso of he squae oo s eldg he fs-ode Mu opeao L ± v p Opeao fo wave avelg ± If we ew phase velo of he wave hs would be ea. Bu hs would eque hee be ol oe plae wave de o he bouda. So assue wave s oal (v p ). Dffeee equao: ( ) ( ) ( ) ( ) Solve fo ( ) Ceeed dffeees aoud / / Mus soe hs value Ca opue hs value fo G. Mu I Tas. o leoage Copabl Vol. MC-3 Robe R. MLeod Uves of Coloado Novebe 98 pp

16 Bouda odos C 66 Nueal Mehods Phoos Radao bouda odo Fs-ode Mu odo (eaple) Lauh a Gaussa odulaed s fo ad opeae a he ag e sep. The pulse popagaes whou dspeso ad appoahes a s ode Mu odo o he a bouda. Gog: 45 e seps (37.5 fs) Gog: 55 e seps (45.8 fs) Goe: 65 e seps (54. fs) Robe R. MLeod Uves of Coloado Jaques Lou NMP W assge 4

17 C 66 Nueal Mehods Phoos Robe R. MLeod Uves of Coloado 43 Radao bouda odo Seod-ode Mu odo Keep e e Talo sees: Tae e devave Ad subsue fo ow soluo o L opeao See Mu pape fo dseao. Reques soage of wo pevous e seps pe bouda ode o plee seod devave e. Also oe ha asvese devaves a o be pleeed a oes so us eve o a fs-ode odo hee. FDTD Bouda odos

18 Bouda odos C 66 Nueal Mehods Phoos Radao bouda odo Pefoae of Mu s ad d Po soue a (55) of a esh. Dashed s ode sold d ode. Te sep 4. Sae as above bu soue a (33). G. Mu I Tas. o leoage Copabl Vol. MC-3 Robe R. MLeod Uves of Coloado Novebe 98 pp

19 C 66 Nueal Mehods Phoos Robe R. MLeod Uves of Coloado 45 Radao bouda odo Reflev vs. agle FDTD Bouda odos s os os s s os os s Mu l l l e e e To fd eflev assue a de ad efleed plae wave o he bouda : Plug o s ode Mu updae equao ad solve fo : Followg sae poedue fo seod-ode BC: s s os os s s b a b a b a Mu K.S. Ku R. J. Luebbes FDTD Mehod fo M CRC Pess 993

20 Bouda odos C 66 Nueal Mehods Phoos Radao bouda odo Tpal eflev D R R Fs vs. seod-ode a oe ell se λ / 5 λ / λ / Fs-ode Seod-ode Seod-ode vs. ell se λ / / θ [deg]. / θ [deg] Robe R. MLeod Uves of Coloado 46

21 Bouda odos C 66 Nueal Mehods Phoos Radao bouda odo aple Fs ode Mu odo opeag a λ / ½ /. The peded eflev s: Mu.3 Saplg he feld vs e sep () a 5 spae seps (6.5 µ) o he spae: Ide Refleed So he foula s oughl gh bu o pefe. Robe R. MLeod Uves of Coloado Jaques Lou NMP W assge 47

22 Soues C 66 Nueal Mehods Phoos Soues Tepoal shape Se dspeso depeds o salles λ us l lages poble. Gaussa: e σ Modulaed Gaussa: Sgle feque σ π e σ σ e s( π f ) σ π e π f σ s( π f ) δ ( π f ) e σ > Robe R. MLeod Uves of Coloado 48

23 Soues C 66 Nueal Mehods Phoos Toal/saeed feld foals Bagoud Vaous po soues wh he esh a be dve a sgle gd pos. Tpall ops we wa a eeded soue de fo fee spae. Defe: To To sa sa Defe ha de feld popagaes fee-spae: o I oal feld ego use oal ehod fo oal feld sa o sa I fee-spae saeed feld ego popagae ol saeed felds o o So sll obes Mawell Robe R. MLeod Uves of Coloado 49

