High Order Scheme for a Non Linear Maxwell System Modelling Kerr Effect
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. High order scheme for a non linear Maxwell system modelling Kerr effect. Armel de La Bourdonnaye. N° 3688 Mai 1999 THÈME 4. ISSN 0249-6399. apport de recherche.
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(174). . p. q. ÄÇ %ÄÃ '.
(175) ª.
(176) . 0.8 initial position order 1 order 2 order 3. 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1. T[aWerfOQ < %_Im5ZhY%xTVO N(Zh`M`«dMrf_a£eY[ON _ar ¸ _ i We`M_ ! ggf_aYaOrt9y. -1.2. . ,Ù. '. ¡ £¢ ¤ ¤. 0. 200. ¡c. 400. 600. 800. 1000.
(177) ª~ ©\^KOZh` _h£M_agfr Oi rlO ar Owª \^KOpZ]oU\XmO _ader \ci O _ar"r.w\^KeNPO T[gl_hgfgfO m gBTVYVY¬\fZ]KM\fO"T[_hh`M_Ugv_ Zi \gl\^_aKMY[OjWU\^£TVZh_am `Zh_]` i \fKM\^OjKZ]gf\XKM_I_am r 9i g O rB}bT[g£¤Op\f\fO r JKeOY[Zhgl\m ZhgfO _h+\^KMTVggfOrfTVO gcT[g\^KeO m_a`MglT i Or^Z]\fT[_h`_h;ZB.Z5aO dZhm ]O\ . ~. . . Yc. /q p. `!a }}
(178) `. } j. .
(179). . vxs. . s . . JgfTVNXKMT[WMgY[Z]m \^ZhO©gfO=KMT[TVhgKB_h rfO TV¨9N(WMdDO`M_amrt\ykXZh£¤`MO mZhOhNPµcgfgFTV`M_ar m O/gf_aYVTV\^OZhrfrlrtkgl\fZrfoIWM0mp\^O WMYVYrlO N(gF_ iJOKMY[g OrfZhOrlgfOWMYN(\^gFZhZhTV`MrfO0YVk gfKMWM_ gfO i`P_a\^`_ ¹aWMrlO,U< nOrfOhµM0Ojm5Zh` glO O\^KMO i T¶¤WMgfTaO O¶¤O mp\n_] _hr i Or glmyKMO NPO]WMrt\^KMOrfNP_arfO]µ . . . . 0.6 order 1 order 2 order 3 0.4. 0.2. 0. -0.2. -0.4. ;TVaWMrfO F©Z5aO dZhm ]O\v_hr ¸ _ i WM`M_ ! gcgl_aYaO r ©_ar Owi Om5rXZh`Z]dM_hdM£Mrfgf_5OoUrlTVaNwOwZ\^\^KTV_aZ]`'\b\^KMJO KMi T[gXOdMOK¶¤ZhO glmT[\X`M=T[gTVT[gX`NZhWMm mmyK_ar iNPZ]_a`Mrfm OwO"O NPT\^dMK¢KZ]\^gfKMTVgfOwO idMKkUglT\^T[mK g \fKM·Ow` i \fKMO OTVr i i µ g\^fTVKM`MOBm O ©\^Z5KMaOO YVO T[` `ei hOp\^oK _hi ;T[NP\fKMTV`MOXTVgfNPKMOOig PT[WMJN KMTVTV`Mg m TVrlg O5ZhglKO Z]gv\0T\^OPK_a\f£MKMglOjO rlhhZ]O"Y[WMZO\ _h\^FKMOP\fKM£OZ]Om Y[Om_h\fÉrfTV\^mBKMO OB`M©OrfZ5hakhO µ dZhm h·`bOp\5\^ KMO `MO§oU\ngfOrfTVO g _]FrfOgfWMY\^gc©O m _hN(dMZhrfO \^KMOxc_IOjgf_aYaOrnT\^Kb\^KMO ¸ _ i WM`e_ gf_aYaOr µ%£D_h\^KT\^K/OTV\fKMO rjqrfO i TVm\^_hrttu©_hrfrfOm\f_ar _ar _hr i Or}xWM`MaO§We\f\^ZgfmyKMONPOh -0.6 200. . 300. 400. 500. 600. 700. 800. 3c. ÄÇ %Äà '.
