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Couplage de méthodes numériques pour les lois de conservation. Application au cas de l'injection.

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(1)Couplage de méthodes numériques pour les lois de conservation. Application au cas de l’injection. Martial Mancip. To cite this version: Martial Mancip. Couplage de méthodes numériques pour les lois de conservation. Application au cas de l’injection.. Mathématiques [math]. INSA de Toulouse, 2001. Français. �tel-00001960v2�. HAL Id: tel-00001960 https://tel.archives-ouvertes.fr/tel-00001960v2 Submitted on 3 Apr 2003. 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..

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(54)  H  46) '#$&J" >"; [$,C>Q;:<-N GAGGAGAGGAGAGGA  @.  

(55)   !"#   $%&''&'&'&'(') *,+  +.- / 0!1 +

(56) 2&23 435  6'&'&'&'(') 7 598  + 3  :;=<  (' + 13  >?'&'(') A

(57) CBD + E "F   

(58)   5G + 35 IHJK'&'&'(').L. H IH ?A1B>J58-K " AGGGGGAGAGGAGAGGAGAGGA @ @@ @ L. H H ].  MN

(59) 23  ,OQPRTSU &''&'('&''&'&'&'(')V 7  - 5 3 B '

(60) 2; 323W 3   + 5

(61) CBD!X'(').Y Z3 + 3[&]\ +  3^&'&''&'('&''&'&'&'(')._ ` ;D3 B!E28 $N3 + a3 !b'('&''&'&'&'(')._ FCc  dH + 5&0

(62)   +- #e&'&''&'&'&'('fLg ` h

(63) 2 +.- [ + 1 i  I35  j&''&'&'&'('fL.

(64) '  * 4 3:D- - +0+SUN0=VW)%+  ] . 'B. ! _X 

(65) 0/e(') E. [t L;/9:Y/.!O2/ a&(- o$‡jd  F  hp;/>"$#3-/Q‡_#!L2H5x>"T$-N&(#9: 4 9:Q.S)  >'&("Pdf;N&(p~/O;/.?.4‡_#!42H5x†„-/.DhFNuQ&(b9:6 ?M&(#b>#- 2*%.1&3"9:C8A) q "?2/"bMo >n&:BC-/>T$- "E&:2/>"TC-N&(#bT$- 42 &( B.4'&:b9 Š.1&L8=&(1&(Y/. 3‚"# -/ 8=&(1&(Y/.L'&(.1&(=F %8=&n&(.q=>P#9:-/#%;/>"MC#E.1&>#o 9_2/L2/"p0„9:.-/" ]m/p2 &( B.4'&:2*P‡_#!=l/'&(b9:6.w5 -/#.M&3;<9:-/b.4‡*#!42H5x†V-/.'F .   

(66)   !"# 

(67)  $. [t "%>#o 9$2/"%C-/>T$- "4&:9_>""%&(-*~Š.9:\2/?"9: 8=&(#9:K9:$#pYN&:>"" -/n."`>"oN&(/q"€2/6n&(#!Q&(-*~ #M&"8€2/"J&( #"`>;N&(M&($#`."`;<9:C#J2/ 2*>#M&(#9:[F. %P2/>R /#m2/9: v2/-*~9:- ),2/9:n&( "cdf 9:#>" E  # E A h[2*-2/9:n&( ˆ"9:O;/.# E&%ZGIHKJbT$-/ c""9:-/8$$#ˆ-/V-/ a'9: PdXTC- a.w5x9:`&(;/;<..M&E'9: B2/B"9:-/;/.1&(q h  9:#>"* E  A F/GI.b9:$#b2/9: 4#.pT$-  " . E (' E A ? E D E () E A ? E  +A ?&* , [. d! F h. c##?'9: ?2/P"9:-/;/.1&(q E  A 2/9:#%-/#9:-/# #?2/?"-/? 9:6$-/..`df89:b 9: .  >q=.q'&(Y/.?;N&(bM&(;/;<9:#%&(--/;/;Z9:#b2/4.1&3"9: 2*#9:‚/#1&(.MW R hF {$-/ˆ""c2/-*~9:- A),2/9:n&( "'l$9:n-/#.%2/"„n&(..1&(q"cTC- ."9: T$- "c2/b#,‡_;< JV.>C#m]m/'F %4 V;<-/#H;N&: &'89: .w5 -/2/"H 9_- 2/[2H5 -/ Vn&(..crc.w5 C#>-/ 2/€.w5 -/ Š2/"&( #"2H5 -/ 6&(-/#Šn&(..=FV^E€;/.- 'lm.1&K#1&(/q=-/.1&(#9: [5x"#O;N&: 2/>q> >>"`d‰&(- -/6#1&(/q=.[5x"#p&(;/.1&(# hF .

