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Spectral Analysis of Defect-Correction Algorithms for Hyperbolic Equations

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(1)Spectral Analysis of Defect-Correction Algorithms for Hyperbolic Equations Marie-Claude Ciccoli, Jean-Antoine Desideri. To cite this version: Marie-Claude Ciccoli, Jean-Antoine Desideri. Spectral Analysis of Defect-Correction Algorithms for Hyperbolic Equations. RR-2831, INRIA. 1996. �inria-00073860�. HAL Id: inria-00073860 https://hal.inria.fr/inria-00073860 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. Spectral analysis of Defect-Correction algorithms for hyperbolic equations M.C. Ciccoli & J.A. De´side´ri. N 2831 Mars 1996 THE`ME 4. ISSN 0249-6399. apport de recherche.

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(89) \ bn$ Tz\ Yµ £AawY-f j™b:l hzjnhAtdY-Y-oo'egbkftnacl `È]¡bntk]^a egl fz]7Y_W+<t3j ]7; j b eg+8£Wbk]^jBac]7hWfW`^Y ewl bnbnœY egl t g˜ \ Ytnjd_zj ¬„¬ SU TW]^Yj ]7j®fWewbŸjn_Wtnhzt™]7j™]7fz‰W—%bOj™]7fW1 o-Y:2 bkTWYihztnYjnY-f?b2œegtdX3_W`7acbn]^egfhAY-tdX]¡bkj®bnTzY:oY-f?bktnac` 1 l ]¡BºÁbYtnYfzo-Y eghRYtkacbnegtŸbke•£RY1hztnYo-egf l ]^bd]7ewfzY l £?mH]¡bkjnY'`^œ–acf l bnTW]7j2]^jbkt™]^˜M]7ac`¡`^m¦eghWbd]7X1ac`  ¬ Æ ewt —8bkTzYeghrbn]^X+a `hztnYo-egf l ]^bd]7egfWY-t+]^jeg£Wbkac]7fWY l œegt \ —8²Tr]7o|T XY-awfzj bkTˆacbpb bkTz1 Y3jnB Y-oegf lr± egt l Y-t®_zh?²]^f lHl ]¡ºÁYtnY-fWo-YieghAY-tnacbkegt]^j®eghWbd]7X1ac1 `¡`^m•2 hztdY-o-ewf l ]¡bn]^1 egfz2 Y l £MmbnTzY°ˆtdjdb ± egt l Y-t–_zh?²]7f ll ]^ºRY-tdY-fzoY:eghAY-tnacbkegt—ˆ²TW]7o|T¶]^jp`^Y-jdjp]^f?bk_W]¡bn]¡˜gYc¬7ˆ egt bkTzYŸœew`¡`^e«²]^fz‰eghWbd]7X1ac`Șcac`^_zYj–TzaN˜wY£AY-Yf•œeg_Wf l !   b 1 B  A B  A   . â%â Örë'óí«ô.

(90)     $!    

(91) 

(92) . QZ. o o oo oo o o o o o o oo o o oo. o. 0. 0.5. %]7‰g_WtnY ?S!8\rhAY-o'bktd_zX ewœ  œ egt b\ . b 1B  ~ ~€. € ‡W¬ ?cZM~? W‡ ¬ª€gWN W‡ ¬Þ~cZz > ‡W¬ª€WN€w W‡ ¬ª€gWN W‡ ¬ Z?€WN€. awf l \. 1. \. .  S%aw£W`^Y€!®hWbn]^X1ac` \awf l eghWbd]7X1ac`ÈjdhRYobntkac`%tka l ]7_Wj l Y-hAY-f l ]^fz‰1egf b “ fHbnTzYXegjdbŸ]^f?bkY-tdY-j™bn]^fz‰„o-awjnY —"bnTzYijnhAY-o'bktka `utka l ]^_zj l Yo-tnY-awjnYjœtdegXè  ~ <eg£Wbkac]^fzY l ²]¡bkT \ bkeK‡W¬ª€WNb.€w+1 ²BTz Y-f¶bnTzY:eghrbn]^X+a `©hztdY-o-ewf l ]¡bn]^egfzY-t®]^j®_zjdY l ¬ “ f bkTzYoQacjnY  —"]¡b l Y-otnY-awjnYjœtnewXè   ~µbne•‡r¬Þ€grN W¬SUTzYtnYœewtnYw— ]^f £AewbkTHo-awjnYj-— bkTzYoegf?˜gYbHtn‰g1 Yfzo-BYÇ ew1 œbk2 TWYKac`^‰gegtd]¡bkTWX.]^jijnegXY'²Tˆacb1awoo-Y`^Y-tnacbkY l £Mm bkTzYµ_zjnYKewœbnTzY eghWbn]^X1ac`ÈhztdY-oegf l ]¡bn]^egfzYt-¬ SUTWYÇjdhRYobntn_zX ewœ oegtntdY-jdhRegf l ]7fW‰¦bne awf l \ \  ]^jihW`^ewbnbnY l egf 