24 Soues C 66 Nueal Mehods Phoos Toal/saeed feld foals Use Suoud eo ego of oal feld wh saeed feld ego. Ide felds ase o bouda RBC RBC RBC Toal feld ego Saeed feld ego RBC Pluses: RBCs ol deal wh saeed (ougog) felds Ofe wa o ow saeed felds so podues auall Ca use sae updae equao houghou ee gd he oe ba ad add/suba de feld o T/S bouda. Muses: Ide feld us be popagaed dsee spae o ah ueal dspeso Sgfa aou of soage s used saeed ego Robe R. MLeod Uves of Coloado 5

25 Soues C 66 Nueal Mehods Phoos Toal/saeed feld foals Ide feld popagao Ide wave us be phase-ahed o he asoop dspeso of he 3D dsee gd. Se epoal pulse shape us oa fe badwdh us ah dspeso a bad of fequees. A eas wa o do hs s o popagae he de feld s ow D FDTD spae. Beg wh dspeso elaos fo D ad 3D: ~ s s D ~ ~ ~ s s s s If de wave s alog pa as a pefel ah b seg D ell se equal o ell se popagao deo. Ohewse solve fo agude of ~ a ee feque ad use hs o p he D ell se. ~ ˆ ~ ˆ s s s s ~ D s s D ~ ˆ Robe R. MLeod Uves of Coloado 5

26 C 66 Nueal Mehods Phoos Robe R. MLeod Uves of Coloado 5 [ ] [ ] Z Z o Z sa X sa Z Z o Z sa X sa X sa Toal/saeed feld foals Ipleeao FDTD Soues Z sa Z sa Y sa Y sa Case : All felds saeed oe: 3 4 Case 3: Feld saeed oe poo of sel oal oe Case : All felds oal oe: Z o Z o Y o Y o [ ] [ ] [ ] [ ] Y X o X o Y sa Y o Z o X o X o Y Y sa Y o Z o Z o µ µ µ µ µ µ µ µ µ µ Case 4: Feld oal oe poo of sel saeed oe Updae equaos sae houghou gd. The oe ba ad add/suba de feld o saeed/oal feld bouda. Saeed Toal

27 Soues C 66 Nueal Mehods Phoos Toal/saeed feld foals aple Saeed Toal Pulse eegg fo bouda. Noe ad phase avellg. Saeed Toal Pulse lauhed. If he sulao was oued ad he RS was a PC he pulse would efle he avel ba hough he S/T bouda eveuall efleg off of. Robe R. MLeod Uves of Coloado Jaques Lou NMP W assge 53

28 Maeals C 66 Nueal Mehods Phoos Isoop o-dspesve aeals aple (/3) Le va spae ad e-deve FDTD updae equao. You ow eed a gd sapled o he sae odes as he ele feld. Apee s law (he updae fo ) s odfed: Y Y Z Z Pobe S T s ode Mu Pulse eegg fo bouda. Pulse ased ad efleed fo.5 4 h slab suouded b vauu. Robe R. MLeod Uves of Coloado Jaques Lou NMP W assge Tased absobed b BC efleed avels hough ST bouda ad o pobe po. 54

29 Maeals C 66 Nueal Mehods Phoos Isoop o-dspesve aeals aple (/3) efleed vs e a he pobe po: Foue asfo of de pulse ad efleed pulse: Refleo oeffe vs feque (ao of efleed o de). Null s a.6 P o λ.5 µ. Fesel oeffes fo fo ad ba have oppose sg so sa eeds o be full wave oud p o. 5/.5 /.4 µ. Robe R. MLeod Uves of Coloado Jaques Lou NMP W assge 55

30 Maeals C 66 Nueal Mehods Phoos Isoop o-dspesve aeals aple (3/3) efleed vs e bu wh 4 slab hess: Foue asfo of de pulse ad efleed pulse: Refleo oeffe vs feque (ao of efleed o de). Now a efleo s a.6 P o λ.5 µ. Fesel oeffes fo fo ad ba have oppose sg quae wave sa ves aga o ae osuve efleo. Robe R. MLeod Uves of Coloado Jaques Lou NMP W assge 56

