Une méthode volume fini implicite en maillages non-structurés pour les équations de Maxwell 3D en domaine temporel
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. An implicit finite volume time domain method on unstructured meshes for Maxwell equations in three dimensions Victorita Dolean — Stéphane Lanteri. N° 5767 Novembre 2005. N 0249-6399. ISRN INRIA/RR--5767--FR+ENG. Thème NUM. apport de recherche.
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(25) $#%& '( )+*-,.+/10 . `FvSq~/mot zApkymo[~|p*mu[^`4`M tw t m zAp^`Yk£¢,[^`Ypl`F~|t p#¢,t mu[ zl~ jasu`¶p^`F¦sotw^k$^soz_ mo[^`4knm~|^t t mnj EztpSmz|®+Mt `Y¢Îmo[^`#~|pXk¢ `s\twk,zMMt zAqkY^b`A`sumu[^`Y `FkukY«lz|mo[^`sW^suzAX`Ysmot `Fk\z|®(mu[^`#su`Fkq^ mutp^ ©_a^tt³m (^ttkup^Xt mu`Y` sokutzzp"q^~|_ap9`VtpMZ\t_9`s`Ikut ^»etzt³mn_"j ~|z|t®Kp6mu[^ Á `W µtp^Z,`Y~»µs) kujM_akmu` `Ymo_ [^zl«|Ãt p*p^z`Ys`Yl¸`s+momoz zE_"`·~|º~`p~|~µj ®Ë~"t `Fs£¥ «£zku_aq[Î~sutw~Akuk*zp9pMq^¢,_at³mo`["sotwmu[^~|` `l^ tw t m eZ,»:_9`mu[^zl¥ ©pVmo[^twk ¢£zAsuºK«¥¢ `"knmoqlj²mo[^`¦kut _a^`Ykm#k`F zAp¡zAso^`sko[^`_a`«}£s~|pºA¨ bWtzwkzAp%«^ zAq^^`Y*muz9mu[` `pSmo`so`Y*¶p^t mu` zA q_9`4_a` mo[^zl^suzAXzSk`F*t"p ¨ × ° ©pk`Fªmutzpl«M¢£`W¶sknm£®¯zso_#q^w~/mo`mu[` }£s~|p^ºS¨©btw zAkuzp"~Ak`F eZ,» _a` mo[^zl¦tp¦mu[`µzpSmu`Sm z®¥mu[^`9¼Y» d²~|M¢ ``FvSq~/mot zApkMZ\[^`Yp%«A¢ `e~p~|j "`,mo[^`µ`p^`YsuAj9zpku`so/~/mutzpa^sozE`sumnj"~|p¦^suz/A` mo[^`¦tpM`skut ^tt³mnj²z®£mo[^`¦t_a^ tw t m#_"~/mosut ¥ ·`*knmoqlj¡mu[^`*^tkuX`YsokutzpI^sozE`sumut`Yk4z|® mu[` eZ,» _a`mu[^zla~|paz_a~|so` mu[^`Y_½¢,t mu[amo[^`,` l^tw t m£ku[`_a`|bWq^_a`sotY~|^so`Ykuq^ mok+®¯zAsq^p^t ®¯zso_½~|pX pzpl¨ qp^t³®¯zAsu_ Asutw^kWzp qXl`mo[^tkµku`Yªmot zAp% ©p¡k`Fªmot zAp²^«E¢£`#®¯zso_#q^w~/mo` mu[^` eZ,» _a` mo[^zl ®¯zsemu[^` kuzqlmot zAp²z|®ymo[^`"» d²~|M¢ `(`YvSq~/mot zApk4~|p²mu[^`YpVkmuq^j±mu[^`ako~|_a`#®¯`F~/moq^su`Fke~Akµt p¡mu[^` ¼Y» Y~ku` mo[~/m tkY«%mo[^`*`Yp^`soj· zpXk`Ysu/~/mot zApVsuzAX`Ysmnj·~|p·mo[^`*tpM`skut ^tt³mnjVz|®\mu[^`*t _a^t m _"~/mosut ¥Z\[^` _a`mu[^zltk\/~ tw^~/mo`Yqkut pa~akut _a^` _azll`¥^soz `Y_ t p±A»# !
(26) $ 2. .
(27)
(28) . Z\[^twk knmoqljatk zAp `Ysup`Y9¢,t³mo[¦mo[^`WpMq^_a`sotw~|EkzA q^mutzp"z|®¥mo[^`mot _a`µlz_"~|tp¦d²~|M¢ `°¤ k)`YvSq~/¨ mot zApktpamu[^so``ku~A `lt_a`pkut zApkÁË[^` mo`soz`Yp^`zAqk( tp^`F~|sytwkzmusoz^tw _a`Yltq^_ ¢,t³mo["p^zkzAq^s `«/¢,t mu[ ku~A `9C~sujMtp^``Yªmosutw X`Ysu_at mmutMt³mnjV~|pV_"~|p` mutw X`Ysu_a`Y~^t t mnj^ÂWqXktp^±~|pVt _a^tt³m ¶Xp^t³mo`az|¨ q^_a` _a` mo[^zl zpq^pkmusoqªmoq^su`F mu` moso~[^`Y^so~¥_9`Fk[`YkYZ\[^` ``Y musote¶`Y E = (E , E , E ) ~|pX mo[^` _"~|Ap^` mote¶`Y H = (H , H , H ) A`sot³®¯ j
(29) x. y. z. x. t. y. z. t. Án¼FÂ. E ε(x) − ∇ × H = 0 ∂t H +∇×E =0 µ(x) ∂t. ©pÏzAsol`Ysamuzl`Ysut`²~¸¼F» `Ysokutzp z|®µmu[^`¡d¡~/l¢£`Y °¤ ka`YvSq~/mot zApk²Á¼FÂa¢ `² zApktwl`Ysamu[~|m¦mu[^` ^ t so`Y mutzpz|®EzX~|A~|mutzp tk k = (k, 0, 0) ~p*mu[~|m,mu[^`4Ezw~|sot "F~/mutzp z|®(mu[^` `Y `Fªmosutwe¶`w*A`Y muzs twkµkq[²mu[~|m KZ\[`#Ezw~|sot "F~/mutzp±z®ymu[` _"~|Ap^` mot ¶`w²A`Yªmozsetk4l`Flq `F ®¯soz_ mo[^`4 sozAkoky^sozlElq=ªm (0,k ×0,EEjMt )`Yltp^ H = (0, H , 0) ld±zAsu`Yz/`YsY«|¢ `e~Akukuq^_a`,mo[~/m E ~|p H ~|so` ®¯qpªmot zApkWz® x ~p t & ^zsµkt_a^tt³mnjA«X¢ ` kut _a^j p^z|mo` E tpkmu`Y~A z|® E ~p H tpkmu`Y~A±z|® H Z\[`p%«lmo[^`"¼Y»d¡~/l¢£`Y ¥`FvSq~/mot zApk\~|p E` ¢,sut mmo`p ~k
(30) t. z. t. y. t. z. z. ∂D − ∂t ∂B − ∂t. îîß. OPQSRQ. ∂H =0 ∂x ∂E =0 ∂x. y. y. Á?Â.
