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Résolution des équations de Navier-Stokes par des schémas de projection Tchebychev

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(1)Résolution des équations de Navier-Stokes par des schémas de projection Tchebychev Olivier Botella. To cite this version: Olivier Botella. Résolution des équations de Navier-Stokes par des schémas de projection Tchebychev. RR-3018, INRIA. 1996. �inria-00073676�. HAL Id: inria-00073676 https://hal.inria.fr/inria-00073676 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 EN AUTOMATIQUE. Re´solution des e´quations de Navier-Stokes par des sche´mas de projection Tchebychev Olivier Botella. N˚ 3018 Octobre 1996 THE`ME 4. ISSN 0249-6399. apport de recherche.

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(105) Cegf_obnZ\egfSi § l Z\ŸkU td‹gU fSoUf[]r^\^\Uk¥œ¶ · •bj`kh—U l U hStd• l Z\obdZaegf£o egfSi}bnZ«bj]SU]Sf£hStdeg‡_^\R TVU l UJ¸pU^aTVQSec^\bn¹h|eg]St D‡ ˆZ‰AŠ ¬Lb~`kf l ZaipF]SU^ · •b~`kh|U l UhStdekuvU obnZ\egf o ekfSidbdZ\bn]SU7]_fhStdeg‡_^\R TVU l U°”†`ktnom Ðeg]hStdeg‡_^\R TVU l Z\ŸœšÍ‹gtj` l ¬ 0 - Q   h|eg]St DˆZ‰AŠ U$b J ˆŒ‰AŠ ¥M˜™U°hStdeg‡_^\R TVU U i}b td• inec^a]JUf l •o eg]_h_^­`cfFb¼^­`Ÿ[Z«bjUinidU<Ub^­` hStnUini}Zaegf'hS`kt»]_fJ`c^\‹gegt}Z\bnQSTVU l Uºb–m[h—U™zp¹‰` ™`_¬ l U™T`kf_Z\R tdU §weg‡rbjU frZat ]_f'hStneg‡r^aRTU l U™b–m[h—UsekZaidinegfVh|eg]St^­`wh_tnU ididZ\egf&¥F¶ · `c^a‹kegtdZ«bjQSTVU l U<hStdekuvU obnZ\egfVo egTVh|egtdbnU h|eg]St o~QL`c[]_U#f_Z\ŸkU‰`k] l U<bnU TVhSi ^a`†tn•inek^\]_bnZ\egf l U l U ]rqJhStneg‡r^aRTUi l U#¸pU^aTVQSec^\bn¹#Ub l&· ]SfVhStdeg‡_^\R TVU l U5b–m[h—U*sekZaidinegf?ino `c^a`cZatdU i¬L[]rZ(inegfœbbjUo~QSf_Z\[]_U TVU fœb5`®Zaid• ip§Jtn• ideg] l tdUk¥|¶U5ino~QS•T` l U5hStdekuvU obnZ\egf Z\Thr^\Z\oZ«bj&U  Q  š OŠ € _ Uidb ]_fJino~QS•T` l egfœb»^\U i»U i}bnZ\TH`®bnZ\egfSi l&· U tdtnU]St¼h|eg]St(^­`ŸFZ\bnU inidU#Ub^­`phStnUinidZ\egfJidegfœb l&· egt l tnU av d  ‚ Q O¥ z fSU5hStnUT'ZaRtnˆ U`kTV•^«Zaegtn`cbnZ\egf l U5^ · egt l tnUUfbnU TVhSi l Uo Uino~Q_• T`oegfSi}Zai}bjU §!Z\fSo^\]StnUJ^\UbnU tnTVU l UhStnUinidZ\eg‹f GKJ l `kfSi†^ · •b~`kh|Š U _ l UJh_tn• l ZaobnZ\egf&¥( †fžeg‡_bdZaUfœb*`c^aektni†]Sfžido~QS• T` l&· egt l tnU a  d°h|eg]St°^­`*Ÿ[Z«bjUinidU5Ub l&· egt l tnU a  dºh—eg]_t7^a`hStnUini}Zaegf6¥S” · ]SfSU5T`kf_Z\R tdU†idZ\T'Z\^a`cZ\tnUc¬ U f6U 6Uobn]L`kfœb l `kf_i™^ · •bj`kh|U l U†hStd• l Z\obdZaekf l U i<U q[bjtj`ch—ek^a`cbdZaegf_i l&· egt l tdU†•^\UŸ[•†id]St™^a`hStdU ini}Zaekf?Ub<^\U bjUtnTVUf_egf£^«ZafS• `cZ\tnUk¬—egf£h|U ]rb5`k]S‹kTUfœbjU tp^­`hStn•oZ\idZ\egfžU f£bnU TVhSi l U iwido~QS• T`ki l UhStdekuvU obnZ\egf&¬ l U T`kf_Z\R tdU§Veg‡_bnU f_Z\t l U iido~QS• T`ki l&· ekt l tdU l U ]rq—¬SbjtdekZ\i Aeg]?h_^a]_i  U fbjUTh_i ¥      . . . . . . .