31 Maeals C 66 Nueal Mehods Phoos Asoop aeals () D Two pobles: σ σ σ Cosuve elao Subsue o Apee s Law σ opoe. Off dagoal opoes of ouple hee opoes of Apee s Law so a spl solve fo updae equao. Do have all opoes of a pope e ad spae ees (Yee se s boe). J. B. Shede ad S. udso I Tas. Aeas Popaga. vol. 4 pp Jul 993 Robe R. MLeod Uves of Coloado 57

32 C 66 Nueal Mehods Phoos Robe R. MLeod Uves of Coloado 58 Asoop aeals () Z Y X Z Y X X Y σ σ σ Fo es o RS volvg oduv (whh ae sapled a he wog e sep) ae e aveage o ge a ½. Appl he oal eeed dffeees fo es o RS volvg. Z Z Y Y X X σ σ σ σ σ σ Repea poedue fo ohe wo opoes of Apee s Law eeed hs sae loao (½ ½). We ow have hee equaos hee uows whh we a heefoe solve. We ll ol eep he e se s a he gh loao. To ge equaos fo ad ee he devao a he appoae loao fo hose felds. FDTD Maeals ad Z Y X Ioduo of esos ouples all feld opoes suh ha he sple Yee gddg of eeed dffeees fudaeall beas. We ll s wh he gd beause of all he ohe good oues bu hs wll be edous hee. The sples wa o udesad he poedue s o osde hee es oe fo eah opoe. The followg s fo he opoe. Repea he ee poedue fo ad. Beg b dseg all hee opoes of Apee s law a he loao ½ ½. The opoe fo eaple would be:

33 C 66 Nueal Mehods Phoos Robe R. MLeod Uves of Coloado 59 Asoop aeals (3) FDTD Maeals [ ] Z Y X X Y Z X Y Z Z Y X σ σ The esul fo he podue eeed a ½ s: Good ews: Ca ow we es of quaes a pevous es. Bad ews: Neal eve feld opoe o he RS s a he wog spaal saplg loao. Soluo: Iepolae as eessa. Fo eaple: X Y Y Y Y Y 4 Repea he ee poedue fo o deve he ohe wo -feld updae equaos. Noe ha he sel s ow spaall lage whh wll pla hell wh bouda odos. Tpal soluo do use asoop aeals ea boudaes. ad Z Y Ths opoe s a he gh loao. The ohes ae.

34 C 66 Nueal Mehods Phoos Robe R. MLeod Uves of Coloado 6 Dspesve aeals () FDTD Maeals R. Luebbes e al. I Tas. o M Copa V 3 N 3 Aug 99 χ δ τ) dτ ( χ(τ) () D() We he sala elave pev as a saaeous (odspesve) ad dspesve e: Whh odfes he osuve elao To dffeee Apee s Law we eed epessos fo D a e seps ad. Tae d D d D χ τ τ χ χ τ τ χ whee he dsee χ s defed b D D χ χ χ Tae he dffeee o ge he e devave of D τ τ χ χ d τ Ca ae χ() o o (see las le) Aba dvso bewee wo es a

35 C 66 Nueal Mehods Phoos Robe R. MLeod Uves of Coloado 6 Dspesve aeals () FDTD Maeals χ χ χ χ Defe χ χ χ Ad subsue o he dsee Apee s Law o fd updae equao fo whee Σ ½ epeses he su of he fou felds he dsee su of of he loop egal. dl Good ews: ave epesso fo updag. Bad ews: Appeas o eque soage of hso of as log as he e eval of χ Soluo: Dffeee he dffeeal equao fo P χ : s χ U e s χ s e e χ P e P P χ χ s e χ Pola dele Dsee fo Defo Updae equao New e

