A New Finite Volume Scheme for Solving Maxwell System
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. A New Finite Volume Scheme for Solving Maxwell System Malika Remaki. N° 3725 1999 THÈME 4. ISSN 0249-6399. apport de recherche.
(3)
(4)
(5) !#"$%# &' ( ) *,+ - %.'/0
(6) 31 2547698:2<;>=@?258A6 BDCFEG>H#IKa JHb_dc@LN_dMOT^GE G>PFQSH RU_fTVMOe WYWYXZG>H\FT[QOWYH gN\]H T^_MSG>MO_VRUT^MSW`X hjidWUklHTnmpofqlrso,t ufR`\]\vWYilT a Hbi^HewCFH idewCFHxX5y{z}|Y~`J Y`J ~` \:R`YH_ !F`} qXT^CFMS_n\:R`\H i vpH\FidH _dH X}TRXFH:XFMTVH#YWYQOPFG>H>_^ewCFHG%HT^Wi^H_^WYQYH TVCFHrR!g@H QOQH P:R!TVMOWYXF_ [BDCFHR`\F\FidW}R`ewCeWYXF_dMS_lTV_,W` RQOHR`\] ¡idWYT^MSG>H_^ewCFHG>H>R`X a a£¢ a a a Y\FR<H i^HWYe_^H G>H X}X}H TVTVT^Hi^H i^MOaH H _RUX T^a CFPNH g¤XidPFH ¡ ¡G>WYWYi^idHG#Hi^MOPFeQ¥R`RNHQ-P]H gN_^BDH\CFH PFidMOXF_ZMSG>_dT^G>H i^X}PFH TVT^e CF_,TVPFW R`ididH Ha MO_\YH pidilM ¡Ha WYQS_ i^Q¦G>_dofPFH a MXTV H R`XFR` ¡WYQOc@i_dMSC:_fRUW`X T^QOCFMSXFH#<_deweCFWYH G>G>\FHKQSH§MOg _ ¨s©}ª¬«dK® !¯°`²± QSHeTVidWYG%R`YXFH TVMO_^G£³rR!g@pHQSQH@PFRUTVMOWYXF_0³´MOXFMTVH{µjWYQOPFG>H³lLTwR`¶FMOQOMOTc`. Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles, B.P. 93, 06902 Sophia Antipolis Cedex (France) Téléphone : 04 92 38 77 77 - International : +33 4 92 38 77 77 — Fax : 04 92 38 77 65 - International : +33 4 92 38 77 65.
(7) &K * / * &K# /s / # " /0.'/A
(8) ) *,+ ²
(9)
(10) t[WYPF_f\Fi^WY\WY_dWYXF_ a R`XF_neHi^R`\F\WYidTfPFX_dewC G%RYWYQOPFG>H _f:XFMO_,\vW`PFifQSR i _dWYQOP]TVMOWYX a Pa _lcN_l¢ TVEG%H a HDrR!g@pHQSQ]¶:R`_ p_^PFi{P]X_^ewC a GRa L]RUP]TVH§³lrsWYPNaTVWYXHXKTVH G>\F_ H@TKP]H P]QOX@P]eH R`_>QOe iP]_^Q PFQTwH RUT^_PNg XPFe GH X} T^i^i MO _PFH H_%X _^H H_di^\:WYR`X}e THU }R`fQOH X]G>H H X}T^T>P \FH i _^HHX}_dTTV R`_¶FMOQSMT RUR`P X}TV_^R`ewYC H G%a RH'e H HT _^Rew`C H e Ga RH _H_d TWY_^GW`XT^e i^W MOH ]_[T@i^H P]QSH RUQOT^e MOWY`XFH G>PFH HX}_ T 9R`MO¶FQSHUFHT_VRKeRU\:R`e MT "!TViVRUMOT^H i a H _D\FidWY¶FQOE G>H _ # ® « % $
(11) & QOH eT^i^WYG%R``X T^MS_dG%H>³ & @PFRUTVMOWYXF_ a HrR!g@H QOQj³µjW`QSPFG>H_#´°MSXFMO_,³ L@TwR`¶]MSQOMOT `.
(12) z.
(13)
(14) "!#$%&
(15) ('*),+#-./
(16) 02143 5 # >
(17)
(18) / 6 798 ® ¯ `4: ® 8 < = © Y:?> 4: ® 8 ®2@ 4A © © 4A ® ¯. ~@~@S ~ DBno CFX:HR`QFc@_^XFMOMO_nT^H¦W` YWYTVC]QOPFHG>XH#PFG>_dewHCFi^H MOG>eR`HKQ_^ewC]H #G>bH ########bb###### N~ ~@S L@TVR`¶Fa MOQSMTcZR`X:RUQOc@_^MO_ #b####b### ~N~N ~@~@ z~ L@mpT^WYP X]_^c H ilW`U RUTVTVMOC]WYHX_dWU\: °R`T^eCFHHbR`X@ee P]PFG%i^R`Hi^eMOc eR`QH#XFH id`c ###bb###### ~N ~@ I L@T^P a c W` TVC]HXPFG>Hi^MOeR`Q a MO_^\H id_^MOWYX$##b### C D © :9Y $ (© E > © : © 8 zNzNS ~ BD± `CFWYH#QOP]idH TVMOe WYTwXR`XFWU` {PFRQ¥RUG>ipW a RHYHWUY PFidM H a _^H W`X:R`#XFebH ########bb###### zN z hjidWY\:R`YRUTVMOWYX W` R%\FP]QS_dH ##b####b### zN I LNeRUTdTVHi^H a R`HR`ei^WY_d_[R a MOH QOH e TVidMSe R`QGR!TVH idM¥RUQ #a b### zN mpWYG>\:R`i^MO_^W`XsMTVCsRr ,LFm :X]MOT^Hb`WYQOPFG>H#G>HT^CFW ### ; H ® 8 % $ # : ® 8 B JI 8 ®@ $ © L¯ K © © 8 . MLMONQPGR#SUTV. ; B. | |. I~ . 6GF. | ~N ~Y~ ~UI. <B <B.