(31) . * . S G
(32) ( . ©pÌ`YvEµÁ°Â « tk*mu[^`V``Y musotVltku^w~`_a`pSmY« twk¦mo[^`·``Yªmosutw±¶X`w¥« tk*mu[`¡_"~|Ap^` mot tplqXªmutzp¸~|pX HD twk9mu[^` _"~|Ap^` mot ¶`Y¥£Z\[^`±~|EEz/` vAqX~|pSmut mut`Yk"~su` tp^ºA`YB Sjmu[^` ®¯zA z/¢,t p zpkmut muq^mut`4~C¢k
(33) ÁËA D(x, t) = ε(x)E(x, t) B(x, t) = µ(x)H(x, t) ¢,[`so` ε(x) ~p µ(x) so`YkuX`Fªmut`Y j²^`p^zmu` mu[^`"lt``Y musot X`Ysu_at mmot Mt mnj¡~|pV_"~|Ap^` mot X`Ysu_a`F~/¨ t t³mnjz|®mo[^`_a`Y^t q^_mo[~/m4~|p±E`#`l^su`Fkuku`Ytp±mu`Ysu_"kWz|®+mu[^`Yt se/~|q^`YkWt p±mo[^`#/~A q^q^_ Á ε ~|pX Â\~psu`Y~|mut`µp^zpl¨©lt_a`pkut zAp~|K/~|q^`Fk Á ε (x) ~|pX µ (x)Â
(34) µ Á SÂ ε(x) = ε ε (x) µ(x) = µ µ (x) `Ykutwl`YkY«Mmu[^` suzA~|S~/mutzpkE``F*twk,At A`p*Mj
(35) v. v. r. r. $D. v r. v r. c(x) = p. )vE'Á°Â\Y~|pX` ¢,sot³mumu`Yp ~Ak
(36). 1 ε(x)µ(x). ∂E ∂H − =0 ε(x) ∂t ∂x µ(x) ∂H − ∂E = 0 ∂t ∂x. zAsY«ltp`Y muzAsutw~|°«lvAqX~kut³¨© zApk`Ysu/~|mut`«A®¯zso_:~Ak ¢,[`so`
(37). B. ∂F (W ) ∂W + =0 ∂t ∂x. Á?Â.
(38). ®¯zAs. x ∈]a, b[. ~|p. t>0. ÁËSÂ. ~ p F (W ) = AW ¢,t mu[ A = 0 −1 −1 0 y~sutzqk£Ezq^p~|sojazp^t³mot zApk\~|p X`µt_9EzAku`Y ~/m mu[^`µEzq^p~|sot `Fk x = a ~|p x = b +©p mu[^twk ¢ zsoº"¢ `µ¢,t'knmoqlj*mu[^`4®¯zA z/¢,t p9z_#t p~|mutzpk
(39) _a` mo~ twEzq^p~|soj zAplt mutzpkWXzmu[²~|m x = a ~|pX x = b
(40) E(a) = E(b) = 0 «t° `mu[^` ¼Y» `Ysokut zApaz|®Emu[^`µ zpXlt³mot zAp ¢,[^tw["tkqXk`Fat pamo[^`eA»Ã~Ak`Á n l`Yp^z|mo`Yk+mo[^`q^pt³m~|soj zqlmn¢\~|s^k p^zAsu_"~| ª« n × E = 0 X`Ysutzlltw4XzAq^p^~suj¦zplt mutzpXk\t? ` E(a) = E(b) ~|pX H(a) = H(b) « ~|p¦~kzAsut p^ Ezqp^~|soj zp^t³mot zAp¦~/m x = a ~p"~ _a` m~| tw,Ezq^p~|soj zAplt mutzp¦~|m x = b Á¯t° ` = 0 ^zs4¢,[~|m zpX `sop²mo[^`a~|kuzso^tp^*Ezq^pX^~|soj² zAplt mutzp%«¥¢ ` _"~º`9qk`9z|® mu[^`E(b) zpM`YpSmutzpk\~^zlmo`Y ^su`YMt zAqkj"t° `
(41). W =. . E H. . , B ≡ B(x) =. . ε(x) 0. 0 µ(x). . . . . í å(î í Û.
(42) .
(43)
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(46) $#%& '( )+*-,.+/10 . |~ p Z\[^`so` ®¯zAsu`A«t³®~V¶sknmu¨°zAso^`s¦hMt A`su¨ d `samnjME`²~|kuzso^tp^·zp^t³mot zAp twk"t_aXzSk`F ~|m"mu[^` XzAq^p^~suj x = a
(47) k 0 k= 0 , E= 0 0 E . mu[^`Yp%«kut pX `. 0 H = H 0 . «l¢£` zAlmo~t p
(48). (n × E + cµn × (n × H)) (a) = 0 n(a) = (−1, 0, 0)t. 0 0 (n × E + cµn × (n × H)) (a) = E(a) + c(a)µ(a) −H(a) 0 0. ~|p mo[SqXk«l¢ ` _#qkm,t _aEzAku`. . (E − cµH) (a) = 0.