(68) '1' . )6 5 5)+.1-.+ -2 -.+:,)+9-O-9 -= -=VW: + . (*)+,-/.10= N02)657 9N:5)+.1-. _{ 9:C#  #Š""€2/-*~ #1&(/q=-/.1&(#9:  F %‹2/>R /#6&(.9:J."J;N&:J2/ 2*>#M&(#9:‚6";N&:"?2/E.1&3n&(/!?-/8=&(C##" C   G J ?

(69)     Cf"N C NG G2D C  A G J ?        CX"N C G G [ †V*R [l*9:`$#9$2*-/#a.1&2*>#M&(#9:Š`#O;  F$t[%;N&:a2/b#O; a"#c 9:#> "  ? :    :  F †V#\9:>"9:-/#%.?;/9:Y/.!p@z- T$-[57&(-6#O; pR N&(. LF {_9:#%2/9:   l*.42//% #M&(C#%'&(.-/.>4;N&(b.4Mo >n&*F : &"8" .   ? " !#   %$& . %. /; >"$#n&($#N&(C#€.w57&(;/;/9"~*n&(#9: 2/6.1&eM9:.-/#9:[l„9'‡/ >"Q-/ oN&:-/ 2/"bn&(.."*". #:,:4 )N VW)>  VW'-=  - ,-/5) + 5 N   . t &B9:.-/#9:L&(;/;/9$Mo >"„;/>"$#„-/ „&(;/;/9"~*n&(#9:32/V.1&a9:.-/#9:[l"9 ‡*)  >"?-/boN&:-/ P2/"Bn&(.."*" 4 + , (5 6 3  

(70) *) " C NG

(71) 3C>COD :  G.- CVD , ( 3  R

(72) ? " + C NG

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(74) '  * 4 3:D- - +0+SUN0=VW)%+ . t &O2*M>#M&(#9:‚2/E.w5x9:;<>M&(#-/p#O;<9:..?>#M&(C#\O;/.C# " 8. 3 8 :. 3 

(75)  . ). '6L.  3 

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(77)  

(78)    C : G [ d ! F! h : R

(79) 

(80)

(81)   . 9 .1&\W‰9: #9: 

(82) C>CBGm;/>"$#b.1&\W‰9: #9: 2*'&(#"3d  9:- Y h 2/b.1&\n&(.. # 

(83)  

(84)    C : Gl "..c2/„.w5 C#8=&(..a2/„#O; 'F(GI."#ˆr%2/>"ˆ.1&b2*>#M&(#9: 2*-Š;/9:Y/.!42/* a&(- o$‡ed  F  hl/2 &( a.E'&:2*42/"bMo >n&:B"9: 8=&(#SWX%r#9: ;Z9:$#'F . .   .  !"# 

(85)   

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(87) M9:#M&(C#Š2/`.1&n&(/! -/8=&(C#ndf89:BR q=-/ ! F  h " AC F DC#" D%$ G D(AC'& F DC#" D)(3GD d ! %F $h C C %C* & D+! 

(88) G-, @ [ . . n e,C. Cr (e). Cl (e). ]mq=-/ ! F  \^ >R /#9:‚2/4.1&L 9:n&(. ${ -/EoN&:-/e2/"pn&(..1&(q" l 9:‚-/#.?-/ 3&(;/;/9"~*n&(#9:e2/"%>"MoN&(/q"42/ n&(#!`$#n."Ww&:"##"P2/"n&(.."'l T$- .w5x9:j&(;/;Z..0 -*~ C-/>T$- =F %.