(106)  

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(108). ^ ^

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(110) _ L Gid D ˆ |~^

(111) } _ M N ˆŒ‰AŠ X G dD ˆ ^ Š F D‡ ˆŒ ‰AŠ M a ˆŒ‰AŠ L. yry{zM|e}#~€. . . . . . . ÐWœ` ÐWg‡.    .

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(113).

(114).

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(116).

(117). ^ ^

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(119) _ L Gid D ˆ ^

(120) _ \E  G _ G J ˆZ‰AŠ \ Š J ˆ \ #_YJ ˆ ^ Š M N ˆZ‰AŠ L Š . . . . . . . 'T . 1½ fU —U obn]L`kfœb]Sf l •ŸkU^\eghSh|U TVU fœb l UºP»`‰mF^\egt l U i l Z —• tnUfœbjU i(Ÿc`ktdZa`k‡_^\U i(`c]_bjeg]_t l ]5bjUTh_i5a egfeg‡_bdZaUfœbp^ · UtntdU ]St l U5bjtdegfSo `cbj]StdU5^ae[o‰`c^\U`kidine[oZ\• U§V^ · •F]L`cbnZ\egf'T  R

(121). . H VDd ¬. Š H Š  H  _ H  D ˆŒ‰AŠ H  _ H  d C$DC ˆZ ‰AŠ \ E1G _ D ˆZ‰AŠ H G J ˆŒ‰AŠ H a ‚ Š HF‚ _ HF‚  da D ˆŒ‰AŠ L G]d D ˆŒ‰AŠ \ N ˆZ‰AŠ \ a Š  H _ H  C$DˆZ‰AŠ \ E1G _ D ˆZ‰AŠ \ a ‚ H ‚ _ H

(122)  ‚ da D ˆŒ‰AŠ L G]d D ˆŒ‰AŠ H   C C  Š

(123) Š   C \ E a<V`\ Š \ #_%d G _ G J ˆZ‰AŠ Š _ C_    H   C  _ \ Š H  _ H  C$D C ˆZ ‰AŠ H a ‚ Š H

(124) ;‚ _ H

(125) ‚  dha D ˆŒ‰AŠ L Gid D ˆŒ‰AŠ H  ?_7d C C  G _ G J ˆZ‰AŠ \  E a Š  Š H a  d L Oƒ .  M. .

(126). . . . .

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(128). ¶ · •F]L`cbnZ\egf Oƒ 

(129) Ceg]_tnf_Z«bV]Sf¾uvU] l U?oegf l Z«bnZ\egfSifS•o U idij`cZ\tnUi!h|eg]StVF]SU^aU?ino~Q_• T` l UhStdekuvU obnZ\egf ÐW  š f  idekZ\b hSdt • oZaip§J^ · egt l tnUV‚U fbjU TVhSi R Oˆg`  ŠH ŠH _H  M ^X Oˆk‡  Š H _ H  M \ VYX ‚ Š HF‚ _ HF‚  M V L Oˆko  .

