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Accéleration de l'algorithme de Newton-GMRES pour les équations de Navier-Stokes

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(1)Accéleration de l’algorithme de Newton-GMRES pour les équations de Navier-Stokes Rémi Choquet. To cite this version: Rémi Choquet. Accéleration de l’algorithme de Newton-GMRES pour les équations de Navier-Stokes. [Rapport de recherche] RR-2287, INRIA. 1994. �inria-00074385�. HAL Id: inria-00074385 https://hal.inria.fr/inria-00074385 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET AUTOMATIQUE. Acc´el´eration de l’algorithme de Newton-GMRES pour les e´ quations de Navier-Stokes R´emi Choquet. N˚ 2287 Juin 1994. PROGRAMME 6 Calcul scientifique, mod´elisation et logiciel num´erique. ISSN 0249-6399. apport de recherche.

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(230) ‘‰]. =  . Uw\‰x h³jdvgezvg\+†\n©LoVXm

(231) µ ‡¸\‰x h mt\n© ‘< tµ§ŒX’X”wŽ 1 ŒXŒ 

(232)  Œ Ž}µ§  X”[Q“ 1 Œ 

(233)  ŒXY ” ŒtµrŽ’gŽg‘‰  XŒ 

(234)  “ ‘Xµ§X ’ X] 1 ’ 

(235)  “}Y  _l0 ‘ P 1

(236) a . 4 (s0 Œ a. ‡¸\‰x h Y cnc ª 1 µ§’XXŒX]X’ 1 ª µ G“}”]X]t‘ 1 tµl‘.X 1 ]XŒ 

(237)  1 ‘Xµ§’gmŽ X ’ 

(238)  1 µ§G“X“ X’t‘n“ ‘Xµ§’X”XŒ 1 ”t‘ 

(239)  1 µ¿“}”X X” ‘Xµ§’ 1 ]”X]XŒgŽ 

(240)  +  C@0 ‘ P 3] a   D0 P ]  œY[´ga¤\‰Y[eIŒ 4 Ý\‰hpoŒ  ‚cnc-\fx an\nx UfYGopjlVXm

(241) µ ‡¸\‰x h mt\n© ‡¸\‰x h Ycnc ‘< ‘Xµ t‘‰”]G“}”X’ 

(242)  ª 1 µ§’XgŽG’X’XŒ Œ tµrŽ‰“X““ 1 “}’ 

(243)  ª 1 µ§ŒX]X”[“ 1 “ ŒXY ‘Xµ t‘‰”Q‘ ]X] ” ]tµl‘ 1 Q‘ 1 ]X” 

(244)  Œtµ§XŒX][“}X1 ’X  

(245)

(246)   _l0 ‘ P 1

(247) a . 4 ( 0 Œ a +  C@0 ‘ P 3] a   D0 P ]  œY[´ga¤\‰Y[eI” 4 Ý\‰hpo+‘  ‚cnc-\fx an\nx UfYGopjlVXm

(248) µ ‡¸\‰x h tm \n© ‡¸\‰x h Ycnc ‘< 1 lµ ‘ 1 Ž}‘ “tŽ}‘‰’ ª 1 µ§’wŽ] 1 ’ 1 “ Œ tŒ µ¿“}XŒŒ 1 gŽ 

(249)  ª 1 µ 1 “}]’G“} Œ X XŒ Y ”tµ¿“}]G“gŒG“tŽ’ ” Œtµ§gŽ]Q‘‰G“-‘ 

(250)  tµ 1 “X“g”gŽ”XŒ  

(251)

(252)  “ ‘Xµ§]XŒX] 1 ]X] 1 

(253)   _l0 “ P 1

(254) a . 4 ( 0 Œ a +  C@0 ‘ P 3] a   D0 P ]  œY[´ga¤\‰Y[ez“ 4 Ý\‰hpo+‘  ‚cnc-\fx an\nx UfYGopjlVXm

(255) µ  ?˜tÅ *W ,/. 4 ( a . Z B ( / 8GHG `\ok\‰hpo , oYG´gal\ / Z#VXm}okU\Z¸YbalWXUt\x etm5mtVXZ7´tU\+v?¦p\nx oYGˆº\‰hAvg\+†\n©LoVm5ˆga¤e-hWXUY[m-v?u}etm5WgYbjlmzhpetU a¤\_mtVXZ7´tU\+vg\ˆtUVtvgegjlohYXcfVX´gjl\nmtª ±X\‰cfo\netUh µ . . . . <"€. ÞßÜyá.

(256) ‘GŽ. 