(49). . . . \Z [^`µlz_"~t p ]a, b[ twk£^tko so` mot "Y`Y9qkut p^ N `wk C kuq[amu[~|m C =]x , x [ ¢,t³mo[ ∆x = l`p^zmutp^²mo[^`*Azq^_a`*z® ®¯zs (bWz|mo`¦mu[X~/m"~¡ `Y \ `YpSmu`so`Y·¶Xp^t³mo` x Azq^l_a`p−` z|mulx`Ftwko soS` j mut#"Y~|mut¢,zp²t³mo["twkµ
(50) ~lzAlmu`F7ÁËku`` ¶CX¼Cª'¿e1 p¡≤`F~j[¡≤N` C «K¢ ` l`¶p^` mu[^`a_a`Y~p²/~|q^` z® j. j+ 21. j. j− 21. j. j+ 21. j. j− 21. j. W. Wj. 1 Wj = ∆xj. ~pV¢£`al`pz|mu`aMj mu[~|mtk
(51) C. Wj± 21 (t). Z. W (x, t)dx. mu[`a~^^sozClt _"~/mo`#/~|q^`az|® W ~|mµmo[^`at pSmo`su®Ë~ `FkµX`mn¢£`Y`p Cj. j±1. 1 Wj± 12 (t) ≈ W (xj ± , t) 2 a. b Cj. 0. '¼. îîß. OPQSRQ. 1. j−1. j. j+1. N−1 N N+1. }£`Y %`pSmu`Ysu`F ltwko so` mut#"Y~|mutzpz|®mu[^`"¼F» zA_9qlmo~|mutzp~%lz_"~|tp. 1/2 3/2 . 2. j−1/2 j+1/2. N−1/2 N+1/2. Cj. ~|pX.
(52) . * . d±q³mot ^jMt p²ÁËA Mj*~ mo`Ykm,®¯q^pªmot zAp Z . B. Ψ(x). S G
(53) ( . ~p t pSmu`Ys~/mot p^9z/`Ys Ω = [a, b] jMt`w. ∂W ∂F (W ) + ∂t ∂x. .
(54). Ψdx = 0. ® |~ ptwk(9[zS`zSwkk`Y`Y¢,p [^muzµ`soE`FÂ)`£momo[^[^``,p¹[Á°~|s ~X `Fmu`Y suzAtw_9knmo`Ftk ®¯
(55) q^pªmot zAp9~Akukuzl tw~/mo`Y4mozWmo[^` ` Á¯t° ` t ® Ω. Ψ(x) x ∈ Cj. Cj. ¢,t mu["
(56). Bj. dWj + Φj,j−1 + Φj,j+1 = 0 dt. Bj =. . εj 0. 0 µj. Á | Ψ(x) = 1. Á?AÂ. . ¢,[`so` ~p ^ `p^zmu`µC~ q`Yk z|®%mo[^`4 zM` ¬¦ t`pSmk ~|p mo[~/m~su`µ zpXknm~|pSm£z/A`s£mu[^`4 `Y © p·`YvEÁËSª« ¢, t mu[ l`Yp^z|mo`Yk4~|p·~|^suzClt_a~|mutzp²z®ymu[` ÀXql¡E` mn¢ ``pVmu[` `k ~p t pIom [^`*ltso`Yªmot zAp¹z|®\mu[`¦p^zso_"~|yzAsut`pSmo`YV®¯suzA_ muz ©pImu[tk knmoqlj«¢£` ~Alz^mW~¦ `YpAmo`so`Y ~|^ suzClt_"~/mutzp ®¯zsmu[^`Fk` ÀXqll`Yk\®¯zA z/¢,t pamu[^`#ko[^`Y_9`#^suzAXzSk`F*MjxytX`Ysup^z ë
(57) µj. Cj.
(58). εj Φj,k Ck. µ. ε. Cj. k = {j − 1, j + 1}. Cj. Ck. . 1 1 1 (Fj + Fj±1 ) = (AWj + AWj±1 ) = ∓ 2 2 2. ~p ¢£`µmu[Mqkzlm~|tp*mo[^`4®¯z z/¢,tp^"ku`_at³¨©ltwkusu`mut#"Y~|mutzp*tp k~A ` Φj,j±1. =.
(59). . Hj+1 − Hj−1 ∂Ej = 0, εj ∂t − 2∆xj ∂Hj Ej+1 − Ej−1 − = 0. µj ∂t 2∆xj. Hj + Hj±1 Ej + Ej±1. . Á AÂ. C. Á¼YAÂ. ©p±zAsol`Ys\muz¦ltwko so` mu#t "` tp mut_a` mu[`#kjlkmu`Y_«¢£` [~CA` ~/mWzAq^sltwkEzAko~|'~p ` l^tt³me'`Y~l¨ suzA ko[^`Y_9`4¢,[^tw[[~k,~ so`Y~Alj"X`Y`p±~|pX~|j+"Y`Y t p"ë!
(60) n+ 1 n− 1 Ej 2 − E j 2 εj ∆t n+1 − Hjn H j µ j ∆t. =. n n Hj+1 − Hj−1 2∆xj n+ 1 n+ 1 Ej+12 − Ej−12 2∆xj. Á¼¼FÂ. ©p±mo[^`#p` Mmµku`Y mutzp±¢ `¢,t +l`A`z²~|pX²knmoqlj ~|p²t_a^tt³memut_9` t pSmo`s~/mot zAp±ku[^`Y_a` ~kWt³m twk\¢ `%ºMp^z/¢,p¦mo[~/mWkuq[ ku[^`Y_a`Yk\EzAku`Ykok`4E` mmo`sWkmo~^t t mnj¦^sozE`sumut`YkY =. í å(î í Û.
(61)
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(65) $#%& '( )+*-,.+/10 . . . . .
(66) . . Z\[^` t_a^tt³m,mot _a` tpAmo`Aso~|mutzp*z®(`FvE%ÁËA so`t`Yk\zp ~¦}£so~p^ºS¨ bWtzwkzAp ko[^`_a` Ejn+1 − Ejn ε j ∆t µj. Hjn+1. ®+¢£`^`p^zmu`4Mj. − ∆t. =. Hjn. σj =. =. 4∆xj ∆t. ". mu[^`YpIÁ¼F|Â\Y~|p E` ¢,sut mmo`p ~k. _"~|musot³¦®¯zAsu_:z|® Á¼YS tk\At A`pSj. ®. Á¼FÂ.
(67). Á¼YAÂ. n n = σj εj Ejn + Hj+1 − Hj−1. σ µ H n+1 − E n+1 + E n+1 j j j j+1 j−1. ~Ak.