(19) I. ) +Q + + . >
(20) " Z
(21) # BDCFH,:XFMTVH,`WYQSP]G%HbG>HT^CFW a _R`i^H¦eWYXF_dM a Hi^H a XFW R`_eQ¥R`_d_^MOeR`Q0T^H ewCFX]MSPFH_[T^W_^WYQYH e YWYi^MXFa _dH e ilH`QSQ]R!TVR`MX Ya HpQSeRWYXF_°_dT^\]i^i^PFWYe ¶FT QOaH G>MO_^e_i^Hq T^T Ha R`H \Fid\FMO`idHWag]MO¶}G%cbRUMOTVX}MOWYTVH XF`_{iV ¡RUW`T^i°MSXFTVC]HTVC]i^HH_^¶:PFRUQTV_^MOMOXFe¦H@MOXPFT^RUH TVYMOWYi^R`XFQN_ T^WH YidH G%i°_R CFMSewC \Fi^W@M a H _#RYH ilc X:RUT^PFiVR`Q°Rc TVW PF_dH>PFXF_dT^i^PFe a TVPFidH a G%a H_^CFH_#T^C:RUTbe W`X] ¡WYi^G TV\FW'i^WYe\WYWYG>_dH \Fa QOH G>gsHYTVH C]W`W G%a H _TV ¡idWYMSHi{_KT^CF[Hf´pr [R!H g@idG%H HQOQQSMOHXF H P: RU z TV 7MOvWYt#XF_
(22) > CFrMOewR C'_^idH H X PFRUMOXi^H[R¦ub\F i^MOG% ¬R`MOidWYcKQ }Yi^M _ a @M R`X I a. a a a PFaR _^%M $vH P:aH idRUH ¶}QNXFcYeidHM
(23) R` ¶:LvX R`a _dHH RH:WYXFX#MTVTVH%CF 7H YOnWYYQO!H PF #QS"nG>R`PFTVHWZXFG>Re c}W`H ³ TVXFµjCF_dWYT^W i^i^a WYPF_KX]e TKW RU0\FR G>\FC}QOH MScNH _da¶]CFi^H TVM _a W BD_^TVewC]CFCFHH H_dG>rHpHTVR HgNe ewWYCFG#H XFQOMOQ¶FMO_dPFXFc@H MS_dX]_{T^H G H RidHD:BDXFR`QOMCF_^TVWHH GRUMSX a iVR¶:R`&e sW` TVCFH_^HKG>H TVCFW a _,MO_nTVC]H a (M 'ePFQTacTVW%:X a T^i^M a MSG>HXF_^MOWYX:R`QG%H_^aC YH XF +HiV RU~ T^ #WY"§i^_* CF)nHTVi^CFHHi^R`_QSQTc@T^CF\HH _:WUH 0Q a :XF`MH TVe H[TVYWYWYid_QOPFR`G>idH,HZG>QOW@HeT^RUCFT^WH a _pR!C:TRYTVCFH,HZR`QS_V_dR`WG>¶HZH H\vXW`MSMSX}X}TVY_H _dT^MSBD}R!CFTVHH _^H GTV%H H ewC]TVCFXFW MSa PF_bHPFTV_dW%HYRUHX TnP]R>\CFMSMO`X Csa WY_^i ewa CFH Hi[G%_^H%ewC] ¡H WYG>i,H`TV¬C]R`HX _^a \:R`ReH'uPFRUXF\FY\FH§i^,³W
(24) ¦gNMOP]GTdTwR!R>TVMO_^WYCFX HG%MHTVC ¡WYRZiTVr C]H,T^L]MSmG> H a a a a Yi^H WYMS_dQSQORUe PFT^idG>MOH`TVHH %M -Qc_^R!ewTVCFMSMOG>WYHX0G>\,WYH _ idB TV WR`X}BDRT#YCFWYmH M _dh HZ-eG>WYT^X]H MSTV ¡G>PFCF_dWH>MSa WYe _X¬WYvG>YM\:YH RUH i^HRH a X]YW`TVW@WTVW Ha :T^X]CFi^MOHH T^_^_dH WYHKQOa P]G>M%T^$vMSHWYH T^idX¬CFH W XF¶]e_¦H P]_TR_^TVrewCFCFH nHc G%LFidmHH _. PF/:MOi^XFH H HMTV RH_ _^ewCFHG%HD ¡WYiH g]R`a G>\FQO0H 1" 2 HD\Fa i^W`\vWY_dHCFHi^H[R,a G>WYi^H_dMSG>\FQOH[R`X a H aeWYXFWYG>MOeR`QN:XFMTVH YQSH WYR`QO\]PF ¡G>i^W`H#G>_^ewH CFTVCFHG%W H> ¡WYC]inMSewT^CsCFHKMS_TV¶:MOG>R`_dH H a MOWY_^e XsidHR>T^M e- H RUX}TVTVMOWYHi^X0H BD_^\FCFR`MO_¦e HG>_^HewT^CFCFHW G>a HKMO_bR`X _^H eRWYX a WYi a H ibR`e§³ e PFi^RUTVHUbR`X a CFR`_sX]W XPFG>H idMSe R`Q a MS_d_^MO\:RUT^MSWYX¬3 H \Fi^Ha _^HXTMSX3TVCFMO_Z\:R`\H iZ_^WYG>H TVCFHWYi^H TVMOeR`Q°R`X:R`Qa c@_^MO_nW` TVCFMO_f:X]MOT^H`WYQSP]G%H_^ewCFHG%HUAR`X pH#YMYH_dWYG>Ha XPFG>H idMSe R`Q H gN\vHi^MOG>H X}TV_nR`X e WYG>\:RUi^MO_^WYXF_MTVCsR'r nLFm :X]MOT^Hb`WYQOPFG>H#G>HT^CFW . . 465LM7468.
(25) .
(26)
(27) "!#$%&
(28) ('*),+#-./
(29) 02143. /p"°
(30) #
(31)
(32) H#\Fi^WY\WY_dHMOXsTVC]MS_,\FR`\vHi[T^We W`XF_dT^i^PFe T¦R`X a _dTVP a cR%XFHFXFMOT^HYW`QSPFG>HK_dewCFH G>H ¡\vWYWYi¦MOX}TVTVCF_HnrR`X R!a g@RUH i^QOHQe_lc@WYXF_dTV_^HM aG H i^H a H%R`_XFW`R`T^X3H>R`T^C:\]RU\FT#i^WR`gNQOMSQ{G%TVCFRUT^H>MSW`:X3HQ aWU _RR`i^H>_dWYQOQOWNP]e TVRUMOWYTVXH a WYR!X TR T^CFYH%W`_VQSPFR`G>G>HH CFMSewCMO_eR`QOQSH a Re H QOQWYiKRe WYX}TVidWYQp`WYQSP]G%HU He WYX]_^M a H i%R`_KeWYX}TVidWYQpYW`QSPFG>H_ TVpCFHH%QSQ H R`QOH \FG>\]i^H WYX}\FT^_KidM¥R!W`TV H>T^TVCFW HTVG>i^H HRU_^TC \F9i^R`W`_¶FMOQSX HG>T^_CFH%MOX FYCFPFH i^TVHZHi^WYY ¡HW`XFi#H W`H§gFPFRU_G%G>\]H QSaH " M¥R] ¶vR`HX ea R`P]TV_^W HTwT^RCFUHH'c MOX}R`TVidWH R`e eWYPFX}TTVCFHbe P]i^i^HX}Tn_dCFH H TV_ . Vj x. Vi. Gj. Gi. ´°MSYP]i^H mpWYX}T^i^WYQ0`WYQSP]G%H .
(33)
(34)
(35) !#"$
(36) %
(37). Hbe W`XF_^M a H iTVC]H¦e WYXF_dH il`R!TVMYH# ¡W`i^G WU °T^CFH#rR!g@pHQSQ _dc@_lTVH G `MO`H X¶}c&. 476. CF< H idH. ". ')( (;:;<>=@? '+*-,/.1032547698. (. <. A. a RUX a = MO_sTVC]H H QOH e TVidMSe FH Q a O M s _ V T ] C H % G ` R Y F X. H V T O M e : H Q NMOT^C. 83BDCFE :. 8H6IJCK6MLCK6N PQQ T QQQ QQ = UWVYX[Z Q U]\^XJZ R (;:< Q T 6 IO8 QQ QQ QQ _ VYXa` QS = _ \bXa`. MLMONQPGR#SUTV. : 9G c QQ. PQQ Qd. Q. QQQ. QQQ QQ QQQ C. e. QQ Q. QQ. 6ML18. (W:f<. R. QQ. QS. QQ. QQ. QQQ. UgVYXJZ c QQ QQQ QQ T Qd =UihJXJZ = _ VYXa` QQQ C QQ QQ T Q _ hJXa` e. 6N18. (;:<. PQQ = Ui\bXJZ QQQ QQ U]h[XJZ Q T R QQQ _ \jXa` QQ QQ = _ hJXa` QS T. c QQ QQQ QQ Qd QQQ C e. QQ Q. QQ.
(38) . ) +Q + + . H T _eWYXF_dM a Hi T^CFH[H QOH G>H X}TV_W` ¬T^CFHG>H _^CR`_je H QOQO_ NR`X ¡WYi^G l p" WYHifH R`ewCZe WYX}TVidWYQ0YW`QSPFG>H 9eH QOQ/" . . ')( '$* .52,. . 476 8. . (;:. .. <. a MOX}TVHYiVRUT^HDTVCFH[e WYX]_^H ilURUTVMYH 7~ ". T. a a TVMYCFHbH MSid_H eWYXFMO_ _dTVT^R`CFX}HfTnW`W`P]XZTH R`RUewi CsP]`WYXFQOMOPFT G>XFW`H i^ G%R`NQ:T^CFTVW
(39) H XZ p HbAWYH¶]T TV_ R`_^MOXZPF\FT^\CFWYH¦_d ¡HfWYTVQSQOCFWRUTjMSX]TVCF>H ¡W`TVMOi^G>GH P]Q¥RUHT^i^MSMW`UX R! ³ ' ( 8 : 8 '$* : , 9z " &% < T "!$# ' 8 : a HXFW`T^H _bTVCFH%XPFG¶H iW` X]H MOYC¶vWYP]i^_W` pT^CFH%`WYQOPFG>( H R`X a )% R ¢ P@g ¡PFXFeT^MSW`XZCFMOewCRU\F\Fi^WYR`ewCZT^CFHbMSX}T^H Yi^R`3Q 476 8 (W: . #. . $. ,. +* . E G> > ®E : ]&: ® 8. H[HUR`QOP:RUT^H[T^CFH ¢ PNgH gNewC:R`XF`H a ¶HTH H XTVCFH[YWYQOPFG>H,R`X a MTV_ XFH MOYC¶vW`PFi^_¶cT^CFH ¡WYQSQOWMSX]\FidWNeH _d_j &% <.47- 698 (;:/0W< 8 ( C ( : # - (;: MS_#R`X RU\F\Fi^WgNMOGR!TVMOWYX W` (;: RUT CFH idH 0 < A 8 021 C 043 :9< # . .. 7 4 9 6 8 7 4 6 8 " * TVe CFH X}HT^MOH Xi^T^H H asil 9¢R`e P@H gZ' MS_ 6a 5 H :' XF 8H a7 @R`¶}X&c a ( MO_jTVCFHDUR`QOPFHfWU vT^CFH[FH Q a WYX>TVCFHD`WYQSP]G%H9 °BDCFH . C. : © ¯L:9Y © &:ED}]&: ® 8. 0: 0: : <8 ;C=; :<?> @8 ;C , > 8<; C A > 8<; C $0 : <60B 47698@; :. MTVC. of TVHiR%_d\:R`eH a MS_de i^H TVM%-RUT^MSW`X0:pHWY¶]TVR`MSXT^CFH¦ ¡WYQSQOWMSX] ) ± .1B * ,GF . .5E * ,HF .. 1 3. 8E 83B. :f< :f<. . T T. 465LM7468.