(89) '%(. (*)+,-/.10= N02)657 9N:5)+.1-. ""9:O;Z9=P.1&n9'‡/ 32/"%>"MoN&(/q"42H5 -/ n&(..P;N&(\M&(;/;Z9:#4r"%89: "  <"#-N&($#p-/ 4"9:PY/N&(9:K"9:$8~*-/B#9:-/#?M&PWX9:C#!=F t &39:.-/#9:&(;/;/9_o >"?2*-ŠMo >n&L0„9:.-/"b]m/aC-/>965 $A5Hd $ ? + D  ha"# 9:Y/#C- P&(-#O;  :   b2/4.1&Ln&(/!-/8:&($#VU A F7" . . 

(90)  

(91)    ! 

(92) D 3 

(93)   D 3  

(94)   &

(95) N< C=3 

(96) G [ ! 

(97) D ?  ""N,C C#"NG G 3 

(98)    D  

(99) 8 &"8" 

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(101)    3 

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(103) G [   

(104) [5x"#m;N&: O;/.>n&($#> 2 &( b."b"9_2/"b2/4'&(.-/.p'&(p.[4O;/.SRN?2 &( p.1&39:O?-/B."p&( #" "  "N C "G ! 

(105) ? @ [ . .  

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(107)    ? ?. 

(108)  3 

(109)  3.         D 3  D 3 " N C G    

(110) 

(111)       "NC G 3 

(112)  D 3  

(113)     . . 

(114)           

(115)  D 3 

(116)  D 3  

(117)      3 

(118) ;  D 3  

(119)  [ . t &LW‰9: #9:  "#p&(;/;<.>" -*~€C-/>T$- 2*-Mo >n&O#B8>SRN#"  . C D 3OD 3GP? <  >C 3BG [ D C D 3OD"!+PG ?  C  D"!D 3BG [    . . . \^ €;/.-  lV9: -/;/;Z9=MŠTC- €=5x"#O-/ €W‰9: #9: tH; Mo/#"/ ‚"9:$#C- =Fˆy%9:-  8 9: p2H5 C#9_2*-/? ."bo >n&:%"9: 8=&(#SWXprL#9:p;Z9:$#'F.

(120) ' T' (*N:5)+.1] ^]. '

(121) . +-._$ `.  . A.       

(122)   . 0 ;  ' T' 

(123) 

(124)  "!$#%'&)(*,+*--&.

(125) t A CFEHG0/ 1%!,2  / #3+465 R C  FG EH  /87 &.

(126) +*-9&:

(127) / +,&.

(128) / ;#:

(129)  / #:=<>#. !!? / @A!B=<C#:! !?#ED  )F G &:@=;&:@ C F I GK &:@";&: 7 &.

(130) +*-9&:

(131) =J F6tNADCFEH"

(132) >C CG , J C T6.G -3T [ L C J GB>C CBG ? d ! F& h /BC NG

(133) S <  MENO  QP . . . . . R. %.

(134). . .  n&(TC- K2/K-/#‚T$- K"##6W‰9:-/.Q"#J#9:-/#€r Ww&(#`O.1&(er .1& "9: #- #9:‚2/?.1&39:.-/#9:‚&(;/;/9_o >"2/9:/ >"?;N&(b."B0„9:.-/"p]m/'F y\9:#M&(#9: " {$„.w5x9: >"#P"##;/9(@"#9: ;<9:-/? 9=?M9:.-/#9: L&(;/;/9_o >"" "9: #M&(C#"%;N&(B9:"'&(- A !  FVURDC  Gc# A ! A FVURDC A G L. A C!  G  A  C ! A G

(135). L. . | 8". ]

(136)  A   9 P P  MENO WYX MQ[1Z \ ?  ] A  

(137)  EP9P. ?. . W ^N O WYX MQ[1Z \. ! 

(138). ! A . "C ) NG D  "C,  C G NG   "  ] " C ) G [. %ŠW‰9:&( H-/ ?"9:PY/N&(9:‚"9:$8~*=l 6 <#*"  "C  ) G ?‹" C  G ].

(139)  A   ? A  

(140) ?. D. d ! FI h. [. . d ! F h. . . . .