(68). ! !# n+1 n+1 n n Hj+1 + Hj+1 Hj−1 + Hj−1 − 2 2 " ! !# n+1 n+1 n n Ej+1 + Ej+1 Ej−1 + Ej−1 1 − 2∆xj 2 2 1 2∆xj. n+1 n+1 σj εj Ejn+1 − Hj+1 + Hj−1. . C. n n = σj µj Hjn + Ej+1 − Ej−1.
(69). ¯® zs j = 1, · · · , N Án¼ S mu[^`Yp¡~"A zA~|su`Y^su`Fk`YpSmo~/mot zAp z®y`YvSq~|mutzpk Án¼ SÂ,twkW¢,sot³mumu`Yp +D. n+1 n+1 n n −AWj−1 + σj Bj Wjn+1 + AWj+1 = AWj−1 + σj Bj Wjn − AWj+1.
(70). Wn = (W1n · · · Wjn · · · WNn )t. D. ¢,[`so` M ~|p P ~su`4^zlºS¨?mosutwltw~|zAp~|X_"~/musotw `Yk,^` ¶p^`FSj M Wn+1 = P Wn. Á¼FÂ
(71). Á¼YAÂ. Mi,i = Pii = σi Bi , i = 1..N, Mi,i+1 = −Pi,i+1 = A, i = 1..N − 1, Mi,i−1 = −Pi,i−1 = −A, i = 2..N.. Å (SÇ ^Ê?Ê ËÉ ÆE / (M #W`so`«K¢£`a~kokuq^_a`mu[X~/mµmu[^`aEzq^p~|sot `Fk ~|p ~|so`#Ez|mo[·zpkut^`so`Y²~AkW_a` m~| tw ~p±mo[Mqk«¥~ su`À`Y mutp^ zpXlt³mot zAp¡twkµt _aEzAku`Yx¡=muz amu[^`a``Yx mu=sotb¶`w Á n × E = 0 tp¡mo[^`9A» ~Ak` ¢,[`so` n ^`p^zmu`Yk mo[^`4q^p^t mo~suj¦zAqlmn¢\~|s^k p^zso_"~|K`FªmuzAsÂ)¢,[^t `4p^z"zplt mutzptwk\t _aEzAku`Yzp mu[^` _"~p^`mutwµ¶`w¥ zs\mo[^`pMq^_a`sotY~|¥mosu`F~/mo_9`YpSmz|®(mu[^`Fk`#zp^t³mot zApk«¢£` _"~|ºA` qku` z|®[^zSknm`k ~p C Z\[^`Yp%«lt pmo`so_ak\z® E ~p H tpIÁ°Â «lmu[^`Fk`zp^t³mot zApk mus~|pkuw~/mu`Fk t pSmu z
(72) C ~p E = −E _9`F~|p^tp^#mu[~|m)mu[`Àql"z® E ~/m£mu[^`eXzAq^p^~sut`Yk x = a ~|p E = −E w t , k k ` , m muz"^ x=b | ~ p _a`Y~|pt p^ H ~p H ~|so`#_at sosozsW/~ q^`FkWz® H ~p H H =H su`FkE`Y mut`Y jA H = H bWz|mu` mu[X~/m9^q^`*moz²mo[^` l`¶p^t mutzp z|®W~¡zpSmusoz£zA q^_a` «(t m9twk#pz|m p`Y `Fkuko~|sojVmuz¡t_aXzSk` ~· zAplt mutzp zp¹mo[^`Àqll`Yk F ~p F +Z\[`p%«(mu[`Yku` zCp^t³mot zApk#_ qkmaE`t p/{n`Fªmo`Y t p mu[^` `l^su`FkukutzpVÁ¼Y . :. < E 6. 0. 9. :. N +1. 0. . 0. . 1. 1. N +1. N. N +1. N. 0. N +1. 1. j. 1 2. îîß. OPQSRQ. N + 21. N.
(73) ¼F. * . } ~ku` . } ~ku` . S G
(74) ( . j=1 σ1 ε1 E1n+1 − H2n+1 + H1n+1. σ µ H n+1 − E n+1 − E n+1 1 1 1 2 1. j=N n+1 n+1 n+1 σN ε N E N − HN + HN −1. = σ1 ε1 E1n + H2n − H1n = σ1 µ1 H1n + E2n + E1n. n n n = σ N εN EN + HN − HN −1. Z\[MqkY«lmu[`4¶sokm~|p w~kmWlt~zAp~| 2 × 2 ^zlºlk£z® M ~|p P _ qkmX` _azllt ¶`Y ~k σ µ H n+1 + E n+1 + E n+1 N N N N N −1. . ¢,[`so`
(75). = σ1 B1 = σN BN. M11 MN N. C1 =. . (+ ËÆ / ËÉ ÆE / (M :. < E <. :. 9. . + C1 − CN. 0 1 −1 0. . n n n = σ N µ N HN − EN − EN −1. ~|p. P11 PN N. = σ1 B1 = σN BN. CN =. . 0 1 −1 0. − C1 + CN.
(76). Á¼C|Â. . :. W`so`«|¢ `W zApkutl`Ys+mu[~|m)E`sot zlltw\XzAq^p^~suj zpXlt³mot zApky~su`~^^t `F9~|mymu[`,XzAq^p^~sut`Yk x = a Z\[tk moso~pkw~/mo`Yk tpAmoz
(77) x=b ~p E = E Á¼YA E =E ~p H = H H =H Z\[^`Yp%«lmo[^` ` l^so`YkoktzpXkµÁ¼YÂ\~su`4_azllt ¶`Y ~k£®¯zA z/¢k } ~ku` j = 1 #. ~p . . } ~ku`. 0. N. 0. N. N +1. N +1. n+1 σ1 ε1 E1n+1 − H2n+1 + HN. σ µ H n+1 − E n+1 + E n+1 1 1 1 2 N. j=N n+1 n+1 σN ε N E N − H1n+1 + HN −1. Z\[MqkY«Emu[^` ®¯z z/¢,tp^ | ~ p P
(78) M. 1. n = σ1 ε1 E1n + H2n − HN n = σ1 µ1 H1n + E2n − EN. n n = σ N εN EN + H1n − HN −1. ^zlºMke_#qXknm4X`9tpAmosuzllqX `Y²t p¡mu[^`9` l^so`YkoktzpXk,z|®ymo[^`9_a~|musot`Yk. σ µ H n+1 − E n+1 + E n+1 N N N 1 N −1 2×2. 1. . M1N = −A MN 1 = A. . n n = σ N µ N HN + E1n − EN −1. P1N = A PN 1 = −A. Á¼ A +C. í å(î í Û.