(40) | CFH idH F 1 R`X a F 3 idH \FidH _dH X}T¦TVCFH#_^\:RUe H a MO_^ei^H TVM%-RUT^MSWYX¬¦´:WYi[T^MSG>H a MS_de idHTVM%-R!TVMOWYX0 pHbPF_dHT^CFHb_^H eWYX a WYi a HiDQSH R`\] ¡idWY>_^ewCFHG%HCFMOewCMO_ a H:XFH a WYXZHRUewCZ`WYQSP]G%H¶}c PQ < B = i* F 1 8 E : R B ¡I " QS 1
(41)
(42) "!#$%&
(43) ('*),+#-./
(44) 02143. E. .
(45) # ;
(46). . <. E. = i*. F. 3. :. 8B.
(47) $ " 1"+ @
(48) %-
(49). HDMOQSQ5_dT^P a c>MOX%T^CFMO_ _dH e TVMOWYX0TVCFH[XPFG>H i^MOeRUQewCFR`iVR`e TVHi^MO_dTVMOe _W` ¬T^CFHD:XFMTVHDYW`QSPFG>H _^_dTwewRUCF¶FHG%MSQOMHTcYa H_^ei^MO¶vH a RU¶vWYHU HH UR`QSPFRUTVHT^CFH_dewCFH G>H WYi a H iR`X a MOT^_ a WYG%R`MOX W` . < < 6 F : $ : ª 8 $ ª :7 ® > ® #:¡&: ® 8 6. . 5´ W`i'TVCFH _^R`H W` #_^MOG>\FQSMOe MTcY,H e WYX]_^a M a H isR G%H_^C3G%R a a H a PF\ W` #\:R`iVRUQSQOH QOH \FMO\H a _ W`¡: °YT^P]CFi^HbH~_^ew" CF:HG>CFH.H id9H#I "pT^CFMS_H#H eX]WY_^G>PF\FidH P]a TVRU¶}TVcMOWYX:T^CFR`H¦Q ¡WYW`QOGQSWRUMSMOXsXFMS_[ePFi^MXTV¶vHi^W`MOPFWYX X H BDC@P]_ 5TVCFH#_dTwRU¶FMSQOMTc hi* 8 " ! 3 , " # 3 , % $ 3 :'& A 7 " C]H i^H h MO_T^CFHG%R!gNMOGPFG R`H\FC:R`_dHs`H QOWNeMOTc MSX TVCFHse WYG>\FP]TVRUTVMOWYX:RUQ a WYG%R`MOX0 ®0®2@®2@ &A © > ® > ® :¡4: ® 8 H#e WY:XF_d< M a H i¦,R'! X]WY"i^G%# R`QOM--H $a : idH Y MOWYX_d\:R`eHMOT^C. . 8 0FC)(C+*. MLMONQPGR#SUTV. 8H0. C(. C+*. Zg<. . R`X a. ` <. . ¬R`X a pH a H XFWUTVH.
(50) ) +Q + + . z. H E. (i,j,k). y. x. ´°MSYP]i^H#~#-h WY_^MTVMOWYXsW` °T^CFH¦:H Q a _[WYXTVCFH¦:X]MOT^Hb`WYQOPFG>H#YidM a. 465LM7468.
(51) .
(52)
(53) "!#$%&
(54) ('*),+#-./
(55) 02143. BDC]H:X]MOT^Hb`WYQOPFG>H#_dewCFH G>HMS_TVCFHXs`MO`H XZ¶c . h 8 0 , 31 1 C(, 31 1 C+* , 31 :< h 8H0), 1 31 C(, 1 31 C+* , 1 31 : 3 A V \ 8H0 , 3 C(, 3 C+* , 3 : = \ 8H0 , 3 C(, 3 C+* = 3 :
(56) , 1 1: = V = 3 A \ V 8 0 , 31 C)( = 31 C+* , 31 : 8 0 , 3 C)(, 3 C *9, 3 A : \ 8 0 , 31 C(, 31 C+* , 31 :< \ 8H0), 31 C(, 31 C+* , 31 : 1 1: = V = 1 1 1 :
(57) A h V 3 3 3 3 3 3 3 , 8 0 , 1 C( , 1 C+* , 80 1 C(, 1 C+* =, 1 : = 3 A V h : = h 3 3 3 3 3 80 , C(, C+* , 80 , C(, 3 C+* : V 8 0 , 31 1 C(, 31 C+* , 31 1 :< V 8H0), 1 31 C(, 1 31 C+* , 1 31 : 3 A \ h 8 0 , 3 C(, 3 C+* , 3 : = h 8 0 , 3 C( = 3 C *9, 3 : , 1 1: = \ = 1 1 1 : = 3 A h \ 3 3 3 3 3 3 8H0 , C(, C+* , 8H0 C(, C+* , : h 8 0 , 31 C)(, 31 C *9, 31 :< h 1 8 0 , 31 C(, 31 C+* , 31 : 1 1 1 1 V A 8 0 , 3 C(, 3 C+* , 3 : = V 8 0 , 3 C( = 3 C *9, ,3 \ 1 1 1 1 = 3 A V \ 3 C(, 3 C+* , 3 : = \ 8 0 , 3 C(, 3 C+* = 80 , : \ 8 0 , 31 C)(, 31 C *9, 31 :< \ 1 8 0 , 31 C(, 31 C+* , 31 : 1 1 = 1 1 A V h : = h 3 3 3 3 3 3 , 8H0), C(, C+* , 8H0 , C(, C+* 1 1 : = h = 1 1 = 3 A h h 8H0 3 C(, 3 C+* , 8 0 , 3 C(, 3 C+* , 3 : V 8 0 , 31 C)(, 31 C *9, 31 :< V 1 8 0 , 31 C(, 31 C+* , 31 : 1 1 : = \ = 1 1 A h \ 3 3 3 3 3 3 , 8H0 , C(, C+* , 8H0 C(, C+* , 1 1 1 1 h h = 3 A \ 3 C(, 3 C+* , 3 : = 8 0 , 3 C)( = 3 C+* , 80 , :. MLMONQPGR#SUTV. 1. 9 ". 3 1. 3 3. :. . :. 1. 1 3 : :. 1. 1 3: 3. :.