36 Maeals C 66 Nueal Mehods Phoos Nolea aeals Sple ehod Fo a saaeous Ke olea ( 3) 3 δ χ δ χ D δ Avod pobles wh olea equao b assug s o oo dffee fo he alulao of he olea pev D ( ) α( ) α ( ) [ ] α X X Y Z X Noes. Noao ad eso fo of olea follow [] bu ould be geealed o oplee hd-ode eso oleaes hough he use of a a veso sep sla o Shede ad udso. Reques epolao o ge he feld opoes a he pope loaos 3. Ol vald fo saaeous oleaes 4. Wll eque ve fe e seps o ae assupo vald. Ths wll u eque ve fe spae seps o avod dspeso. Thus pall ol useful fo D pobles. G. W. Zheg ad K. S. Che I. J. Ifaed Mll. Waves vol. 3 o. 8 pp Robe R. MLeod Uves of Coloado 6

37 Maeals C 66 Nueal Mehods Phoos D Nolea aeals Moe ople ehod To deal wh boh dspesve ad eso dspesve oleaes defe P L P NL Tea lea dspsesve pa as befoe. Followg ha appoah fo a paula phsal sse ha geeaes he olea we he dffeeal equao ha desbes he evoluo of P NL ad use FDs o fd a updae equao fo P NL. Fo eaple (see ef) Ke Raa The Raa e fo eaple follows he followg D Robe R. MLeod Uves of Coloado 63 P. M. Gooa ad A. Taflove Ops Le. B 7 pp. 8-8 Feb. 99.

38 Mawell-Bloh C 66 Nueal Mehods Phoos Mawell-Bloh quaos D FDTD sulo Feld Aos of egesaes > > Mawell Cul quaos: µ o o o P Shodge quao: h ˆ ( ( )) p ˆ µ Couplg: ˆ µ eˆ P Np Robe R. MLeod Uves of Coloado Ma Sowe NMP 4 fal poe 64

39 Mawell-Bloh C 66 Nueal Mehods Phoos Ao Polaao Quau Mehas Shodge quao: h ˆ ( ( )) w/ sae spae { } elds soluos of fo: ( ) ~ ~ Fee ao: / : [ ] e h ( ) > > [ ] e e Noe ose oss e: [ ] os Osllaes a ao feque Robe R. MLeod Uves of Coloado 65

40 Mawell-Bloh C 66 Nueal Mehods Phoos aple doge eleo wavefuos Goud sae: Y-Z Plae S ( ) a 3/ e / a ed sae: a o Y-Z Plae p P 3/ / a 3 ( ) e os( θ ) a a 3 π a o Osllag dpole oe deo: p eˆ os Robe R. MLeod Uves of Coloado 66

41 Mawell-Bloh C 66 Nueal Mehods Phoos Louvlle quao ow o hadle eseble of aos Laos o Shodge quao: Does odel a eseble of aos Ca o easl lude deoheee No spoaeous esso Des Ma: ρ ˆ ρ ρ ρ ρ Louvlle quao: seble aveages * * * * peao value of he posve o egave feque opoe of p Populao pobables goud ad eed saes & ρ [ ] ( eq ) ˆ ρ γ ρ ρ ˆ h Fo Shodge Pheoeologal dea es Robe R. MLeod Uves of Coloado 67

42 Mawell-Bloh C 66 Nueal Mehods Phoos Devao of Bloh equaos & ρ µ [ ρ ρ] γ ρ Populao sae h γ & ρ [ [ ] h ρ µ ρ ρ ρ h * ρ ρ Coheee bewee & ρ Wg ou eah e of he Louvlle equaos ρ Bloh veo epeseao: Subsue: Re[ ] ρ I[ ] ρ 3 ρ ρ Lage ae hage of oheees Cosevao of populao & & & Bloh quaos T µ ( T h eq µ 3 3 ( T h ) 3 ) Robe R. MLeod Uves of Coloado 68 3

43 Mawell-Bloh C 66 Nueal Mehods Phoos Fal se of Ds Subsue polaao Mawell: p Nµ & & & & ( ) µ eq ( ) h & µ o o T T T T Nµ o h µ Nµ 3 o Bloh sphee: 3 [ ] Robe R. MLeod Uves of Coloado 69

44 Mawell-Bloh C 66 Nueal Mehods Phoos Nueal poble Need o advae sae veo b d: C & 3 C C T & T w/ µ & 3 T h eq µ & T h 3 FDTD updae sel: 3 3 / 3/ Robe R. MLeod Uves of Coloado 7