(79) ¼A¼.
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(81)
(82) !"
(83) $#%& '( )+*-,.+/10 . Å (SÇ ^Ê?Ê ËÉ / YÆ ÆE / (M ©p mo[^tkW~ku`«l¢ `~kokuq^_a`emo[~/m,mo[^`Ezq^pX^~|soj twkzpkut^`so`Y~k\_a` m~| tw"Á ®¯zAst pXknm~|p`« mo [^`so`*twk9~|ptp zA_at p^V¢ ~CA`*~/m mu[^`Ezq^p~|soj xx==ab«'moso~CA` tp^±®¯soz_mo[^`*` ® m9muzVmu[^` sut[SmªÂª qXksutzA¶Xp^_ soakm~ mo¨ [^zÀ`*sqllpM`q^s¦l_a`YhM`tzsotw_a`~|suX¨©)zSd Xkt ^zAmut tpS z`Ym#p*sazt mnp®\jSkuEM^`²tt`su¢ `F~|"X«'k®¯muzAso[^zsutw^_ktp^EmoIz[^q^` pXzAhS^pmu~|l`Ysot jVmu`tsuz¨ p¸z·~ptlk¦sut mu_at t_aztp^p¹E zAtwkuk ¼`Y¼ mo¸%su`Fku~/~/m"[mo`Y`mo·_a[^`²`µt pE~²z~Aq^ku¢£pkutw`F~|~|~|ºVsoj j9ku`xqXpXkk`F=`"¦Mtapj mo[^` zA_9qlmo~|mutzp~Àqt ljMp~_9twk\zpSmu`SmF #z/¢ ``YsY«SkqX[~#Àql*l`F zA_9EzAkut mutzp*so`YvSq^tso`Yk£mu[^` ©^p t~zAsozlpX`Y~|s t mu"Fz"~/multzz"p¡kzXz|«l®)¢£mo[^` `4_"{~~ºz`µ^qtw~|k` pVz_"® ~/
(84) musot V~kokzlt~|mu`Y²muz±~ zApk`Ysu/~|mut`®¯zAsu_ q^w~/mutzpVz|®\`YvEyÁËSª. :. < E E. :9 L. 9. Q=. . 9. D B. Z\[^`Yp%«^`FvXÁ?|AÂ\~|pE` ¢,sut mmo`p±~k. . :. = BW =.
(85). . ε 0 0 µ. . . E H. ,¢ t mu[ Z ≡ Z(x) = 1 Á°|AÂ − ~|pX x = b twk\t`Yp Mj
(86) Z\[^` pMq^_a`Ysutw~KÀql*¢,[^tw[± zAsuso`YkuXzAp^k)muzamo[^` XzAq^p^~sut`Yk x = aε(x) . ∂Q ∂G(Q) + = 0, G(Q) = ZQ ∂t ∂x. Z\[^` _"~|musot³ Z Y~|pX`^t~zpX~|t "Y`Y*~Ak Z = T ΛT ,¢ t mu["
(87) ~|p T −c 0 1 1 Λ= , T = 0 c cµ −cµ ·` p^z/¢Ìz_a^qlmo` t³mk,p^`YA~/mot A`µ~p XzSkt mut`4~smk
(88). −. 0. 1 µ(x) 0. − + − F 12 = (−Z)+ 1 Q1 + (−Z)0 Q0 , F 21 = ZN QN + (Z)N +1 QN +1 . −1. Z ± = T Λ± T −1 , Z + =. . 1 c 1 2 − ε. Z\[^`z_a^qlm~/mot zAp z®'mu[` pSq_9`Ysutw~EÀql ku` m (−Z) Q = 0 mo[^`"p
(89) − 0. Φ1,2 + Φ1,0. OPQSRQ. F 21. ,. Z− =. =. 1 2. . . 1 cε 1 −cε. 1 −c 1 2 − ε Q0. . 1 − µ −c. so`YvSq^tsu`Fk)mo[^`4[^zSknmkmo~|mu` kukuq^_a`4mu[X~/m,¢£`. 0. bWz/¢Ãmu[`4muz|m~|¥ÀXql¦®¯zs j = 1 twk. îîß. 1 − µ c. −1.
(90). F 12 = (−Z)+ 1 Q1. 1 1 = (AW2 + AW1 ) + (−Z)+ 1 B 1 W1 = − 2 2. H2 − c 1 ε 1 E 1 E 2 − c 1 µ 1 H1. !. Á°l¼FÂ.
(91) ¼C. * . j¡qXktp^±`YvEyÁ°l¼Fµtp·mu[^` zpSmo` Mm z|® mu[^`}£s~|pºA¨©btw zAkuzpVku[^`Y_a`a®¯zs _azllt ¶`Y ~Ak ®¯z z/¢k
(92). S G
(93) ( . j =1. «¥`YvE)Á¼YSÂ4~|so`. °Á  zs\mu[` _9`mo~ t4Ezqp^~|soj¦~/m j = N ¢£`4^sozl =`Y`Yµ~k\(σtp ku−q^ku`Y mutzp±+l ^¼/ ©p±kuq^_a_a~sujA«Mmu[^` ¶Xsokm~|p*mu[^` w~kmWltw~|Azp~ 2 × 2 ^ zlºlk\z|® M ~|pX P _#qkmE` _9zllt ¶`F~Ak
(94) ( ( M = σ B + D P = σ B − D Á°|AÂ M = σ B − C P = σ B + C ¢,[`so`
(95) ε1 (σ1 + c1 )E1n+1 − H2n+1. = ε1 (σ1 − c1 )E1n + H2n. µ (σ + c )H n+1 − E n+1 1 1 1 1 2 11. 1. NN. N. 1. N. D1 =. . . . . .