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(258)   “ Ž}µ ” 1 “ 1 ][“-‘ 

(259)   1 µ  t‘ 1  1 ” ‘Xµ X]Xt‘‰ X” 

(260)   1 µ 1 ŒX’X’X]Q‘‰Œ XY “-µ“}”X]X]G““ 1 

(261)   ] ”tµ{‘n“-‘X‘n“QŽ}‘ 

(262)   1 µ ]gŽ 1 ’G“gŒX ]XY Œtµ{‘‰”X”XXŒ[“tŽ 

(263)   Ž tµ XŒ 1 ‘b”X 

(264)   XYc ±\‰c “Q 1 “ 1 1  _l0 “ P 1ba . 4 ( 0 ‘ 1

(265) a .Z B ( 0 ‘ 1

(266) a +  C0 ‘ P ]'a  I0  P 3]  YG´gal\‰YGe 4 

(267) \‰hpo_‘  ‚cncº\fx a»\nx UYGo¥j¤Vm

(268) µ  )  n6<9Ý ƒ|¦p\nx oe-vg\Dvge—ˆtUt\‰x cfVm-vj¤o¥j¤Vmtmt\nZ\nm}oˆtUVˆ-V}hg\Ix vtYGm-h¾al\¨cŠ-Y[ˆgj¤okU\¨”£m

(269) ¦§Y£ˆ-YXh¸Z#VXm}okUw\zx ckadYbjlU\nZ\nmgo v?¦§Yb±GYGm}ofYGWX\‰h|hpetU&al\U\‰v&\nx Z¸YGUUYGW\Xµ‰ ¼a YGe-vgUYj¤o U\‰cfVm-h³jdv\nx U\nU&a¦p\nx oke-vg\LalVXUhksXet\adY+ˆº\nUo\vg\+cfVXmg±X\nUª WX\nm-cf\YGe7U\‰v\nx Z¹Y[UUYGWX\&\‰hpoh³jlWXmgjrqNcnYGopjl±X\XµtT|YGUcfVm}oUk\Xuna¦§YbalWXVUpjloŠtZ\vg\†\n©LoVm;Ycnc-\fx a»\nx Ut\Ax YZVmtª oUt\Ax vg\‰h>Ut\‰x h¥ega¤ofYGoh\nm-cfVXetUY[WX\‰YGmgohnµG a

(270) hp\nUYbjlohpVetŠ-YbjloYG´gal\v?¦p\nx o\nm-vgU\¯a¤\‰hijlm}±X\‰h¥opjlW}YGopjlVXm-hY+¡ v?¦§YGetokU\‰h ˆtUV´ga¼\n¡ Z\‰hnµt„YGm-h&al\ etoetUbutVXmzcfVXm-h³jdv&\nx U\nUY_al\+Z \nX Z\o }ˆº\+v?¦§YbalWXVXUpjloŠtZ#\+ˆ-VetUvg\‰h [ ªhckŠ \nx Z¸YXhnµ  F,F ˜ 

(271) ˜    Æ  Á  E à  ›£˜   ÅXÅ ˜t– Æ ˜ –  tà E Á F ˜ à  Á › D Æ{˜ E Ø –  )"%$ &)(+* ,¹” 4 ‚cncº\fx a»\nx UYGo¥j¤Vm ,  a a a 6  i H a  i H a   i H / B B B H ‘<. ‡¸\‰x h mt\n© 1 µl‘ 1 X’ 1 1 Œtµ§] 1 ’]XŒ X] 

(272)  ]tµ§]t.‘ Q‘‰ŒXŒX’ 

(273)  ’tµ¿“}X G“X“tŽ 

(274)  ‘Xµ§ ” t‘4XgŽ’ 

(275)  Œtµ 1 1 ”]XG“} 

(276) . . . . ™. 8;:  8;H :  H  8;: i  : r w  r i w  i H 8;D.  . . . .  6  i H 8;: r i w   . . . i H 8;:.   . . `Vm-hpopjloetopjlVXmzvge¢ˆtUVX´ga¼\n¡ Z\+Y[ˆtˆtUVwckŠ \ x , Œ 1 /;PqPPqPqPqPqPPqPqPqPqPPqPqPqPqP . 0   H &  i H  0@6  i H   i H A 0 )   0 0

(277) 0 > 0   i N 0 ,mz`]+6  i H P 6  ( i 0 ,K, A a

(278) H /a aq, A a

(279)  i. ˆ-VetU µ‰µ‰µ. U€ƒ‚„„ †. ‘ a µ‰µbµ a .W) ‘. H/A H am &  i N N / w.