(58) . ) +Q + + . H a HXFW`TVHg PQQQ QQ (. % % . 8 , \ 8 , V 1 8 , h \ 1 8 , 1 8 , V 8 , h. QQ QQ QQ QQQ QQ <. QQ QQ QQ. . QQQ QS. . f_^MOXF>TVCFH´:WYPFidMSHiTVi^R`XF_d ¡W`i^G%RUTVMOWYX0NcNMOH Q a (. . 1. 1 3 C)(, 1 3 C)(, 1. 1 3 )C (, 1 3 C 9 * , 31 C(, 31 C+* , 31 C(, 31 C+* , 3 C(, 3 C+* ,. 1. : c QQQQ. 3 3. 13. 1 1. QQ QQ QQ QQ. 31 3. :. : :. 31. QQ. :. QQ QQ QQ QQ. : e. QQQ Q. _[TVCFH¦ ¡WYQOQSWMOXF>_dc@_dT^H G. % % ( <. 1 3C9 1 * , 3 C *9, 1. . 7| ". CFH idH 1 3 T A R`i^H%TVCFH'´5W`PFi^MOH iR`XFYQOH _sBDCFHR`G>\FQSM:e RUTVMOWYX G%RUT^i^M g % % C idMOT^CH _[ MSX
(59) TVC] MS_C eRU_^ Hg PQQQ 1 3 c QQQQQ T QQ 3 ( 3 = QQQQQ QQ . T QQ QQQ = = T QQQ QQ Q % % < ., QQQQ T = = T T T QQQQQQ " QQ T T T QQQQQ T QQ QQ = T T T T QQQ QQS T T T T e CFH idH 1 < = \A ^0 3 3= VA ^0 3. 3 < h A \ b0 1 ^0 3 < h A V ^0 1 ^0. < 0 VA ^0 . < 0 \A b0 3 . 465LM7468.
(60) Y.
(61)
(62) "!#$%&
(63) ('*),+#-./
(64) 02143 . 3 3 h A ^0 1 = VA b0 < \ A V ^0 3 ^0 . . < 0 h A ^0 1 < = h A ^0 3 1 = \ A b0 3 3 _^P '%e MOH X}T _dTVR`¶FMOQSMTc e W`X a MTVMOWYX ¡µjWYXtfHPFG%R`XFX 3<. =. DB CFH<XFH eH _^_^R`idcRUX a e W`X a MTVMOWYX " i^MTVH _ . 1 3 :
(65) T C A C 1 % 3 % % % %
(66) % % & 8 C C. CFH idHbT^CFH¦UR`QSP]H _
(67) % % i^H \]i^H _dH X}T[TVCFHbH MOYHX UR`QOPFH_nW` TVCFHR`G> \FQOM:eRUT^MSW`X GR!TVi^M gv H¦:id_dT[e WYX] _^M a H iDTVC]HHMS`H X}UR`QSP]H _nWU °T^CFHbGR!TVi^M g < = MOT^C . .. H T. C. *. 8. . :f<.
(68) =. 8
(69). < :f<. 3. 3:
(70) 3. A 3
(71).
(72) , A
(73) ,8 A , , , *3 3 3 i * 3. *3 3 1 i i 3 3 ! ^0 , "# ^0 , -$ 3 b 0 , 3 3 3 3
(74) , A
(75) ,3 8 A , :
(76) 3 , A
(77) , 8
(78) :f< 8
(79) ,
(80) , : 3 8
(81) :f< T
(82) ,
(83) , < T
(84). SP. A © : © 8 ® 8 ¯L:¡4: ® 8. % % 3 ce. HbC:R& YH_dTVR`¶FMOQSMTcW` TVC]H_dewCFH 1 G>3 HCFH X & H¦YMYHCFHi^H¦TVCFHb\FidWNWU ¡WYi FTVC]HHMS`H X}UR`QSP]H _
(85) % W` TVCFHbG%RUTVidMg R`i^H¦YMYHX¶}c .
(86) 1 % 3<. =. 0 = 3 A 3 8 A = : ,8 . ,
(87) < CFMSewCsMSG>\FQOMOH _TVC:RUTDT^CFHb_^ew CFHG>HMO_[_lTwR`¶]QSHU <. MLMONQPGR#SUTV. . =. 3: <. . ..
(88) ~ . C. ) +Q + + A © 8 © © F ª ® 8 ¯L: 4: ® 8. H´5W`i \FidW `HbFCFH idHbHC:T^RC:YRUH TD ¡WYi ] TVCFHb_dewCFH G>HMO_MSX]_dTwRU¶FQSHU b 3 = = LNPF\F\WY_^H¦T^C:RUT^.
(89) 1 %3 <. A. <. .
(90) & , 3 = & . ,. TVCFCFMO_MSew_dCTVP MO_a RUc¶FTV_^CFPFH¦i a ¡WY_^QOMOQOXFW eH MOXF>_dTVR` ¶FMOTVQSCFMTHc i^H e ¡W`WYX i^H[a MTVTVCFMOWYHfX _^ew CFHG%H¦MO_pMOXF_lTwR`¶FQOH` H a H a PFeHn ¡i^WYG 3 i* 8 , ! 3 ^0 1. 1 C 3 C :
(91) T A . BDCFMS__dTVR`¶FMOQSMTc e idMOT^H i^MOWYX MS_bTVidPFH>MSX \:R`idT^MSePFQSR`i¦ ¡WYi 1 <3 3i<3 RUX a <3 ,. 3 % # 3 ^0 3. ,. 3 % $ 3 ^0 :& A. 8. CFMSewCsMSG>\FQOMOH _TVCFH#m´ e WYX a MOT^MSWYX " . < < < ¯ ª 2® @ 4 A © 9>°: © :}% : ª ® > ® # :¡& : ® 8 < BDCFHb_^ewCFHG%H 9I "pMO__^He WYX a WYi a H i[R`e ePFiVR!TVHbWYX _dT^i^PFe TVPFidH a `i^M a _ = © Y :?> & : ® 8 ® @ 4 A © ® ¯ : © ¯ © ]4 : ® 8 BDCFH#G>W a M:H a H P:RUT^MSWYXG%H TVCFW a CFMSewCsR`_fMSX}T^i^P a P]e H a ¶c R`i^G>MOXFR`X a cYH T^T jMO_KR _^P]MOT^H a T^H ewCFXFMOPFH' ¡WYi¦T^CFH'RUX:R`QcN_dMS_#W` T^CFHW`i a Hi#W` RZXPFG>Hi^MOeR`Q G>HTVC]W a TVeWRUT^i^PFHi^_^H#WYQH Yi^H#idWYRKi^_QOMS+ XFH R`H#iTV:HidG>_dT¦\vidWYH ideH RUQAQS\FQ°id¶FWYi^¶FMOH QOH ¢ G>cZ_ TVjC]BDHCF¶:MOR`__dG>MSeHi^TVPFC]QOW H _,a WUR` QOQSTVWCFMO_D_,TVG>W Ha T^CFH i^W MaYHb7´:TVC]WYH,inTVG>i^P]WYXNidH ³ a HTVR`MSQO__^H H #"§AH T _#eWYXF_^M a HiR QSMOXFH R`ibHYW`QSP]T^MSW`X \:R`ilTVMSR`Q a M($¬Hi^HXT^M¥RUQ HP:RUTVMOWYX MSX TVCFHbG>WYXFW a MOG>H XF_dMSWYXFR`QAeR`_dHO ' A < ' ! 9 "
(92) . 465LM7468.
(93)
(94)
(95) "!#$%&
(96) ('*),+#-./
(97) 02143. z. HR`\F\]i^W}R`ewC TVC]HHP:RUTVMOWYX 7 "p¶}c RUXsH§g]\]QSMOe MTfFi^_dTl³WYi a H iPF\}MOX a _dewCFH G>Ha . 1. H T . . . = * i. h. . <. . . T^CFHR`G>\FQOMO:e RUTVMOWYXse W@H!'%e MOH X}T a H:X]H a ¶}c . h<. . h. . . BDC@P]_ :TVCFHbG>W a MOFH a H P:RUT^MSWYXZW` TVCFHR`¶WYH_^ewCFHG>HidMOT^H _j . . CFH idHW. A. ' '!.
(98) <.