(96) . . 1. 1. c1 )H1n. 1. 11. 1. N. NN. N. . c1 ε1 0. 0 c 1 µ1. 1. N. E2n. 1. N. . zzAp q©kupÎtw_9l``·muso[^`YZ\tw k t _a
(97) k `F`·zAªmup»WtkzzA`YpÎsu_a/¢£~|~mut `·p tzknpÁ mo qz l®KejÏ~4Z,lk»4zAtwku_aÂ*su`¡_9`muz|`` ®#mu®¯[^zmuzlso[^Î_ `V^z_asu®zA~|mumuX[[^zS`,`Yk`_"`F¸`Y~/ªmu^motwsusozA~`_"eMt ~|zAsuqzApkuX ` jAmu`Y£tws£moZ\t ``F[^p^k*`¡`Ysuz®¯A® zjmu«C [^z/t pM¢,` At`p^©_as7k^t^~A twkt Et t mnm)`Yj moz|k®Etp^mu~|t [^muso``` t_a^tt³m_"~/mosut #[^s~ªmo`sot Y" t pmu[` µ Z,»¸_9`mu[^zl9~|p `/~ q~|mutzpz®mo[^`\pSq_9`Ysutw~lltkuE`sktzp% (+ S( Ç ?ÉÉAÆ / ( l Ç ËÆ < ; 6. . 9. L. +:. 9 &:. &:. 9. A`sot³·`¶X`Y,ksuku`Fz_a~ `#Mmo[z~/pmFku«|`t sop9/~/momu[^t`,zp `Y`Fp^vA`qXs~/~|mu%tzÁ?p »4Á¯Â+x+ z/zjMpSpAmomot t pMpq^zA¤ qk\kmu«C[^tw`Ykzzsomu`soz_¦^Â\tw\®¯zAs~Akmo`A[^«C` mu[^d²`,~|`Yl `F¢£ª`Ymu so°z¤ kW_"`Y~vSqp^~|`mumuttwz)p`Yk,p^¢,`sot mu[j pzaq^sosu`YpAmF^Z\[^twk mu[^`Yzso`_:kmo~|mu`Yk mo[~/m
(98) Z Z Á? S d P · nds = 0 Edv + dt ®¯zAsW~|pMj zAku`YzA q_9` V ¢,t³mo[¡~"so`Aq^~s,XzAq^p^~suj ∂V «X¢,[`so`4mu[^` ``YªmosuzA_"~|p` mutw4`p`soj E ~p*mu[^` x+z/jMpSmutp^¤ k\`FªmuzAs ~su`µsu`FkE`Y mut`Y j¦At A`pMj
(99) D. V. ∂V. 1 t ε EE + µ t HH , P = E × H 2 E H. ©p±mo[^`mo[^so`` ¨©lt_a`pkut zAp~|~ku`« ~|pX ~su` A`Y muzsk,¢,[^zSk`#``_a`pSmoke~|so` mu[` « ~p z_aEzp^`YpAmk+z|®Kmu[`,``Yªmosutw,~|pX _"~p^`mutw ¶`w^kYAbzmu`\mo[~/my®¯zAsy~4t`Yp9_a`mo~|t\Ezq^pXx^~|ysoj« E ×z n=0 ©¢,p·[`mo[^p tk#mu~|[^kpX`4`F ªmomut x(_atzz/p'`jM«¥pS`Y¢£mutz`"p^Xqlk¤momuk\t qzAmolp*[^j¡`z|zAmu®(su[^`Ymu`"[_1` EjM``[Xt``Y~CwªS^motk,suzzAmoq_"[sµ~/~|zmW®£pmumo[^` [^mu` tw`"W``¶`Y``Yªw momususozAtz_"k\_"A~|~|t AAp`p^` p` mumotwSµt j¦``p^mup[^`Y`su` Asotj*_aj²tw^kt p·tw` lt mumW~A[^ª`¦komo l[^j tw`ko _a zAso`"p` mukÁn`"`Y¼CsuYAª~`Y ku¥` E=. í å(î í Û.
(100) ¼F.
(101)
(102)
(103) !"
(104) $#%& '( )+*-,.+/10 . /Æ )ÆE ËÇ ËÆ GM. +: +:. 9. .
(105) . #K (
(106) S
(107) ( 1
(108) K(. 6. 1 . N. En =. 1X ∆xj εj (Ejn )2 + µj (Hjn )2 2 j=1. °Á Â
(109) . . .
(110) /Æ%Æ ®¢ `_#^q mut^j*mo[^` ¶skmW`YvSq~|mutzp z|®Án¼C|Â\Mj E + E ∆x ~p mu[^`9k`F zApzp^` Mj 2 H +H M « o m ^ [ ` ² p A ~ ^ ¦ o m ^ [ ` o s Y ` u k ^ q ³ m k l « o s. ` © ¨ | ~ o s s | ~ ^ p A , ` u m ^ [ 4 ` o m ` o s " _ k,~|p kuq^_ z/`Ys,~| j ¢ `µA` m
(111) ∆x 2 X Á°|AÂ E −E = (P −P ) ∆t ¢,[`so` P tk^` ¶p^`FSj
(112) ( S .