(130). . . 798 y:7<;.

(131) ˆ. 

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(133) ^ · egt l tnU  U f¤bjUTh_i*idegfœb

(134) CegtdT•U ihL`kt#^\U i• F]L`®bnZ\egfSi Oˆ  `cZ\fSi}Z([]_UhL`ct<^aUi •F]L`cbnZ\egfSi R }‚„g`  Š H _ H  M ^ X ‚ Š \  ‚ _ H ‚  M ^ X d‚„k‡  V`\ Š \ #_ M ^`L }‚„ko  ¶U io ekf l Z«bnZ\egfSi*fS• oU inin`cZ\tnU ih—ek]St5F]SU'^aUVino~Q_• T` AW  š 'f  idekZ«bh_tn• oZai§

(135) ^ · egt l tnU  U f¤bjUTh_i*idegfœb

(136) CegtdT•U ihL`kt#^\U i• F]L`®bnZ\egfSi Oˆ  '¬ }‚„  `cZ\fSidZ(F]SUhL`ct^\U i•[]S`cbnZ\egfSti R }‚g‚„`  Š H _ H  M ^ X ‚ Š H

(137) ;‚ _ H ‚  M ^ X d‚g‚‰‡  }‚g‚‰o  Š H #_ M ^`L ”wU°h_^\]Si ¬M•bj`k‡_^«Zaidinegf_i^ · UtntdU ]St l U1bjtdegfSo‰`®bj]StdU°id]St(^aUioegf l Z«bnZ\egfSi»`k]rq†^\Z\T'Z\bnU i ¥c½1fh_tnekuvUb~`kfœb(^ · U  ¥ OŠ  in]_t™^aUp‡—ekt l Ub<U f

(138) bnU fL`cfFb™o egTVh_bnU l U AWg‡  ¬F^\U i™o egf l Z\bdZaegf_i#`k]rqH^\Z\T'Z\bnU i<id]St™^a`ŸFZ\bnU idinU†b~`cfS‹gU fœbdZaU^\^\U inekfFbtR _ _ |~} D ˆZ‰AŠ L  Mja ˆZ‰AŠ  \  a<V`\ Š \ #_%d C J C ˆZ‰AŠ \ v a Š H #_d C C J ˆZC ‰A Š H av#d~X d‚ Q  Š Š ^\U i#o ekf l Z«bnZ\egfSi`k]rq!^\Z\T'Z\bnU i#id]St<^­`ŸFZ«bjUininUfSektnT`c^\U†•bj`kfœb l egfSfS•U i#hL`c&t ÐWg‡  ¥L †ffSekbnU tn`J[]_U d‚ Q  fSU

(139) Ceg]Stdf_Z«bhL`ki l Uo egf l Z\bdZaekfSipid]ShSh_^\• TVUfFbj`cZ\tnU iph—ek]St#^­`'hStd• oZ\idZ\egf l ]-ido~QS• T` ÐW  š 'f  ¥. . . . .

(140) . . . . . . .

(141). 9,38/0 &!p0`<,3 ) "!$#&%(';:). . . . . .