(99) . . '! '. MS_DTVC]Hb ¡W`i^G%R`Q0_dH i^MOH _[H gN\:RUXF_^MOWYXWU °T^CFH¦ ¡PFXFeT^MSW`X 8. . i* ]*. ,. . h h :F: 8 . ®0®2@®2@ &A © > ® > ® :¡4: ® 8 A ´5W`iTVCFH_VRUH#WU {_^MOG>\FQOMSeMOTc`vHMOQSQidH _dT^i^MOeTfWYPFiMOX`H _lTVMO}RUTVMOWYXT^W%T^CFHWYXFH a MOa G%HXNa ³ _^MOWYX:R`QDe R`_^HU H \Fi^W`\vWY_dHTVW \Fi^W`H T^CFHZR`¶W `Hs\]i^WY\WY_^MTVMOWYX<PF_dMSXF TVC]HZG>W M:H a H eR`_dP: H RUT^ MSZ WYX%< TVHewR`CFX XFa MO@` P]H`< "YH[ i^HeR`QOQ5CFH idHfT^CFHD:XFMTVHf`WYQOPFG>Hn_^ewC]H G>HnMOXT^CFHfX]WYi^G%R`QOM -H . . . PQQQ QQ R. Q. QQ QQ. QQ. QQ QQS. MLMONQPGR#SUTV. i=* 1. = * i. <>=. <>=. A " #. SP. A " #. PQS. . = ce. = ce Q. l ".
(100) ) +Q + + I B°W[WY¶]TVR`MSXR`QOQYTVCFHjUR`i^MSR`¶FQOH _RUTTVCFH _VR`G>H TVMOG%HUUHP]_^HT^CFH BRc@QSW`i°_dH i^MOH _°H gN\:R`X]_^MOWYX . <. . <. <. A. ,. . . i*. i* =. . A. i* A. ,. ' ' * $. '+*. ,. ' ' * $ ' ' * $. i* 3 ' 3. i* 3 ' 3 '$*. ,. . . ,. 3. i* 3 ' 3. i * : , D8. 3 , 8 i* : *3 ' 3 i i* '+* 3 , 8 '$*. ,. 1 < = ]A * '$' * . 8 i * : 3. ,. :. a M:H a H P:RUT^MSW`X'TVHewCFXFMO@P]H¦R`X a TVCFHfR`¶W `H a H YHQSWY\]G%HX}TFHfYH TT^CFH ¡WYQSQOWMSX]G>W a MOFH a H P:RUT^MSWYX) f_^MOXFTVCFHfG%W. ' ' * $ PQQ. R. QQQ QS. QQ. QQ. ' ' * $. . < <. 3 3: i* "#$: ]* : % # ' # , 8 , D8 , D8 '. =. =. '. '. #. ,. i* "#$: 8. 3 ]* : ,D8. 3: % # ,D8. CFMSewCs\Fi^W`HT^C:RUTDTVC]H_dewCFH G>H l "pMO__^He WYX a WYi a H i[R`ee PFi^RUTVHbMOXs_d\:R`e H#R`X a T^MSG>H` < < C H ® 8 © ]]4 : ® 8 ® @ 4 A © 8 © :7Y$ © 8 © 4K ª H a H :XFH¦TVCFH a MO_^ei^H TVHHXFH id`c¶}c A < 8 : 3 , l` " . © F4 I BDCFH ¡WYi^G#PFQSRUTVMOWYX lY "°MS_ R,P:R a iVRUT^MSeR`X a%a H:XFMTVH\WY_^MTVMYHD ¡WYidG ¡WY i
(101) < "i#* & A 465LM7468.
(102)
(103)
(104) "!#$%&
(105) ('*),+#-./
(106) 02143. ®0®2@ . A . <. . 8. . 3 : ,. H a H a PFe HbTVCFRUT^ . A . PQS. . PQS. =. =.
(107) = : cQ 3 =
(108) 3 = : 3 8 e 8 .
(109) = : cQ 3 8 e. ¡ W`i
(110) & < A < HR`QO_^W>C:RYH A T < T RUX a < BDCFH¦`RUi^MSRUTVMOWYX W` TVCFHbH X]H i^Uc MS_YMY HX¶}c <. A . A. . < <. <. T. 1 : 3 8F8
(111) 3 8
(112). = 8. . 3. : :. : 3 = = = 3
(113) A 8 3 8 = : =
(114) 3 = 8 , =. . . ,8.
(115) .
(116) 3 : :3 ,8 8 T. =. C. = 8 = . ( " . : : =
(117) A 8 1 = 1 : . :
(118) 8 = : =
(119) 8 1 = 1 : A , : < T l~ ". CFMSewCsMSG>\FQOMOH _TVCFHbe W`XF_^HidURUTVMOWYXZW` TVCFH a MO_^ei^HT^H#HXFH id`cY < < ; ¯ ª 2® @ 4 A © 8 © :9Y $L¯ :9 > © U : ® 8 H,\]i^WY\WY_^HfCFH idHnTVWH UR`QSPFRUTVH[TVCFH a H YidH H,W` AT^CFH a MO_^\H id_^MOWYX W` ATVC]H,_^ewCFHG>H`´MSid_dT pHbi^HeR`QOQ0TVC]H a MS_d\vHi^_^MOWYXsi^H QSRUT^MSWYXZMOXZT^CFHbe WYX}TVMOXPFWYPF_[eRU_^H`:MT _YMYH Xs¶}c 3 3 3 lz " < * . MLMONQPGR#SUTV.
(120) . ) +Q + + . B DC]MS_[i^a HQ¥RUa T^MSW`XsMO_DYH a idMOFH a ¶}c XFW`X a MO_^_dMS\FRUTVMYHRYH_fMTVC R`H#\FCFR`_^HbYHQSW@e MTc H H PFe HbTVC]H MS_de i^H TVHbR`H#H@PFRUTVMOWYX ¡i^WYG T^CFH¦:XFMTVH¦YW`QSPFG>H_^ewCFHG>HO % % ! 3 % % ! % % ! < 3 SP 3 % % ! h 3 % % ! 3 % % ! c e A l I " 3 3 3 SP % 3 % ! % 3 % ! 3 % % ! 3 c % % ! % % ! % % ! e \ V , , pHbe W`XF_^M a H iTVC]HCFR`i^G>WYXFMOe¦R`OH h \
(121) V ! A
(122)
(123) , ! " # % $ i * : < . 8( C C C qXUklH e TVMOXF%T^CFMO_[H§gN\Fi^H_^_^MOWYXMOX l I §" ]c@MSHQ a _j * < 3 PQQQS ^0 3 * h,! b0 3 * \ "# ^ 0 3 * V %$ c QQQ 3 i *3 ,! 3 , %# 3 , % $3 e ^0 A i. . l ". q Ti^H\Fi^H_^HXT^_TVCFHfXPFG>H i^MOeRUQ a MO_^\H id_^MOWYXi^HQ¥RUT^MSW`X0BDCFHnB°Rc@QSWYip_^H idMSH_H gN\:RUXF_^MOWYXW` l "p`MO`H j_ 3 3 3 h ,! \ "# V %$ 3 3 i* 3 :F:< 3 3 * , * , * :F: l " 3 8 ,D8 . * 8 ,D8. * CFH idH 3 3 3 3. * < * h , * \ , * V BDCFHpidH QSRUTVMOWYXF_dCFMS\ "A_^CFW_T^C:RUT TVCFHXPFG>H i^MOeRUQ]G>HT^CFW a \Fi^H_^H X}T^_jRf_^He WYX a WYi a Hi a MS_d\vHi^_dMO`HHi^i^W`i He WYG>\:RUi^H>XFW T^CFMO_Hi^idWYi,TVWT^CFH a MS_d\vHi^_dMO`HHi^i^W`i¦YMO`H X ¶}c H H _j_^ewCFHG>0H 7 f_dMSX]bT^CFHDB°Rc@QOWYi_^Hi^MOH _jH gN\:RUXF_^MOWYX0TVCFH a MS_d\vHi^_^MYH[i^HQ¥R!TVMOWYXF_^C]MS\ YMO`H XZ¶}c ¶vW`T^C G%H TVCFW a _ # HH _[G>H TVCFW a RUX a T^CFH¦:XFMTVH¦YW`QSPFG>HG>HTVC]W a "pidMOT^H j_ C A © 8 : © ® $ © L¯ :7 > © !# : ® 8 3 3 . 3. 8. . ]*. =. 3 * 3. QQS. PQQ. . =. A. 8 ]* : :f< ,. !3 h , * 8. ,. #3 *\" 3. *. ,. $3 *V% :. ! h " * ,D8 ,. # *\% 3. *. ,. $ *V-. c QQ : QQ e. l| ". 465LM7468.
(124) |.