(280) ‘‰. =   iN 0.  .  )

(281)   )

(282)  . 0.  . 1. 1. ‘. (. bµ µ‰µ a1

(283) a ‘ a1/ (  ™   `VXm-hpopjloeto¥j¤VmzvgezˆtUVX´ga\n¡ Z\+YGˆtˆtUVtcŠ \ x , ŒXŒ /;PqPqPPqPqPqPPqPqPqPqPPqPqPqPqP  `AYbadcfega>vg\ : 8;: r  i N w r  i N w VUoŠtVXWXVm-Ybal\o\faiset\ : ZU0 ZA N + H 0 ] ( ) ^ h0L6  i H .  . ,.1

(284) a. . 0. :. .  0 :  i H ]          |   ™.3  

(285) UpjdYGmtWega{Y[UpjdhYGopjlVXmvg\  Z¸YGoUpjdcf\+vg\ \‰hhp\nmg´-\nUkW¾hpetˆ \nx Upjl\netU\ P  -YXcfo‰µ /( ‡žvg\  0  Yb±X\‰c VXUoŠtVXWVXm-Ybal\\no oUpjdYGmtWXegadYbjlU\7hpetˆ \nx U¥j¤\netUk\<vg\+UfYGmtW7² 0  ™!89  ¬5jlmgjlZ7jdhYGo¥j¤Vm·vge5ˆtUkVX´ga¼\n¡ Z\ ] - N PqPPqPqPqPqPPqPqPqPqPPqPqPqPqPPqPqP  H . . . 0. ‡ \nm-vgU\ -. . . , N H a µ‰µbµ a N  / (  i H )  i N. \nx oY[ˆ-\ }® Y tˆ ®}vgVo ®}hkcnYba ÝalVXˆ-h h‘ Œ ” Œ ” Œ ] ‘ h, Œ ,/})h,/ h” / .`) .`) .`) ‘‰” ,/.`) Œ , / .`N  ) ) ‘‰Œ / ,/.W) Œ / h¥“ Ý, VXoYb/ a Œ .`) ” Œ .`) ” .`) Œ ‘b” ,/.`) Œ /N N )-,/.k/ YG´gal\‰YGeI] 4 `VXZˆgal\

(286) gj¤ot\+x vg\a¦§Ya¤WVXUpjloŠtZ\+v?¦§Ycnc-\fx a»\nx UY[opjlVXm

(287) µ .  FF ˜ ݘ   Á š   ˜t™. <"€. ÞßÜyá.

(288) ‘‰’. 

(289)    !". 0. 10. -2. 10. -4. Residu. 10. -6. 10. -8. (25). 10. GMRES. -10. 10. (10/5) -12. 10. 0. 10. 20. Ýj¤WetU\¹‘. 30 40 Produits Jacobien-vecteurs. 4 . 0 .  . . H. u. 50. 60. 70. 0L1

(290) P ’. 0. 10. -1. Residu. 10. -2. 10. (10/5) -3. 10. GMRES (25). -4. 10. 0. 10. 20. 30 40 50 60 Produits Jacobien-vecteurs. ÝjlWXetU\+Œ 4 Ý\‰hpo‰‘u-`|ƒ 0 ‘Xµ 1 U€ƒ‚„„ †. 70. 80. 90.

(291) Œ1. =  . 0. 10. -1. Residu. 10. -2. 10. o(10/5) x(5/15) -3. GMRES. 10. (25). -4. 10. 0. 20. 40. 60. 80 100 120 140 Produits Jacobien-vecteurs. ÝjlWXetU\+” 4 

(292) \‰h¥o‰‘Xuº` |ƒ 0 “-µ 1. 160. 180. 200. 0. 10. -1. Residu. 10. -2. 10. (10/5;10) -3. 10. (25) (10/5). -4. 10. 0. 10. 20. 30. 40 50 60 70 Produits Jacobien-vecteurs. 80. 90. 100. ÝjlWXetU\“ 4 Ý\‰hpo‰‘u-`|ƒ 0 ‘Xµ 1 utY 0 ‘ 1 <"€. ÞßÜyá.

(293) Œt‘. 