(142). "! ,3&) ,;`)8` _0(8. 1½ fª]_bnZ«^\Z\ij`cfFb!^aUi

(143) o egf l Z\bdZaekfSi4Oˆ  š }‚g‚ 'l ]ªhL`ktn`k‹gtn`khSQSU-hStn•o • l U fœb‰¬<oegfSi}bjtn]rZaidegfSi]Sfµido~QS• T` l&· egt l tnU l U]rq—¬§‘€ frZ\ŸkU‰`k]rq l UžbjU TVhSi R<^aUi• F]L`cbdZaegf_iЈ  ¬ d‚‰  `cZ\fSidZ†[]_U  M ^ Ub ‚  M ^ l •bjU tdT'ZafSUfœbw^\U i#Ÿc`c^aU]Stni]SfrZaF]SUitR. M V X ‚Š M X ‚_ M \ VL ypU l Z\inh|egin`kfœbwF]SU l U^ · • F]L`cbdZaekf d‚‰ko  h|eg]St l •bjU tdT'ZafSUt Š Ub #_ ¬S^\U ]StŸc`c^\U ]Stf ypek]Si `„ŸkegfSi o egf_idZ l • td•Uf-hL`ctdbnZ\o ]r^\Z\U t#^\U i l U ]rqino~QS•TH`ci l&· egt l tnU Q in]_Z«Ÿc`kfœbjitR

(144). Š M   X Š M \  X . _. . · U i}bwhL`ci ]_f_ZaF]SUc¥ ¶U™hStdU T'ZaUt¼Uidb»]SfJino~Q_• T` l&· egt l tnU l U ]rq l U°b–mrh|U l `kTVi <`kidQ

(145) CegtdbnQ'½1]_^\U t¼tdUbj`kt l • l ]JidU oegf l  ¬ l `kf_i°^\U F]SU^6Uidb7U —U obn]S• Uw]SfSU Uq[bjtn`kh—ec^­`cbdZaekf l&· egt l tnU ]Sf l U^a`hStdU ini}Zaek4f  Š M V egt l tnU l Ub #_ M^  ^aektni l U5^ · •b~`kh|U l UhStn• l ZaobnZ\egSf R fR  ½ b~`ch—U l UhStd• l Z\obdZaegS . . . . 

(146). . .  D‡ ˆŒ‰AŠ \ S D ˆ H‹D ˆ ^ Š \FE1G _ D‡ Œˆ ‰AŠ H GKJ ˆ H a D ˆ L Gid D ˆ   yry{zM|e}#~€. \ a D ˆ ^ Š L GOd D ˆ ^. Š M N ˆZ‰AŠ X D ‡ ˆŒ |~‰A} Š M a ˆZ‰AŠ L. }‚„€g`  d‚„€k‡  .

(147) ‚„. h )#  . ½ b~`ch—U l UhStdekuvU obnZ\egfSR. .  D Zˆ ‰AŠ \ D‡ ˆZ‰AŠ H G J ˆŒ‰AŠ \ J ˆ M ^ X   . }‚‰Wœ`  G L]D ˆZ|~‰A} Š M ^ X d‚‰Wg‡  D ˆZ‰AŠ L €  M a ˆŒ‰AŠ €;L }‚‰Wgo  ¶U1inU oegf l U idb(]_fSU1Ÿk`ctdZa`kfœbjU l ]ino~Q_• T`}‚„€  š d‚‰W  ¬ l `kfSi^\U F]SU^FUidbU —U obj]S•Uº]SfSUºUq[bjtn`kh|ek^­`®bnZ\egf l&· egt l tnU l U ][q l ]bjU tdTVU l UhStdU ididZ\egf  Š M  Ub #_ M \ V <l `cfSi#^ · •bj`kh—U l UhStd• l Z\obdZaekf&¥ fR  ½ b~`ch—U l UhStd• l Z\obdZaegS . . . . .  D‡ ˆZ‰AŠ \ S Dˆ HFDˆ ^ Š \ $E G _ D ‡ ˆZ‰AŠ H G a  J ˆ \„J ˆ ^ Š d   H a D ˆ L GOd D ˆ \ a D ˆ ^ Š L Gid D ˆ |~^ } Š M N ˆŒ‰AŠ X . }‚ k` d‚ c‡.  Df   Df . D ‡ ˆŒ ‰AŠ M a ˆZ‰AŠ L. ½ b~`ch—U l UhStdekuvU obnZ\egfSR.  D Œˆ ‰AŠ \ D‡ ˆŒ‰AŠ H G J ˆZ‰AŠ \  J ˆ H J ˆ ^ Š M ^ X   . }‚„Šg`  G L]D ˆŒ|t‰A} Š M ^ X d‚„Šk‡  D ˆŒ‰AŠ L €  M a ˆZ‰AŠ €3L }‚„Ško  s»ek]StpoegTVTVU fSoU tp^ · Zafœbj•‹gtj`®bnZ\egf-Uf-bnU TVhSi  M ^  ¬LfSek]Siph|egidegfSi D g^ Š3MjD g ¬LUb^\Uo~QS`kTVhSiBJ g U i}b inecZ\b'•‹œ`c^™§¦_¬inekZ«b'hSt}Zaio egTVTVU