(93) ]7‰W¬ ?r¬ pewbnY+bkTzacb]^f³‰wY-fzYtkac` ܜegtawf?m '—

(94) ]¡œ \ \  —

(95) bnTzYµfzewfz¢-Ytne¶Y']7‰gYf?˜wa«  `7_zYjewœ  actnY®j™]^bn_ˆacbkY l ]7fbnTzYŸo-egXhW`^Y'qhW`7b awfzYŸb eg1 fµbnB 1Tz YŸ o]^tn4Kbo'`79 YŸecœ 1 o-Y-f?bnY-t  ^ awf l tka l ]7_Wj    1  ab+ ¬ X= b –e w bkBY+ bkTzX= acb b2+ bkTzY1Bhˆ+ awtn; awb X YbkYtnj \•acf l oeg1 f?˜gb Ytn ‰g Y+a tkawhW]BIl + `¡m; bkb e bk"TzY' ]7Bt2+`¡]7XiX= b ]¡bk+ j:acj2BbnTz+ Y ; b  f=_WX3£AY-t®ewœ l ]7jdo-tnY'bn]^¢Qacbd]7ewfhRew]^f?bkj 3.]7fzotnY-awjnYj-—ÈawjyjnTzeN²fegf %]7‰gj¬®†acf l >1œegt \ awf l tnYjnhAY-o'bn]¡˜gY`¡mg¬ ˆegt

(96) Y'qWawXhW`^Yw—gœegt 3   bnTzYUtnY'`ša bn]¡˜gY l ]¡ºÁYtnYfzo-Y£AYbC²Y-YfbnTzY. 1. . é%âæ.

(97)  ?. 

(98) 

(99) !"

(100) # $!%'&(! )*,+-. 0.59 0.58. optimal theta. 0.57 0.56 0.55 0.54 0.53 0.52 0.51 0.5. 

(101) ]^‰g_ztnY†!"®hWbd]7X1ac` \ l Y-hAY-f l ]^fz‰+egfKbkTzY2fM_zX3£AY-t–ewœ

(102) hRec]7f?bkj´ 0. 10 20 number of points (N). ge hWbn]^X1ac`Á˜cac`^_zY®œegt°ˆfW]¡bkY 3”awf l nb TzY®`^]^Xi]^b \  ]7jfzY-awt œegtUbkTzY2jdhRYobktnac` tka l ]7_Wj–fWYQawt  ¬ =(B 2. B2. 30. . b1 B . acf l o-egtdtnYjnhAegf  l ]^fz‰w`¡m.    .'*)&% $&%('  % +  %(- % +