(294)    !". 0. 10. -1. Residu. 10. -2. 10. (10/5;10). -3. 10. (25) (10/5). -4. 10. 0. 20. 40. 60. 80 100 120 140 Produits Jacobien-vecteurs. 160. 180. ÝjlWXetUk\W 4 Ý\‰hpo‰‘Xuº` |ƒ 0 “ºµ 1 u-Y 0 ‘ 1. 200. 0. 10. -1. Residu. 10. -2. 10. -3. 10. CFL=0.5 CFL=4. CFL=1. -4. 10. 0. 20. 40. 60 80 Produits Jacobien-vecteurs. 100. 120. Ýj¤WetU\<] 4 

(295) \‰hpob‘XutcfVXZˆ-YGUYj{h¥VXm5ˆ-VXetUvj\nx U\nm}ofh` |ƒ U€ƒ‚„„ †. 140.

(296) ŒXŒ. =  . 0  bA ÝL   9 Q L` ’X” R ‡µAY[UU\no‰uX¬£µ&\nUU XuµQ`Š-YGm

(297) u}Qµ}„\nZZ\fauttµg„VXm-YGo‰ugtµ}„VXmtWgYGUUYtuT7µ}­j¿€p²}ŠtVeto‰u ‡µXTV nVºug`µ‡¯VXZ7jlmt\XutYGm-

(298) v +µ TYGm¸vg\nU T|VXUhpo‰µ .\ N  &L q  = Lq =!"  !"   L .Lq.\ L P  ! X ! Z Pq .L q h!   ! #\  =.XL»µ ®X ³‚¬ +mt\noparjl´

(299) uœ‘‰’’X”tµ Q ®}’ 1 R Tµ§† &UVG©LmYGm-v ®}YXYv?µ g´tUpjdv²gU a¤VG±7Z\noŠtVtvth VXU&mtVXmgarj¤mt\‰Y[Uh thpo\nZ¹h V

(300) \‰se-YGª opjlVXm-hnµ  W  !    \ N = u

(301) ‘‘ , ” /4 Q“ 1 “g}‘uX¬IYj>‘b’X’ 1 µ Qr`A­’” R ‡Â`ŠtVtset\noYGm-vz#­|UŠt\faµ®XVXZ\+cfVXmg±X\nUWX\nm-cf\Uk\‰hpegaloh VXU oŠt\_mt\n©LoVXmtª WXZ#U\‰h YbalWXVXª UpjloŠtZzµ&‡ \‰hp\‰YGUckŠz‡ \nˆºVXUo†LV5Ž tug ‡ k®X‚<uÝ®X\nˆto\nZ7´tU\¸‘‰’X’X”tµ Q§„_®}X” R tµ§­¢„\nmtmgjdhœYGm-v_‡µ °®}ckŠtm-YG´-\faµ 9Q\3!"   =4X L   Lq  ! kAX N ! \3!  !" + X h ku!"  W3   !" Lnµ7T|U\nmgopjdcf\nª Yayahp\nU¥j¤\‰hjlm~`VXZˆtetofYGopjlVXm-YbaA¬YGoŠt\nª Z¹YGo¥j{cnh uN‘b’XX”tµ Q§„eto’t‘ R ƒ|µr` „etooVºµ  Št\7\ 

(302) \‰cfo V VXUvg\nUpjlmtW¸VXmzˆtUk\‰cfVXm-vjlopjlVXmt\‰vWXZUk\‰hYbalWXVUpjloŠtZ Y[ˆtˆgayjl\‰v oV5hpVGal±X\7oŠt\7cfVXZˆtU\‰hhj¤´gal\¸m-Yb±Xjl\nUªhpokVX²X\‰h\‰sXe-YGo¥j¤Vm-hVXmetm-hpoUe-cfoetUk\‰vzWUpjdvthnµ+Gega  ‘‰’X’Q‘Xµ|®Xet´tZ7jlook\‰v5oV ³m}obµ-tµ VXU†etZ\nUpjdcnYbai¬¢\noŠtVtvth&jlmI­>mtWGjlmt\n\nUpjlmtW-µ Qr« Tƒ

(303) X’ R «<µ 8«VGalet´ Y[m-v~`µ 2TYGm~ƒ?V}YGm

(304) µ   !   \ N   !" L L  XVX !"!  -µGVXŠtm VXˆt²Xjlm-hnuݑ‰’XX’tµ Q§¬ Ÿ ’X” R µ¬YGmgo\ne 

(305) \fa Y[m-v8tµ Ÿ okoV-µ Ÿ ˆtopjlZ¹YbaI\‰segjl±Ybal\nmgo—ˆtU\‰cfVXm-vjlopjlVXmt\nUhnµ  h %9Q\3 h yut” 1b4 Ž’ 1 t‘‰ŒtuXGetmt\¸‘‰’X’X”Qµ Qr®}YXYX’X” R 8®}YYXv?µ‚ V-

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