(125)
(126) "!#$%&
(127) ('*),+#-./
(128) 02143 A ©©© %A © © ¯ :9 > © !#: ® 8 C. 3. . 8. . 3 *3 i. =. 3 * 3 PQQQQS . . A. =. on_^_dPFG>HT^C:RUT. 8 i* : :f< ,. * ,! 3 , * \ "# 3 , * V %$ 3 h ,! * : 3 A 8 ,D8. *. h. ,. *. \. "# . *. 3. ,. *. V. -$ . c QQ : QQ e. l ". . MTVC < A MS_[R\H i^MOW a BDH i^CFidWYH X0i [CFpWHHe `R`H X i#aT^CFH Ha P] e H H H TV_C:_^R!ewTCFH¶vG%WUTVHC C:_^R`ew_bCFHQOMOG>TdTVH QOH_ZQO\FH i^_dH_#_^H XX}PFTsG>R H idMS_^e HR`e Q WYXa aMO_^\WYH i i^a _dH MSW`i X a TVMS_dCF\vR`HX i^_^T^MCFYHH :XFMTVHYW`QSPFG>HWYXFHUBDCFHi^a H ¡WYi^HMT{a MSQOQF¶HDMSX}T^H i^H_dT^MSXF¦T^W¦a e WYXF_lTVidPFeTa R,C}c@¶Fi^M a _dewCFH G>H e R`WYi^HG#TV¶FC]MSX]HMSXFRU¶FMSQO¶MTW`c T^C WU G>T^HCFTVH%C]W FXF_MOT^H%MSX `WYWYiQOPFG>H iHTVW_dewCFH§gNH G>\FQSHW`MOT^TKWZT^CFT^i^H H RUT(M $ve H WYi^G>HX}\FT%QOH R g UYR`HX}WYTwG>R`H`TVH id_ MS°H_ CFR`MSX ewCa TVCFHf_^a G%R`QOQSa HiR`G>WYPFX}TpW` R#XPFG>Hi^MOeR`Q a MS_d\vHi^_dMSWYXa WU 0TVCFH H H __dewCFH G>H¦MOX'a TVC]Hn_dT^i^PFe§³ TVMSG>PFid\FH QOH G>Yi^H MX}T _ He R`X XFW`T^MSeH>T^C:RUT¦¶W`T^C G>H TVCFW _bR`idHH e W`XFWYG>MSe R`QjR`X _dMSG>\FQOHTVW D"° * + D"° >
(129) / H`MO`HCFH idHD_^WYG>HXPFG>Hi^MOeR`QFH§gN\vHi^MOG%HX}TV_jMSXW`i a HiTVWnUR`QOM a RUTVHTVCFHpFXFMOT^HpYW`QSPFG>H aa a a a TVG%H H _lTKTVCFeW WYXF_^MO_dH T^_^_Kei^MOMOX ¶vH e WYG>R`¶\FWPNYTVH`MOXFBDTVC]C]HH'G>H`H WY_dQOCFP]H TV_MOWYPFX _^H W` RZR`idG>HW PFa X]H_daTVW`idPF DeidT^H PFa_^i^W`H X:R`XFa WYeXFHH_ MSX BDRsCFeH>R@:MidT_dcYT H'R`QS_dWYMYHRUX H g]R`G>\FQOHW` [R`X HQSHeT^i^WYG%R`YX]HTVMOe':HQ \]i^W PFe H ¶}c R_dCFH H TW` e PFidi^H X}TfWYH ifRe idWY_^_[_^HeTVMOWYXsW` {R>i^HeTVR`XFYPFQSR`iDpR`H YP]M a HUpofXFW`TVC]H iTVH _lT[e WYXF_dMS_lTV_ MSX RP:RUQSMTwRUT^MO`HR`X:R`Qc@_^MO_DMSX R`XC]HTVHi^WY`a H XFHWYPF_DeRU_^H`jq TDidH \Fa idH _dH X}TV_[RK\FidWY\:R`}R!TVMOWYX W`_^MO #GR<PFQS\]RUTVPFHbQS_dTVH C]HMSX}_dTVeWRUTdR TVHG%i^H RUa T^H pi^RMSR``Q,HKeR`WYeG>i^WY\v_dW`_[_^R H a MSW`H #QSHeR TVidUMSR`e R`e P]QPFG%G RUTVHR`i^X MSR`Q7 R YQSR`_d_ R`_lTVQcYfH @. . ]*'& . M . .
(130) #
(131) # @")
(132). >H YMYH>CFHi^HTVCFH>H YWYQOP]TVMOWYX W` pR G>W a H>W` i^aH _dWYX:R`X]e H>MSX RG>HTwRUQSQOMSe>e R@MTcY%BDCFH X@P]G¶H inWU {\WYMSX}T^_f\H iR`H QOH XFYC}T¦MO_fR`i^idWYPFX A * Hb:i^_lTneWYG>\:R`idHT^CFH#XPFG>Hil³ . MLMONQPGR#SUTV.
(133) ) +Q + + SM e R`Q_^WYQOP]T^MSWYX TVW TVCFH%H g]R`e TW`XFH :YPFida H'z MO_R - W@WYG$W` TVCFH%XPFG>H idMSe R`Qj_^WYQOP]TVa MOWYX "§ HZeR`X XFW`TVMOe H TVCFRUTHCFR`H R YWNW i^H _dWYQOP]TVMOWYX ¡WYi T R`H QOH XFYC}T^_ fR`X T^CFH a a TVX@CFP]H%G%e HWYi^X]MOe_^R`H QDilU_dRUWYTVQOP]MOWYTVX MOWYXWU pMSTV_>C]XFH%W`XT PFG>MS_dH\vi^MOHei^R`_^QMYH HUXFHXFi^`HMOc`T^CFH BDi CFMO_M($¬MOP]_b_^GMYR!H`TV<ewCFBDHCF_#H TVW :TVYC]PFH%idHT^CFI H "WYid_^HCFT^WMSe R`_ Q i^H _dPFQTV_[YMO`H XZMSX TVC]H¶H `MSXFMOXF%W` TVCFMO__dTVP a cY 1 EXACT APPROXIMATE 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1. a a SQ´HMSXF`YPFC}i^TVH>_ z BDMOG>HH YWYQOP]T^MSWYX W` pRZG>W H>W` pidH _dWYX:R`XFeH¶HTH HX T `R X T p R`H ³ 40. 45. 50. 55. 60. 65. 70. 465LM7468. 75.
(134) .
(135)
(136) "!#$%&
(137) ('*),+#-./
(138) 02143. 0.2525 ENERGY 0.252 0.2515 0.251 0.2505 0.25 0.2495 0.249 0.2485 0.248 0.2475 0.247 0. @ .
(139). #
(140). " . . . ´MS`PFi^H¦I$ n MS_de idHTVHH X]H i^Uc 10. 20. %
(141)
(142). 30. 40. 50. 60. 70. 80. m!pWY< XF_^M a Hi>Re PFidi^idH X}T ? WYH iKT^CFHei^WY_d__^H e TVMOWYX W` fRidH eTVR`XFYP]Q¥R`i#RYHYPFM a H WYX T ¡FYPFi^H# "p`MO`H XZ¶c < A ^0 8 # : 58 * : l " . q XZ< T^CF MS_H§g] \H idNMSG>BDCFHXH#T G>H_^H¦C]P]gN_^H H asa a CF< Hi^HG e WY:idR`i^XH _da \vT^W`CFX Hba _D\Fi^TVWYW \FR`}RU\vT^W`MSWYMSX}XTV_ ¡id\H H PFiH pX]ReYc H QOH X]`TVC0 BDCFHa T FYPF i^a H_ 7 a "AR`X a | "¬i^H \]i^H _dH X}T{id H _d\vHeTVMYHQOcTVC]H HQSa HAeT^i^MOe :HQ a R`X a TVCFHjG%R`YX]HTVMOe :_^WYH QOQ P]TV\]MOWYi^XFW _PFeWYe G>H \:¶}R`ci^H TVa C]H[T^W>ePFT^CFi^idHbH X}H g]TR`e TfHW`XFeH R`_X% _^HH[T^CFHYW@W P:R`QOMOTc>WU ¬TVC]HfRU\F\Fi^WgNMOGR!TVH MLMONQPGR#SUTV.