(141)    . [. t &;/9(@"#9: &( E2/>R /K8>SRN‚2/9: ‚-/ ‚;/9:;/>#>‚#!"JO;<9:#M&(C#=l "9: "N&(C#\.1&3"9: 8=&(#9:‚2/E.1&3n&:M " ,. _. .     . L. . A C!  G  C>CG. %2*#%&(.9:bT$-[5x..?"#b"9:. . M8:&(#8=F. - C ? ,   ! 

(142) 

(143) >C CG [ _ 

(144)   .

(145) '. (*)+,-/.10= N02)657 9N:5)+.1 .    #    !   .      

(146)  .   +    . | ;N&(#O2/`"`;<9:C#'l 9:j&‚"9: 2/!>€T$- .€2/9:n&(  E "#32/>""9:-/;<>J 2/-*~ŠM9:- A),2/9:n&( " E  #HE A T$-/H;<9=!"2/C#p-/ EC#"#9:K 9:Š8$2/ E  A F. % 2/>R /#.1&€WX9: #9: 2/n"9:-/;/.1&(q F  GIH J r€.w57&(2/2H5 -/ WX9: #9: TC- E.w5x9:6;/9:.9:/q;N&(b"9:C#$-/#>Lr3#9:-/# E 2/4.1&PW‰& "9:‚-/8:&($# " . . ! F! F. p. 4. C>CBGP?. . . . +. C G. 5 6 @ >C @. . XE FXE  C FXE C. +. . C. F. E A. 

(147)  D  D

(148)  [. AE. A A. d ! F! h. %;/>"$#M&(#9:Š;Z9=Y/.p2/p"##%W‰9: #9: "#ˆ2/9:/ >"E2 &( ˆ.1&4R q=-/ . 1 α (x). 1- α (x). 0 Ω. 1. Ω. Ω. 2. ]mq=-/ ! F ! ] 9: #9:‚2/?"9:-/;/.1&(q 1,2. / #b."b2/-*~Š9:- A),2/9:n&( "Ldf8$#M&(-*~€;Z9:-/B.1&3"9:$8q " hB2/ E 9 .1&LWX9: #9:‚2/4"9:-/;/.1&(q?8=&(-/#B8$M&($# ". %62/>R. E A E. R ?   ? . A. . . XE  A D C=CG ? FXE  A D =C CG ?. C F. . C. . @. . . +. d ! FK h.

(149) ' T' (*N:5)+.1-. '. %&(;/;/9_o. Š &(.9:P"##W‰9: #9:2/`"9:-/;/.1&(q`2/n.1&Š n&(/!JT$-  ;Z9:-/p."B0„9:.-/"p]m/ " . . . %‚&O&(.9:. ,. +. 7C>CBG- C ? L  C G

(150) D , 7C>CBG- Cd? L C G [   .  "NC NG +  ? "C G

(151) ?. . .

(152). . . . d ! F Y h t[O;/ ;<3q=.9:YN&(.v2/3.w57&(.q9:#o/`"#E.O-/8:&($##"<9: ;N&(#?2/"?9:.-/#9:  &(;/;/9$Mo >""PrJ-/e #M&($# :  2/9:/ >=F %e-/#.3."2/-*~QMo >n&:?$-/>TC- " >;N&(>C#3dX9:;Z>M&(#9:6T$- E.w5x9:Š 9:#?&(.9:B";<"#8C#  # A hl*;/-/p9:  <"#- 3.1&J"9:"#9:e;N&(E.3"9:-/;/.1&(q6df;/9(@A"#9: L A  #6L  A hlZ;<9:-/E9:Y/#/ ."E9:.-/#9: ?r.w5  #M&(C#?-/8=&(C# :    F %e&n;/>"$#>Oo >n&(#T$- C#"" >#M&(;Z"%2 &( B.1&PR q=-/ ! %F $*F .

(153) D  F  @ D + . . . . n. v1n C. v2 D. S1. S2 n+1. u1 C. Π 21 v 1n+1 C. u n+1 2D Π12 n+1. v 2D. ]mq=-/ ! F%$ %| .q9:#o/4q=.9:YN&(. " Mo >n&:b;/-/b;/9(@A"#9: .