(113)
(114) #
(115) $'
(116) $#%K S
(117)
(118) E n+1 ≤ E n S 1(
(119) $#A ( ( S
(120)
(121) n+1 j. n+1 j. n j. n j. 1&
(122) . 4. (. j. j. n+1. N. n. j=1. n+ 21 j+ 12. n+ 12 j− 12. n+ 21 j+ 21. n+ 1 Pj+ 12 2. " n+1 # n+1 n+1 n n + Hj+1 Ej+1 + Ej+1 Hjn+1 + Hjn + Ejn Hj+1 1 Ej · + · = 2 2 2 2 2. ~|p·E`aku``Yp·~Ak ~ ltko so` mo`9zq^pSmo`so~|sumµz®)mo[^`"x+z/jSpSmot p^`Fªmozs ~|p¡mu[`"`YvSq~/mot zAp °Á |A mo[^`ltwkusu`mu`µ®¯zso_ z®'mu[` x(z/jMpSmutp^a`YvSq~|mutzpIÁ? Aª £vXÁ?|S ~|p E`kt_a^t³¶`F muzajMt `Y
(123) Á°A| E −E =P −P ∆t mymo[^twkyEztpAmF«¢ `_#qXknmm~|º`,tpSmuz#~Azq^pSmmu[`,mu[^so``zpl¶Xq^s~/mot zApk+z|®¥Ezq^p~|soj zAplt mutzpk ~Ak, zApktwl`Ysu`F¦tp kuq^ku`Y mutzpk^ ¼emuz*^ M Å (AÇ ^ÊËÊ ËÉ ÆX / (S V` zAlmo~t p
(124) n+ 1. Pj+ 12. D. 2. n+1. :. :. 9. n. n+ 21 N + 12. n+ 21 1 2. . E1n+1 + E1n H1n+1 + H1n 1 −(E1n+1 + E1n ) H1n+1 + H1n n+ 21 P =0 · + · = 12 2 2 2 2 2 n+1 n+1 n+1 n+1 n n n n + EN HN + HN −(EN + EN ) HN + HN 1 EN n+ 12 =0 · + · PN + 1 = 2 2 2 2 2 2. ~p*mu[Mqk E = E (+ ?Æ / ËÉ ÆX / (S V` zAlmo~t p"
(125) n+1. :. :. n. 9. :. . îîß. n+1 n n n + EN 1 EN H1n+1 + H1n E1n+1 + EN H1n+1 + HN n+ 12 = · + · P 12 2 2 2 2 2 n+1 n+1 n+1 n+1 n n n n n+ 1 E 1 + E 1 HN + H N 1 E N + E N H1 + H 1 2 · + · PN + 1 = 2 2 2 2 2 2 OPQSRQ.
(126) ¼. * . +D. mo[~/mtwk P = P ~p mu[Mqk E = E Å (AÇ ^ÊËÊ ËÉ / YÆ ÆE / M( Z(~|ºMtp^at pSmoz¦~zq^pSm,`YvE'Á°A£®¯zs ku[^z/¢,p mu[~|m
(127) n+ 21 N + 21. n+ 21. :. 1 2. 9. n+1. :9 L. n. :. 9. S G
(128) ( . . j=1. «^t³metk,`F~kut j. E n+1 − E n n+ 1 n+ 1 = PN +21 − P 1 2 2 2 ∆t. ¢,t mu["
(129). i c1 µ1 n+1 1 h c1 ε1 n+1 n+ 21 n 2 n 2 ≥0 P (E + E ) (H + H ) + = 1 1 1 1 1 2 2 4 4 n+1 n+1 n+1 n+1 n n n n 1 + EN HN + HN −(EN + EN ) HN + HN 1 EN P n+ 21 = =0 · + · N+ 2 2 2 2 2 2. ~p*mu[Mqk E < ;. ≤ En. . (+/ ?Ê ËÇ-* Æ Ç ( Ë +Ê ?É ËÇ# lÇ . 79. <. . n+1. +:. : :. :. ?. M : 1:. : H. `FvE%Z(Án¼F~|ºMÂ\tp^~| pIt pSmuÁn¼Fz ~Â\Y ~zAp q^pSEmµ`mol[^`F`9 zmu[^_asoE``"zAku`Yzpl~¶k Aq^
(130) s~/mutzpXkWz®£Ezq^p~|soj¡ zAplt mutzpkY«Emu[^`"_"~/mosut . M. z®. Á°|AÂ. ¢,[`so` D wt k~9^zMºS¨©ltw~|zAp~|E_"~/mosut ~|pX M tk,~akuº`¢\¨©kujS_a_a` mosutwe_a~|musot³¥ Å (AÇ ^ÊËÊ ËÉ XÆ / (S V` [X~C`
(131) ltw~| (σ B + C , σ B , · · · , σ B , σ B − C ) D = (+ ?Æ / ËÉ ÆX / (S V` [~CA`
(132) ltw~| (σ B , σ B , · · · , σ B , σ B ) D = Å (AÇ ^ÊËÊ ËÉ / FÆ ÆE / (M V` [X~C`
(133) ltw~| (σ B + D , σ B , · · · , σ B , σ B − C ) D = /Æ )ÆE ËÇ ËÆ
(134) . /Æ%Æ Vhlt p` ®¯zs`Y~A[ z®mo[^`~|Ez/`\ zApl¶qso~|mutzpk'mu[^`,_"~|musot³ M twkykuº`Y¢\¨ kujM_9_a`musot)¢£`\[~CA` mo[~/m,®¯zAs~|pMj X ∈ R
(135) M = Dm + Mm. m. :. :. 9. m. . m. :. :. 9. 1. :. 1. 1. m. :. 2. 2. N −1. N −1. N. N. 2. 2. N −1. N −1. N. N. N −1. N −1. N. N. N. . 1. :9 L. 9. m. 1. GM. 1. :. 9. 1. 1. . 2. +: +: 9 < #
(136)
(137) ( / 1
(138) #( '( G (
(139) . 2. "
(140) # (
(141) . . N. ". .
(142) K 1 G
(143) . m. 2N. X t M X = X t Dm X. í å(î í Û.
(144) ¼C.
(145)
(146)
(147) !"