(143) ~. ) +Q + + j. d. 0. ´°MSYP]i^H#o _^C]H HTfW` °e PFidi^HXTfMOXsR>i^HeTVR`XFYPFQSR`ipRYH `PFM a H métal. 1 EXACT SOLUTION APPROXIMATE SOLUTION 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8. ´°MSYP]i^Hb LNHeTVMOWYXZWYXc3 W` °T^CFHbH QOH e TVi^MOebRYH -1. -6. -4. -2. 0. 2. V. 4. 6. 465LM7468.
(144) ~N.
(145)
(146) "!#$%&
(147) ('*),+#-./
(148) 02143. 0.8 EXACT SOLUTION APPROXIMATE SOLUTION 0.6. 0.4. 0.2. 0. -0.2. -0.4. -0.6. ´MS`PFi^H#|# LNHeT^MSWYXZWYX c3 W` °T^CFHbG%R`YXFH TVMOebpRYH -0.8. . -6. @ . . . -4. . 1
(149). -2. 0. 2. \. 4. 6. a a W` H! _dMS< G#PFT QSRUR`TVX HDa TVC]R HDYHQSYR`W`_^QS_#P]a T^RUMSTbW`X%TVC]WUH ¬RfQOH\F T PFQO _^Z H< MSX}T^ Wb°R,R`XG%a H ` MSRb< eWYG>" \vW`BD_^HCFH%a W`G% 5HU_^R`C ePFP]PF_^G-H a R!MSTX i^MOTVYCFC}MO_T H gN\vHi^MOG>H X}T e WYidi^H _d\vW`X _{TVW,T^H X\WYMOXT^_a \vHipR`H QOH XF`C TMOXKTVCFH a MOH QOH e TVidMSeG%RUTVHi^MSR`Q9 BDYWNCFW Hna RU\FP:\FR`i^WQOMOgNTc MOGWUR! °TVT^H[CF_^HbWYQOX@P]P]TVG%MOWYHXi^MOMSe_ R`e QWY_dG>WYQS\:PNRUTVi^MOWYH X TV Wb:YTVPFCFidH[H H g]" R` e TpW`XFH`NR`X pHfeR`X%XFW`T^MSeH[T^CFH. MLMONQPGR#SUTV.
(150) ~Y~. ) +Q + + 0.8 EXACT SOLUTION APPROXIMATE SOLUTION. 0.6. 0.4. 0.2. 0. -0.2. ´°MSYP]i^H h idWY\:R`}R!TVMOWYX W` {R\FP]QS_dH#MOX}TVW%R`XZCFHT^H i^W`YH XFHWYPF_[G>H a MSR -0.4. -3. @. . ". f5#
(151)
(152) . -2.
(153). -1. 0. " #
(154) %
(155). 1. " $ " i. 2. #
(156) $ . 3. . H H,MO\FQOQSi^PFW`G>\vWYMOX:_dHRUT^C]HbH RKi^H%e W}TVWZRUT^H_^MOa Gt[PFQSo[RUmpT^Ho RZ_deRUTdTVHCFi^WYMOXF_^H_\FG%idWYH¶F_^C QOH G e W`i^MSi^X H_^R`\X WYX CFa H_T^H T^i^W W`YH XF\HWYWYPFMOX_KT^_De R`\v_^HHUi % RYHQSHXFYC}T :Ya PFidH# "D¶}cZR>G>WYXFTW@T ew CFA i^WYG%RUT^MSebpR< `H\F QSR`eH a R!Ti^MOZ YC}< TnW` TVC]H#a WY` ¶@klH eT HZR`_d_^PFG>HR MOH QOH e TVi^MOeQSRcYHiW` fTVCFMO&e XFH _d_ R`X K " T
(157) MOT^C BDeR`CFX H%XFFW`YTVPFMOe i^H>H TV CF RUTb"ni^TVHCF\FHi^HX@_^P]HG%XT^H_i^MOTVeC]R`H%Q _^MO_^W`W`QSP]QSMOT^XFMSH W`_#X W`a W@TVH CF_dH%X _^T#e \FRUT^idT^H H _^i^HH X}a THR`QSX}Hc eTVWYidMS_^e>eMSpQOQSRURTV`MOH`WYX RUR!X Ta T^CFHH MSX}TVHid 9R`eHb¶vH TpHH XT^CFH a %M $vH idH X}TfG%H a MSR] . . 465LM7468.
(158)
(159)
(160) "!#$%&
(161) ('*),+#-./
(162) 02143. ´MOYPFidH²BDCFH#tfo[mpo. TT. . A. G>H _dC. ´MOYPFidH²BDC]H#_deRUTdTVH idH a H QOH eT^i^MOepRYH MLMONQPGR#SUTV. ~`z.
(163) ~UI @. ) +Q + + . . .
(164) $ #. . . $. . . #
(165)
(166) -
(167) ) @. HCFMSew\]C'i^WYMS_ \¶:WYR`_^H_dH aT^W W`X'e W`CFG%MOY\FC'R`i^P]H \WYP]MOX ia G>_dewHCFTVC]H WG>a H,MOX%MTV_^C \:R`R<eH`r ]R`nX LFa m Ru:PFX]XFMOYT^H§H³,
(168) ¦YWYP]QOT^PFTVG>RH_dewCFG>H HG>TVC]H,W MOaX TVMOG%HDMTVC%a TVCFidH H_dT^H \F_BDCFMS_ _^ewC]H G>H,MO_jT^CFMOi a WYi a Hi R`e ePFiVRUT^HWYX_lTVidPFeT^PFi^H a YidM a _ Hse WYXF_dM H iR _^MOG>\FQSHeR`_dH`CFMOewC i^H\Fi^H_^H X}T^_TVCFHs_VR`G>HH§gN\vHi^MOH XFeH R`_'MOX T^CFH \:a R`iVRUYiVR`\]C 7zNSa "§ HnPF_dHnTVC]Hf_^R`G>H,G>H _^C ¡WYi T^CFHfTWKG>HT^CFW a _ ¡_VR`G>H,XPFaG#a ¶vHipW` HQSH _^Y_idTVH C:HR`W`X {TV ¡C]idH HH r WY,G LF" m FH#XFe MOR`T^HX`XFWYWUQOTVPFMOG>e HbH#TVWYC:X]RU HT[TV C]:H#YPFXFidH H>Y:" XFMTVBDHbC]YHWYT^QOPFMSG>G>HbHKW`G> °He T^WYCFG>W \FP]TV(MRU$¬TVP]MOWY_^X H YMS_[MO`R`H idXi^WY¶}PFc X a TVC]HK_XFdH H e W`G%X a H _ TVCFW BDa CPFMO_,_ 5RUpi^i^HW`PFeXRUXsa XFA WUTV_dMOe H H¦e W`TVX C:a R!_ T[0T^R`CFX Hba XF¶}Hc TVG%C]HKH TVr CFW na LFYm MO` H _[G>¶vHH TVC]T^T^W H a i mh \H il ¡WYi^ G%R`XFeH¦TVC:R`X TVC]Hr ,LFm FXFMOT^H`WYQOPFG>HW`XFH` . 1 EXACT NEW FINITE VOLUME FINITE VOLUME RK3. 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1. ´MOYPFi^H>YO mpWYG>\:R`idMS_dWYXZ¶vH TpHH XTW%FXFMOT^H`WYQOPFG>HG>H TVCFW a _ 0. 20. 40. 60. 80. 100. 465LM7468.