(148) ]SfS^ U!hStnUinidZ\egf g o ekThS`cbnZ\‡_^\U`„ŸkU o D 0  ‚Df   ¥ ”wU!h_^a]_iJh—ek]St^\U ino~Q_• T` d‚Df  š }‚„Š  ¬œf_eg]S_ i™h__ tnU f_egfSi.J Š •‹œ`c^6§&J ¥rs»ek]St7^\U i l U ]rq!ino~QSUT`ki#oZ š l Uinid]Si#o U^­`JtdUŸFZaUfœb§ o‰`®^ao]_^aUt a D Š X J Š d™Ub a D X J d`‰ŸkU o*]Sf-ido~QS• T` l ]-h_tnU T'Z\U tegt l tnUc¥  ¶U i°U inin`cZ\i™fF]STV• t}ZaF]SUi l ]hL`ktj`c‹gtj`kh_QSUWTVegfœbjtdU fœb7[]_U ^\U i l U ][qino~QS•TH`ci l ]Vb–mrh|U l oZ š l U inid]Siwidegfœb l ]-idU oegf l egt l tdUU fbjU TVhSi h|eg]St<^a`'ŸFZ«bjUininU`cZ\fSidZ(F]SU*h—ek]St<^­`'hStdU ididZ\egf&¥ . . . . . . .

(149)  .

(150). 9,38/0 &!p0`<,3 )  !2#8%('e) *0`,3`(i,3) *0+ . . . ™˜ egfSi}bjtn]rZaidegfSi*]Sf¤ino~QS•TH` l&· egt l tdUV€_¬§

(151) Wf_Z«ŸgU `k]rq l U'bnU TVhSi ¥(¶Ui5• F]L`cbdZaegf_i Ð ˆ  š d‚g‚ pl •bnU tvš T'Zaf_U fœbp^\U i]SfrZaF]SUipŸc`c^\U ]StditR

(152). Š M VSV  X Š M \  X . _. M  X  M . \. V  X ‚ Š M  X ‚ _ M \  X ‚  M VYX Š M  X . . ?_ . M.  ¬ l `kfSi ^aUF]SU^¼U idbwo egf_idZ l • td• UJ]Sf_UUq[bntj`kh|ek^a`cbnZ\egf l&· egt l tnU ™˜ Uido~QS• T`

(153) Uidb†]_f¦ino~QS•T` l U fbjUTVhSi l U^a`'hStnUini}Zaegf-Ub ]SfSUU qrbntj`kh|ek^a`cbnZ\egf l&· egt l tnU*€Uf?bnU TVhSi l ]bjUtnTVUf_egf^\Z\fS•‰`®ZatdUH¥ fR  ½ b~`ch—U l UhStd• l Z\obdZaegS . .

(154). 798 y:7<;. \ Q. VL.

(155) 

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(157) _ \ E1G _ D ‡ ˆŒ‰AŠ H : VYV D‡ ˆŒ‰AŠ \@V  D ˆ  ^ Šd ˆ ˆ  G a J \ J  v

(158) H  a D ˆ L G]d D ˆ \  a D ˆ ^ Š L GOd D ˆ ^ Š H a D ˆ ^

(159) _ L G]d D ˆ |~^

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