45 Mawell-Bloh C 66 Nueal Mehods Phoos Solo pulse esuls 8.8 π Seh Pulse Feld R Poso (µ) π Seh Pulse Feld R Poso (µ) Robe R. MLeod Uves of Coloado 7

46 Fa-feld poeo C 66 Nueal Mehods Phoos Fa-feld poeo If he poble s a saeg obe suouded b ep spae FDTD s a effe algoh fo he ep spae. Rada s a good eaple he RF doa. Mosop s a eaple he opal doa. Defe a sufae C whh oas he saeg obe. Ths sufae wll pall be ousde he oal/saeed feld bouda so oas ol saeed felds bu sde he adao BCs ( s he gd). Saeg soues sde he sufae a be eplaed b sufae ele ad age ues whh adae he sae felds ousde: ah ageal feld s ow a dpole wh a ow adao pae a a po eeo spae: Te doa Feque doa Robe R. MLeod Uves of Coloado M.J. Bah R. R. MLeod R. W. Zolows J. M Waves ad Apps V6 N 99 7

47 Fa-feld poeo C 66 Nueal Mehods Phoos Fa-feld poeo Copaso o MoM Copaso wh ehod of oes Copaso wh sple aalal ase M.J. Bah R. R. MLeod R. W. Zolows J. M Waves ad Robe R. MLeod Uves of Coloado Apps V6 N 99 73

48 Fa-feld poeo C 66 Nueal Mehods Phoos Fa-feld poeo Copaso o Me heo Noe ha FDTD peds esee ad agude of whspeg galle ode bu o shape due o dsee sufae. Copaso vs. e Copuaoal spae usg PC ad PMC bouda odos as se plaes. Copaso vs. agle a oe Robe R. MLeod Uves of Coloado 74

49 Fa-feld poeo C 66 Nueal Mehods Phoos Fa-feld poeo Copaso o epee Saeg fo ee ube. FDTD epeed o do well hs ase se obe fs eagula gd. Robe R. MLeod Uves of Coloado 75

50 C 66 Nueal Mehods Phoos Robe R. MLeod Uves of Coloado 76 Sabl ow o eep he ode fo eplodg l l l l l l l l l l l l l l A geeal ehod o deee he sabl ls s o subsue a assued soluo ha ludes he sabl he solve fo he odos whh he sabl does o gow (vo Neua aalss). Fo eaple sag wh ou FD epeseao of he 3D wave equao (geealed fo pevous D veso): Assue a soluo of he fo [ ] l l e α Subsue o he FD equao ad dvde b [ ] [ ] [ ] [ ] { } [ ] [ ] [ ] [ ] os os os osh osh os os os e e e e e e e e α α α α whh s he Coua odo! l Valdao Sabl Re[Cosh - ()] The epoeal gowh α wll be deall eo f he RS s bewee - ad. The dageous ase s - whe π / whh s Nqus.

51 Valdao Covegee C 66 Nueal Mehods Phoos Cosse & Covegee ow we ow he aswes ae gh Reall ou epessos fo eeed fs ad seod-ode FDs. ( h) f ( h) O( h ) f f h ( h) f f ( h) O( h ) f f h Tuao eo A FD equao s osse wh he ogal PD equao whe he uao eo goes o eo as he esh se goes o eo. A eaple of a osse FD shee s a eeed-dffeee veso of he hea dffuso equao (DuFo-Fael shee): ul ul ul ul ul ul u u B spl wg he Talo sees fo he FDs we fd he leadg es of he uao eo ae: u 4 l 4 l u 6 u 3 l 3... Does go o eo f / s a osa! A FD appoao s sad o ovege whe he soluo of he FD equao oveges o he soluo of he PD as he esh se goes o eo. La equvalee heoe Gve a popel posed lea al value poble ad a fe dffeee appoao o ha sasfes he osse odo sabl s he eessa ad suffe odo fo ovegee. PARTIAL DIFFRNTIAL QUATIONS Robe R. MLeod Uves of Coloado 77