(180) ª.
(181) . Z. \^J _wKM\fOKMglO dgfZ]O \fmT¬Zh_aY` Z]i dM_adM`erfO_5oU_hTVNw;\^ZKM\^O TV_adM`«rfOpITVT[Y[_aYWe£Dg O gf\fOKMrfOjTVO g _ar i .O \rn}(_a`eOhµ©0JOBKeWMO gl¹O rlgl \c\^ Ogl\nm _arlrfO gl d¤_h ` i g L . c x O f g M W Y ^ \ c g h Z l r O a _ ` ¹ a M W l r O U ~ L nO rlDHO0OXrlO N(ZhrD\fH KZ]\vxc _IOZh` i ¸ _ i WM`M_ m _hT[`MmT i OXdDO rlOm\fYVkhµ±gf_(\^KZ\n0OXN(ZhTV`MYk s. . #prqts. .
(182). n. p. . qts. hGs. .
(183). #. . . . . q. s. v. s. s. . . m. 1.5 Godunov+Pred. Cor. Godunov+RK Roe+Pred. Cor. Roe+RK 1. 0.5. 0. -0.5. ;TVaWMrfO ~ Éu©_hN(dMZhrfTVgf_a`b_h;xc_IOjZh` i ¸ _ i We`M_ TV\fKxc _hrq¡rlO i 0u©_arF m m _a_aNPrlrfO dmpZh\^rl_aOnrKeTVgO rfNPO_a\^rfKeOOcZ]\fm T[NPm WeOr^Z]gf\fmyO.KMONP_ar;O \^g KeÉOJgfK9dDWMO g O µIi \^g0KMZ]Oc` gfi Om Y[Op_aa` O i YVgÉrf_hON(|\^ZhKMrO glTVKMgÉ_U\fm K9Z]g¡\¡£M\^WeKe\ÉOnZ]dMY[rlglO_ i NPTVm_a\f_arlOr _agfmT[JYVY¬ZKe\^Ov_a`MrtkaO§oU\©\^O gt\.m_arfrlO gfdD_a` i g©\^_ \^KMOc\^KMTVr i _h%dMrlOIT[_hWMg.gfOrfTVO g .\ µ9©O WegfO. DH 0H x\^cKOZ]gf \jWMY\f\^KMg©O"Zh rfqL OnrlO_ai `wTVm¹\f h_aWMrt rfu©O _aU rf rlO vma\f_aZhrXT[`wTVg ©OnN(rf_hO rfN(OwZhZhr m (mWM\fr^KZ]Z]\f\©O"xcZ]_I` Oni ZhNP` i _aL rl¸ Ow_ _ai glWMm TV`MY[Y[_ Z](\f_am rl_ak/TV`M \^mKMT i ZhO` Zh\f ` KMi O xcWM`eJaKeOpO·jY[ZhWUgl\f\j\yZ pgl¥ myKM\^O O NPgt\cOh T[gcZh£D_aWe\\fKMO.Z5aO dZhm ]O\ .\ . `!a
(184) ` Zh` i \fKMOrlO gfWeYV\fg ZhrfO(_a` ¹MaWMrfO"e Ow_a£MglO rlhOZhZ]T[` \fKMO"dDO rtO mp\ Zhmm _ar i Zh`MmO"_h -1. 0. . 200. 400. 600. 800. 1000. Oc. Jc. s. . m. s. JZ. p. ,Ù. '. ¡ £¢ ¤ ¤. /q. . }}. . #. p. q. s. . n.
(185). . p. q. v8s. . j.
(186). . vxs. . s. qts. } . h. . . s . . .
(187). . qis. . # . v. s. .
(188) ª. ~ . 0.8 Godunov+Pred. Cor. Godunov+RK Roe+Pred. Cor. Roe+RK. 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1. ;T[hWMrfO Éu©_aNPdZ]rfTVgf_a`b_h;xc_IOjZh` i ¸ _ i WM`e_"TV\fKxc _arqrlO i 0u©_ar9 . -1.2. 0. 200. 400. 600. 800. 1000. {Z3c. ÄÇ %ÄÃ '. .