(103)  "*' =©qWawXi]^fW]^fz‰3ac‰?ac]^f

(104) ]7‰W¬ ?r—g]¡b©]^j8tdY-X1awtUcac£W`7YUbnTˆacb©bnTzYjn_z£WjnYb8ewœÆbnTzYjnhAY-o'bktn_WX X1a l Y ewœÈbkTzYyfzegfz¢Y-tneiY']7‰wY-f?˜cac`7_WY-jewœÈbnTzYŸawXhW`¡]^°zoQacbd]7egfKX1acbntd]Gq ì]^j‚˜gYtdmµ`^e=oQac`¡]7¢Y l —zawf l ]7fbkTzYÇfzY]^‰gTM£AegtnTze=e l ewœ®a¦tdYQac`–fM_zX£RYt-¬ “ f jd_zo|Ta¦oQawjdYw—acœ bnY-t1awhzhr`^]^oQacbd]7ewf ewœ bkTzYi£ˆacjd]^o]¡bkYtkacbd]7ewf¦bneKtnY-XeN˜gYbkTWY1YtntdegtŸo-egXhAegfzYfMbnjac`7ewfz‰KbkTzYY]^‰gYf l ]^tnY-o'bn]^egfzj awjnjdero']šacbnY l bneµ¢-YtneY']7‰wY-f?˜cac`7_WY-j-—"bnTzY]¡bkYtkacbd]7ewf¶oQawf„YQawj™]^`¡m„£AYiawoo-Y'`7YtkacbnY l £?m„eN˜wY-t <egt©_Wf l Y-Mt 

(105) tnY'`ša«qza bn]^egfi²TW]7o|To-awf£RY‚˜M]^Y²Y l awj a®hˆawt™bn]^o-_W`7awt©o-awjnYecœÁachzhW`¡]7o-acbn]^egfiewœ awf•Y']7‰wY-fzXe l Y3acfzfW]^TW]^`7acbd]7egfbkYo|TzfW]^¤M_zY ÜjnY-Y2œegtYqzacX1hr`7Y¥ †c¨ '¬ ]¡bkTzew_Wb–o|Tˆawfz‰w]^fz‰ bkTzYŸ°WqrY l hAew]7f?b-—zbkTzYŸ]¡bkY-tnacbn]^egf]7jbnTzY-fÇtnYhW`šaco-Y l £?m ! dE ?X .  ]7f ²Tr]7o|TbkTWY¶X1acbkt™]¡q ]^j 'l  Y°z fzY1 l 4 awj a 9' + 1—acf l bkTzY•fM_zX3£AY-t ]^j+bnTzY tnY`7a qWacbd]7egf hˆawtkacX1Y'bkY> t <ewt+hzjdY-_ l e ± bd]7XY-j™bkY1h '¬ù  \Ma ]7fzoYw—bkTWY„Y]^‰gYfM˜cac`^_zYjKewœ awtdY â%â Örë'óí«ô.

(106)     $!    

(107) 

(108) . N† 0.5. optimal spectral radius. 0.4. 0.3. 0.2. 0.1. 0. 

(109) ]7‰w_ztnY> ! ® hWbn]^X1ac`zjdhRYobntkac`Ætna l ]^_zj l Y-hAY-f l ]7fW‰egf:bnTzYfM_zX3£AY-tuewœAhAew]^fMbnj-´ b 1 B   ^ jd]¡bk_ˆacbnY l ewfKao']7tdo`^Y®ewœÈoY-f?bkYt   awf l tna l ]^_zj   cÜ ‰w]¡˜gY-faw£ReN˜w1Y ' —=BN]­a¬ Yw¬uajn1 X1acB `¡`Á+ fMb_zXa £R=XYb t-—= ]^b8 ]7Bj+ fˆac= bnb _ztk+ ac`Æbne3BIhW+<]^o;  b  ! 1  d Q†:. 1 B ²TW]7o|TÇ]7j–aitnY-ac` fM_zX3£AY-t¬ 0. 10 20 number of points (N). 30. »pj–atdY-jd_W`^b-—ˆbkTzY2jnhAY-o'bktnac` tka l ]7_zjewœ%bkTWYfWY² ]¡bkYtkacbd]7egfK]^j‰w]^˜wY-f•£Mm!. dE>X            Ba K 4 J b N 9 1   ²TzY-tdY   . 3 U  c a n t Ÿ Y k b z T Y  M z f N e  ² „ f ' Y 7 ] w ‰ Y f?˜cac`7_WY-j–ewœ ¬ ®fzY2Tˆacj !   .  1 BXA A aHB    46 ]  46  .        Ba  1 Ba 4 +  ? + ?0AFC 9 1  bI4  a B+ ; ^ bJ9   bI4 a  ; bJ9  B < + I B + = b + I B < + ; b 1 1 BI+ b +34`B+ bJ9 BI+ ; b  = X b BI+eb a BI+ X= b + B+ ; b  é%âæ .