(169)
(170)
(171) "!#$%&
(172) ('*),+#-./
(173) 02143 5 / . ~Y. DH C:R`H[\]i^H _dH X}TVH a MOXKTVCFMO_j\:R`\H ijR,_dH e W`X a W`i a Hi{:XFMTVHYW`QSPFG>H_^ewC]H G>H[T^W¦i^H_^WYQYH TVX@CFP]HfG%rHi^R!MOeg@R`QNH H QOgNQ5\H H i^P:MORUG>T^H MSWYX}X]TV_ _{BDH idCFHHWYMOXi^`H TVH MO_leTVR`MO}Q5RUR`TVXFH R`a QOc@_dBDMS_ CFW`MO_ _lTVTVCFP Ha G>c#H_^C]T^CFW Wa _°pTVC:R`RU_jT \FTVi^C]HMS_^_ H TVX}H T^ewH C]a XF]MSR`X PFa H W` 5R`\F\]i^WgNMSG%RUT^MSWYXMS_ pHQSQ]R`\F\FidWY\Fi^MSRUT^H T^Wni^H_^WYQYHTVCFHH QOH ea T^i^WYG%R``XFHT^MSe\]i^WY¶FQOH G>_{MOaX Re WYQSG>MOXF\:H R`R`idiH ea RU_^T^H`W ]¶FT^CFP]HsTDXFWUWUTVCFTHXFi^H_e H F_dXF_VMOR`T^ilHsc`CFWYWYQSP]G>G%WY`HsH G>XFHHWYTVPFC]_ W ja BD_'CFR`H,i^HZR MTVU_'R`X}_dTwMSG>R``\FH QO_MOe W`MT Ac¤T^CFMOMOG%_\]G>QSHHG%TVC]HW XN³ Tw:RUH T^Q aMSWYX¬`AH e R`TVX WYa id_KMTVR`_ idH¢ H§Q¥R`g]e MO¶FRUMOTVQSH MTa c RUTVT¦WT^CFCFR`H>X _Va R`QOG>HH%MO\vT^CW`MSX}eTVWY_G> \F QOMSe H%RUTVeHR`a X YR`HQO_^WYW G>XFHT^W`i^T^MOMSH e_bH%_^T^MOCFXFH>e H%XPFR`QOG>Q°H T^i^CFM H ³ eR`Q°e WYXF_dH il`R!TVMOWYX W` jTVCFHH XFHi^`c` )nMOXFTVW'T^CFH _dH\Fi^W`\Fi^H TVMOH _ AMT¦_dH H G>_¦TVW'¶vHT^CFH G%W`_dTDa RU\F\Fi^W`\Fi^MSRUTVH[G>HT^CFW a aTVW¶HneWYG¶]MSXFH a MOT^C'TVC]Hf:XFMTVH a (M $¬Hi^H X]e H,G>HT^CFW a a. MSa XWYG%W`i R`MOHXFif_ jTVWonWYXZ¶]C}TVc@R`MS¶FXi^M R'a RUC}T^cNMSWY¶]X i^M TVHew_^CFewCFXFHMOG%PFHKH ¡MOW`_i[MSXZT^CF_dH>T^P ra cYR!g@ H QOQH@PFRUTVMOWYXF_,MOX e W`G%\]QSMOeRUT^H ' [ (
(174) qMO_^CsT^W%T^C:R`X ni W`PFQ¥R%´:H -WYPFM0 ¡WYiDCFHifP]_^H ¡P]Q idH G%R`i @_fR`X a _^H YHiVR`Q R a @MOe H_ q pWYP]Q a R`QO_^W>QS%M `H¦T^C:R`X ni´:i^H a H i^MOeh°WYP]\:R`P a R`X aZ± i^MOe nPFeHR`P ¡WYiTVC]H MOiDCFH QO\] ¡PFQ9 DUD" p/ O! u mR`i^\H X}T^MSHi :o a H¦QSRpWYPFi a WYXFXFRc`H`n R`i^idWYP]T^PFi^WYP¬
(175) $U
(176) %+Q ! ?
(177) Q , +Q !#$
(178) +Q +Q1&3 3 !
(179) 0 +Q$ $U +QL?
(180) 3 0ufR`\@³ \vWYilT a Hbi^HewCFH idewCFHm ± ufrsqlmDLvXFW A Y`I "§. 3& 3 !#$ 3&%&
(181) (' ~ ] h mpMOWYXFM9 j ´ !- WYPFM9 q_^_^R`P]TVMOH i ' $ $ "
(182)
(183) +Q
(184) ),+#-./
(185) +Q 3 R uH ewCFHi^ewCFHo i^W`_^\:RUT^M¥RUQSHUµjWYQ[Np\]\0 z]!³z}~ lYUI " z ´p [H idG%HQSMOXFHU / +Q$U
(186) %Q
(187)
(188) 3 3
(189) (' +Q +Q1 $ 0 ! 3 3J!#$
(190) +"$ 9-.
(191) +Q ! +3&%# 3 3& +Q +3&% ) +#-./
(192) +Q 3 ] mpW`G%\¬ hC}cY5µjWYQ T F\F\ = YYz "§ . . MLMONQPGR#SUTV. . .
(193) ~` . ) +Q + + I tfMOH Q9 5 rR a. a d _. H 0 X @ u S M w e : C U R i ¬MOWYQ%`W_ }c` $
(194) 3 +Q )
(195) Q
(196) L$J),+#-./
(197) 3 G+Q 3 ± QOH e TVi^W`GRUYXFHT^MSe_ `WYQ = A :\F\ = T T FXFW #r uH G% RM9 ´ !- WYP]M7´p h° W`PF \:R`P a 0 %Q +Q 3& + 1 %Q 3. 3 +" +Q - G+Q 3 ),+#-./
(198)
(199)
(200) $ ' 0 :iVR`\]\vWYilT a H i^H ewC]H i^ewC]Hq tnuqo XFWzYz}NY kdR`X}@MSHi#` #L fon+t
(201) ¦o,u µKF³ fo ´pF³D r ) fo,rrso fqo,t o# S
(202) +Q
(203) Q $ +Q 3&%Q$ !#$ $ +#'. 0 3& +Q $U
(204) (' $ $ & Q
(205) (' ? ]µjWYQ |`|NF\F\0 | Y!³l|`~NYFt[WN l "§ | bo BR ¢ W `H`5r ± pi^W a MOX0 $U +Q , ! 2 +"Q 1 2 +Q $ +#' 0 +Q $
(206) (' $ 3 3
(207) (' ?
(208) +#-./
(209) 3 +Q 3 q ± ±j± B iVR`XF_ ofX}TVH X]X:R hidWY\:R`}R!TYWYQ9 rZBDBp³ A X]WF \F\¬ A ³ OT onPFYP]_dT l}|` " u ´p R`idG>MSXF]´p cYH T^T Q +Q +" $ +
(210) 3 +
(211)
(212) 1 +Q * + $ + 1 +Q +Q1&3 3 !
(213) Q $ 0
(214)
(215) 3 ] mpWYG>\0 hC}cN_S ~ "§ \0 }|UI "§
(216) > Lv H HU $ +Q 3& !
(217)
(218) +Q +Q$U1 %+Q2 $
(219) 3& $ Q + !q ±j±j± B°i^R`XF_ ofXT^H XFXFRnhi^WY\FR`}RUT !`WYQ7 o,h³ !XFWF z]!\F\0 T A ³ T @!rRc " O
(220) > Lv H HU ] Lv mpCFH X¬ #!#$U +Q(
(221)
(222) $ 0
(223) +Q
(224) +Q
(225)
(226)
(227) +Q
(228) ]q ±j±j± "§ OY!
(229) > Lv H HU ] Lv mpCFH X0 +Q 0 +Q$U1 Q
(230) 2
(231) +Q
(232) ? " ! 1 $U Fq ±j± ± " . 465LM7468.
(233) Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - B.P. 93 - 06902 Sophia Antipolis Cedex (France) Unité de recherche INRIA Lorraine : Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - B.P. 101 - 54602 Villers lès Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot St Martin (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - B.P. 105 - 78153 Le Chesnay Cedex (France). Éditeur INRIA - Domaine de Voluceau - Rocquencourt, B.P. 105 - 78153 Le Chesnay Cedex (France).
(234). . ISSN 0249-6399.
(235)
Documents relatifs
constat, unanime en Colombie, que la résolution des conflits par élimination de “l’opposant” est l’une des caractéristiques de la violence nationale. Première
Automatic test data generation, structural testing, constraint solving techniques, global constraints 1.. Structural testing techniques are widely used in unit or module testing
It is obvious that the steel showed the plastic strain memory effects at each test temperature, in other words, in spite of the hardness of the steel cyclic stress,
Les cités et leur territoire d'influence sont administrées par de puissantes familles, celles de l'aristocratie terrienne et marchande, qui tirent leur légitimité de
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des
In this paper, we have defined the steady-state control problem for MDP, and shown that this question is decidable for (ergodic) MDP in polynomial time, and for labeled MDP
Trois autres échantillons du pool CHAT étaient également gardés au Center for Disease Control et ont été inclus dans cette analyse (CHAT pool 13, CHAT type 1 Wy4B-5, CHAT 1FL)..
Un habitat de La Tène moyenne et finale à Berry-au-Bac au lieu-dit “ Le Chemin de la Pêcherie ” (Aisne).. Jérôme Dubouloz, Michel Plateaux,