(189) ª.
(190) . 0.6 Godunov+Pred. Cor. Godunov+RK Roe+Pred. Cor. Roe+RK 0.4. 0.2. 0. -0.2. -0.4. T[aWerfO Éu©_aNPdZhrlT[gl_a`b_hFx_IOjZh` i ¸ _ i WM`M_ T\^K«xc _arqrfO i 0u©_ar F9 -0.6. 0. . '. ,Ù. Oc. ¡ £¢ ¤ ¤. 200. 400. 600. 800. 1000.
(191) } cix O _INPO_aZ]`M` gti\^rfZ]¸ \^O_ i i WMKM`MO_ rf7O(£9gf_ak=Ya\fOKMrfO g ÆZhmJ\jKM\fO«KZ]£¤\ Op\fTV\^\ Or"TVg¨9YVWO glZhg YVT\·i k TV¶¤_]WM gfT\^aKMO"O«ZhdM` rfi O i £¤TVOpm\f\^\f_hO r"r m rf_hO rfMrfO Omm\f\fg _arw\fKMTVOg h`MZ]_ rf T[Z]N(\^TV_aZ `]Ov_h\^±Ke\^O KMOvmyKM©_aZ5TVm aOOY[_hO`MhWe\^gfK TV`M\^ KM_aZh`M`"YkP\^Ke\^O KMOxWMde`MrfO ai O§T[mp\^_aWer.\f\^mZX_arfgfrlmyO KempO \^NP_arOh< |We_arlrÉ\^Ke\fKMO rfTVg.NPrf_aO rlZhOhgfµU_agf`TV`M0m OO \^©KMOjOcN(xZ _UhOOjgl_a\^KeYVhO O rÉmyKeT[g¡_aT[NmOWMmy_]K";WMmyKMglT[O `eZh(dDO \fKMr¡O\^K¹Zhrf`Pgt\n\^Ke_h`MO Oh¸ _ i WM`M_ B_h`MOcZh` i gfTV`Mm O\^KMOpk m _hT[`MmT i Ohµ . #". ~. . . " -0+.@ 92 8D@. ·` \fKMT[gglO m\fT[_h`'µ]0O0i derfO glO `\;\·0_\^O gt\+m5ZhglO g ;JKMOɹMrfgl\+_a`eOÉi TVg+Z©Z5aO0dZhm hOp\ \fKMrf_hWMaK O\^ r^T\^Z5KMhOO rYVY[TVZ `MY[jTV`M\^O KMZhrlrF_aWMNPaO KPTV\^WMKMN Ogf_aZhrNPZ O`M\·_akU` dDO YVTVgÉ`M_hO5¤Zhr;NP_aO `Mi OhT[Ze FJJKMKMOO gfOi m _h_aNw` Z]T[m `(Zh_hgfO¤m TVg_aNPZ dMWeZ]WM\^Z]gf\^glT¬TV_aZh`P`(£DT[gFO5\fZhKMN O rfZhO ` mpi \yZh\^`eKeaOXY[O \^TVNP OBgl\fO d= TVg L Ý+JKeO©gfdMZ]\^T[Zh Y9 ¢gl\fOBO dB£D_hO aiTV`=T[gl£m rlkO\^Z"TVg^©Z\^Z5TVa_aO` dT[gZh m ]O\ dMrl_adZhZ\^TV`M` F ` F rfTVaK\·.Zhr i .\ ©O gfOp\ . s. s. s. . s. qs. q. q. s. Rq L `!a }}
(192) `. 7s. } . p. qr7s. `. }}. . ` } a . j. .