(148) $#%& '( )+*-,.+/10 . )~A[±~ku`µz|®+XzAq^p^~suj*zp^t³mot zAp twk\p^z/¢Ìzpkutwl`so`Yk`Y~|s~/mo`j Å (AÇ ^ ÊËÊ
(149) ËÉ ÆE / M( AEhMtp ` ~|pX ~su`Ez|mo[¡kuº`¢\¨©kujS_a_a` mosutw4_"~/mosutw `Fk¢ `zlm~|tp ®¯zs :. :. 9. C1. CN. X 6= 0. X t Dm X =. N X. Xit σi Bi Xi > 0. ~p M wt k,XzSkt mut` l`¶p^t mu`~p mu[Mqk\tpSA`sumut^` (+ ?Æ / ËÉ ÆX / (SA ·` zlm~|tp*®¯zAs X 6= 0
(150) i=1. :. :. 9. :. X t Dm X =. N X. Xit σi Bi Xi > 0. ~p M wt k,XzSkt mut` l`¶p^t mu`~p mu[Mqk\tpSA`sumut^` Å (AÇ ^ÊËÊ ËÉ / FÆ ÆE / (M ·` zlm~|tp*®¯zAs X 6= 0
(151) i=1. :. :9 L. 9. :. 9. Xt Dm X = X1t (σ1 B1 + D1 )X1 +. ~p M twk,XzSkt mut` l`¶p^t mu`~p mu[Mqk\tpSA`sumut^`. N X. Xit σi Bi Xi > 0. i=2. FÇ / * Æ ,Ç ( ( (+ ËÉ%Ê / Ë (+/ ?Æ Z\[^`4t_a^ tw t mWko[^`_a`"Án¼C|Â)®¯zs « « ÁËzpkmo~pSm ε ~p µ tp*mo[^` ¢,[^z`4lz_"~|tpX ~p*®¯zs~9q^p^t ®¯zso_:_a`Yku[ ∆x = ∆xε =¢,suεt mu`Fµk
(152) = µ ∀j < ;. E. . ?. :. 9. : +M. j. +:. 9. j. j. n+1 − Ejn Ej ∆t n+1 − Hjn H j ∆t. =. 1 n n+1 n+1 n ) ) − (Hj−1 + Hj−1 (Hj+1 + Hj+1 4ε∆x 1 n n+1 n+1 n (Ej+1 + Ej+1 ) − (Ej−1 + Ej−1 ) 4µ∆x. Á° A C. )vE+Á? AÂ,Y~|p²~|wkuz¦X` ¢,sut mmo`p²~Ak,mu[` ltwkusu`mut#"Y~|mutzp±z|®y~¦ku`Yzpl¨°zAsol`Ys,¢ ~CA`4mnjMX` `YvSq~/mot zAp ®¯zAs E .zs\mu[tk\^qsuEzAku`«l¢ `4¶sknml`Flq `4®¯soz_ mu[^`4¶skm\so`w~/mot zAp z®`FvXÁ? AÂ
(153) 1C. =. Ejn+2 − 2Ejn+1 + Ejn (∆t)2. C. = =. îîß. OPQSRQ. ! Ejn+2 − Ejn+2 Ejn+1 − Ejn − ∆t ∆t " n+2 n+1 n+1 n Hj+1 − Hj+1 Hj+1 − Hj+1 1 + − 4ε∆x ∆t ∆t # n+1 n+2 n+1 n − Hj−1 − Hj−1 Hj−1 Hj−1 − ∆t ∆t 1 ∆t.
(154) ¼F. * . mo[^`p'«^Mj¦qkut p^9mo[^`k`F zAp*so`w~/mot zAp ¢ `4`m Ejn+2 − 2Ejn+1 + Ejn (∆t)2. S G
(155) ( .
(156). n+1 n+1 n+2 n+2 (Ej+2 − 2Ejn+1 + Ej−2 ) (Ej+2 − 2Ejn+2 + Ej−2 ) + 2 2 2 4(∆x) 4(∆x) n n − 2Ejn + Ej−2 ) c2 (Ej+2 2 4 4(∆x). ". c2 4. =. #. Á?A ¢,[t[²twk4~ zpkutwknmo`pSmµ~^^sozCMt_"~/mot zAp±z|®mu[`9¢\~C`#`YvSq~/mot zAp±®¯zs E j²lztp^~*kut _at~se_"~|pt³¨ q^~|mutzp ®¯zAsmu[`#` l^tw t mµko[^`_a`¢ `#A` mmo[^`®¯zA z/¢,t p¦ zpXktwknmo`pSme~^^sozCMt_"~/mot zApz®mu[^` ¢ ~CA` `FvSq~/mot zAp
(157) H − 2H + H − 2H + H Á?^¼F c H = (∆t) 4 (∆x) ±( (+ ËÉ ^Ê / Ë (+/ ?Æ Æ ÎÇ ( ? +Ê ËÉ ¯Ç¸YÉ (M ( ®9¢£`·t pSmosuzllq`It pÌmu[^`IpMq^_a`sotw~| ¢£`4A` m
(158) ko[^`Y_9`"Á?Â\~p [~su_azAp^tµ¢\~C` E = E e +. n+1 j. :. : +M. +:. n j 2. 9. :. ?. n j. n−1 j. 2. M : :. ?. 0. n j+2. n j 2. n j−2. . i(kj∆x−ωd n∆t). c2 (eiωd ∆t − 1)2 = (eiωd ∆t + 1)2 (e2ik∆x + e−2ik∆x − 2) 2 (∆t) 16(∆x)2. mo[~/mWY~|pX`kut_9 t ¶`Y ~Ak.
(159). . ωd ∆t sin 2 (∆t)2 2. . =. c2 cos2 4. . ωd ∆t 2. . sin2 (k∆x) (∆x)2. ®¯soz_¢,[^tw[±¢£`Y~|p²l`Flq`#~|p±`Msu`Fkukut zAp®¯zsmo[^`s~/mot z"z®(mo[^`#pMq^_a`Ysutw~%^q^wku~|mutzp ω moz"mu[^` `^~ªm,^qko~/mot zAp kc ~k,~ ®¯q^p mutzp±z® ∆x ~|p mo[^` } +VpMq^_ X`Ys α = c∆t
(160) d. ∆x. 2 = ± arctan kc∆t. ωd kc. . c∆t sin(k∆x) 2∆x α 2 arctan sin(k∆x) = ± αk∆x 2. . }£`Y~suj«lmo[^`#~Xz/A` so~|mutz"jSt`w^k~* sot³mo`sot zAp*®¯zAsW`/~ q~|mutp^amu[`pMq^_a`sotY~|'ltwkuX`Ysokut zAp z|®+mu[^` /z A`s~|lku[^`Y_a`% ®¥¢ `l`pz|mu`WSj K = k∆x ~paqkut p^Z(~CjM zAs` l~pktzpXk+z|®Kmu[^`mu`so_"ktpSAz`F ¢ `4zlm~|tp
(161) Á?A ω α +2 2 + 10α + 3α q (α, K) = = 1− K + K + O(K ) kc 12 240 Z\[^`#X`Y[~CMt zAq^s,z|®+mu[`#vSq~|pSmut mnj twk,so`^so`Yku`pSmu`Fzp¶qsu`# ®¯zAselt ¾K`so`pSm/~|q^`Yk z®'mo[^` } +IpSq_#E`s α «~k,~ ®¯q^p mutzqp z|® (α, K K) ·`~pku``µmo[~/m
(162) . 2. d. 2. 2. 4. 4. 5. . í å(î í Û.
(163) ¼/.
(164)
(165)
(166) !"
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