(154) '. (*)+,-/.10= N02)657 9N:5)+.1-. 4^ &( "##3R q=-/=l ."?>#M&(;Z"  # A "9:";<9: 2/C#O&(-*~e2/-*~eMo >n&: C-/>T$- "P#P."?>#M&(;Z""L  A #0L A  ;/>"C#$#P.";/9(@A"#9: L#?."9:-*) ;/.1&(q=F t„57&(.q9:#o/O2/3"9:-/;/.1&(q32/>R /#E2/-*~‚ 9:-/8.."4-/#" C! 

(155) G  # C

(156) ! A  G  TC-/[9:$#p2/9:/ >""b;N&(p."aWX9:P-/."p-/8=&(C#" " . 4. (( ! 

(157)  ((.  ?. ((. 5(. (( ! A  ( 6((. . . . . . d ! F  h. . .  ?. ((. &"8". C + 

(158) G L A  3 A  

(159) ; "NC,G !    D "C NG   

(160)   C +   G 3 A    L  A 3     ",C,G ;!    [ "NC  G A  

(161)  

(162) 3   

(163). . ?. . . ?. . . . . . !    ! A  . %n&(T$-.  ? .  ? .

(164) 3  

(165)  C + 

(166) GK3 A   D ;  3  

(167)  C +   G3 A   [ . ;. . J BTC- `.w5x9: &O;/.#$#n"9: #-/#3.Jn&(..1&(q"O2/"LC#A) "#9: j2/"bn&(.." 2*-n&(..1&(q"  #  2*-n&(..1&(q ! lN2 &( b.1&'9: P2/ "9:-/;/.1&(q=F cP 9:-/8'&(-Qn&(..1&(q3"#%-/Q9:-/#. "C#.m;<9:-/%.1&`2/>9: #M&(#9: 2/?"9:C8q "P#b.1&3"9: #- #9:$-/>TC- P2/4.w57&(.q9:#o/=F. % ;Z-/#€2/9: ‚2/>R /Š2/-*~ 9:.-/#9: Š&(;/;/9$Mo >""6&(- M `2/"J0„9:.-/" ] /p-/ E  df";<"#8C# E A h " . 4. (5 ( !   C>COD : G(?   ! 

(168) 

(169) C>CBG 

(170)  

(171)    C : GSD 

(172)   6 !   C>COD G(?    !   C>CG [ 

(173)  

(174)    C : G (( :      A   . d ! F ! h. †V#b9:€-/#.4 M&O&(- <."a9:.-/#9: p&(;/;/9_o >"" ,C#>"2*1&(" ?-/8=&(C#" " . (( 3   C>COD : G(?   3 

(175) 

(176) C>CG 

(177) 

(178)   C : GSD 5 

(179)  6 3   C>COD G(?    3  C>CBG 

(180)  

(181)    C : G [ (( :      A    . . d ! F  $h.

(182) ' T' (*N:5)+.1-. †V*R [l 9:62/>R /#p.1&39:.-/#9:‚&(;/;/9_o >"9'‡/ >" ". '6<. C G. 3  =CED (? :. C=CG !   C>COD : G C + . ;. . C=CG G !   C COD : G . [. d ! F  k$h. (&  m.7ˆoN&(;/# $bT$- v"##c2//!ˆM9:.-/#9:3&(;/;/9_o >" "9:C8q8b.1&39:.-/#9:ŠWw&(Y/.?$#9:;/TC- =F. %L8:&p2/>9:C#V2.  . .  .      . D- WV 03 UN,-.+ -.)N +

(183) )6  -.+ c 9:OP 9:- b.w57&"89: \~*;/.TC- >?‚C#9_2*- #9:[lN."9:-/;/.1&(qP2/?Mo >n&M W‰&(#\&:#- ..C#\rO;N&(#p2/E#9:b2/>""b2*"#""\2* H>C#"*" t["cq=.."a;N&(a9:"'&(- dX9:- ,q=2`;N&(#Mo//q p)V89: xƒb-/Y !!  #

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(204)    C G C  : ;   R       ! A    C>CBG 

(205)  

Références

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