(193). 7ss. . vxs. . hN(Z ` O i i TV0WMN OdMrl¹MO MglBO ` \"§µ±\fKM0OO(NP_h£M_ gfi OWMrlaYVWMOPg"\^KM_hvZ]\ \^KM\^KeO«O O YVdO Zhmm \fhrfTVOpm\X¹gldMOYrfi O Z Zi \ gjT\^gfOYV©L Zh` i \f·K` Z]\j\^KMT\^ObgYVN(T[`MZO Zho9r T[T[Ngjm WM_hN `Mm TVO`9`9\f\fO r^`MZ]gl\fTVO \·ik Z]i Y[T[_aNP`eTV«`MT[Tgl\^KMg O ZgoUTVgX·`7Zh` \^iKeOT\^`Mg _aN(`7ZY[oUTV`MT[NO ZhWMrPN*N(T[O `i \fTVO WM`MN glTV\·k¢¹MT[Mgj arl O5 Z]§\fµ¡O \frKM\^OKMZhd` Zhm \f]KMOO\ T[`MT\^T[Zh_aY%r._a\f`eKMOhO JZhKMWMTVglgngfT[T[ZhZP`«ZP£D\·O5kIZ]dMNbTVm5µZ]Y%0£¤OOWeKgfZ5OUTVZh_agcrZh_h`«FZ(TV`Ma_hOT[rf`Mr(NP£DO _ai WMTV` WMi N Z]rlkm_a` i T\^TV_a`«Z] \. Z£¤h`/OYV\nO YVr^O Z mi \frfT[WMTVm gc¹TVgnO Y myi KM_hdDgf_aO Y[`ZhrfZhT g O i ZhY[_h`M\^KMµ|O gl _(Z\fKoUTVZ]g \c \^JKMOKMO Zh`M©aZ YVhO O _hYVO `MZh]dD\^O K rt\^TVWMg rlOj_hF\^KeO£Dµ±O5\fZhKMN O L. T[g V. . . . ^. q. s. V. ^. qs. 4. }Rq. q. m. #. 4 ¡. . } q. . p. 7 s. 4. . . q. . qs .q. `
(194) .
(195) #.. . \^_ TV\fK ª Ý OL _a£Mgf%OrlhOP_a\fr KMi OBOp\y£DZhO5TVZhY[g N Zh£¤Z]\_hWe\ Z]WMgfglT¬Zh `=£D`O5Z]¹N(hWMgrfµ±O_a`ehOBXm5\^ZhKM`OX£¤rfOO ZhONrv_adernrf_aT[d`MgtZ]\yZhZ]`e\^m OOg ZhdMdMi rl_ oUTVN(Z]\^OYVk T\^K \fKMO Zh`MªaYVO=_hZhdDO rl\fWMrfOI .\\fKMO/m _a`\^rfZhrlkhµvTV` \^KMO/O rlr N(T[` _UOm\^WMKMTVgfWMOjTV`MN O§oedD0Op¶±O OrlOT[mm5NP\ Z]nO``9Zh\fgf` g O i O \fKM¹O M`=gl dMY[Tj\^gv\^KMT[`Z]\^\X_w\f\·KM©O_£D¹O5ZhY[ZhNNPO ¹`rf\fgtg\ Zhm gv_h`MT\ m OTV`9g \fmr^Y¬Z]Zh\fglO gfg TVm5ZhTVYV\ YkTVgX_a£M\fKMglO OrtagfOO YVi q. X. \. . . . 4. V. ^. q. 7sss a. V. ^. ÄÇ %Äà '.
(196) }e.
(197) . T[aWerfO . Z5hOdZ]m hO\nTV`Z YVT[`MO ZhrNPO i TVWMN . Oc. ;TVaWMrlOP © Z hOXdZ]m hO\nTV`«ZP`M_a` YVT[`eO5ZhrNPO i TVWMN . ,Ù. '. ¡ £¢ ¤ ¤. c.
(198) }ª. ~ . ;TVaWMrlOPa ZhWegfgfT[Zh`b£¤O ZhN TV`Z YVT[`eO5ZhrNPO i TVWMN . c. ;TVaWMrlOP ª ZhWegfgfT[Zh`«£¤O ZhN TV`Z `M_h`«YVT[`eO5ZhrNPO i TVWMN . c. ÄÇ %Äà '. .
(199) }a}.
(200) . X . b
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