(110)  >. 

(111) 

(112) !"

(113) # $!%'&(! )*,+-. SUT=_Wj. . .  bI4 a BI+  ; b 9 . 4 J b N 9 1  B—8+ bkTzYÇb tn+O ˆegtYqzacX1hr`7Yc—Uœegt Y l 4_zBIobd+e]7ewbJfá9 œÜawBIo'bk+<egt;<b]7j   ¬SUTW]^j  ]7XhW`¡]7YjŸbkTˆacb2ewfW`^m ?]¡bkb Ytkac1 bd]7ewB fz jawtnYijd_W¯1o]^Y-f?bbnetnY l _zo-YibkTWY14`YBtn tdeg9 tŸewœ‚2 †2 ewt l YtnjŸewœ   X+ac‰gfW]¡bk_ l Yc¬. .  -0% á%(K ì %‚3   '%'3K) ) Ç -U' /10  %'3. p» œ bkYteghWbd]7Xi]^¢]^fz‰öbkTzY‰c`7eg£zac`y]¡bkYtkacbd]7egfÜegthzjdY-_ l e ± bnY-XhRewtkac`®]^f?bkY‰gtkacbd]7ewf1bkTzacb m=]^Y` l jbkTzYjdY-¤M_zY-fWo-Y  —u²‚YÇfzeN² bn_ztnföbkeHbkTWYK¤=_WY-jdbd]7ewf ewœpTzeN²*]^jbnTzY`¡]7fzY-awt jdm=jdbnY-X jnec`^˜wY l a byai‰w]¡˜g' YfÇ ]¡bkYtkacbd]7ewf"¬8V SU TzY2jdm=jdbnY-X bke£AYjdew`¡˜gY l ]7j !. V. . . . ²TzY-tdY !  awf l  Z[ ¬1 ® fzYµhReg1Ojdjd' ]^£W ]¡`^]¡ bCm ]^j:bke¶1 jdew4 `^˜wYa £Mm a„9X3' _r `^bd]7‰gt™] l bnY-o|TzfW]^¤M_zYÇawj3]^f ¥ >N¨U²TW]^o|T¸]^j:Y'œš  °ˆo]^Y-f?b:£W_Wb3oegXhW`^Y'q ]7f³awhzhW`¡]7o-acbn]^egfzjewbkTWY-t2bkTˆawf Xe l Y'`‚hWtneg£W`^Y-Xj¬Ç»–`^bnY-tdfˆacbkY'`^mw— ²‚YhztneghAegjdYTzYtnYbkeµjdew`¡˜gYbnTzYjdm=jdbnY-X £MmÇtdY`7a qWacbn]^egf"—A_zjd]^fz‰Kac‰?ac]^f•£z_rby]^f¶a+XegtdY V‰gY- fzYtkac`uœegtdX—ajdbntkacbnY-‰wm¦ewœ‚Y]^‰gY-fzXe l Y+awfzfW]^TW]¡`šacbd]7ewf"¬ Å YtnYw—%bkTzY1Y]^‰gYfM˜cac`^_zYj3ewœ awtnY2j™]^bn_ˆacbkY l ewf„aijnY‰gXY-f?b–]7fbkTzY2oegXhW`7YqÇhW`7awfzY !. : \  \  46 <jnYY

(114) ]7‰W¬:'¬ ˆew`¡`^e«²]^fz‰ ¥ †c¨­1 —ÈbCB ²a e•oeg+ XhW`^Y'q¦4 o-ewa fwvx_z9 ‰?acbn? YihzjnY_ l e ± bn]^XY-jdbnY-hzj awtdY jnY`^Y-o'bkY l jn_zo|T bkTˆa bÇbnTzY]^tK]7f?˜wY-tnjdY-jactnYtnYhztnYjnYfMbkacbn]¡˜gY³ewœbnTzY aw£ReN˜wY¦jdhRYoA bntn_zX— fˆawXY`¡m !  \  : \  \  .  ²TzY-tdY

(115). ¬¶SUTzYf"— ²1 ]^bnTzegB_Wa b3o|Tˆawfz‰w]^fz‰¶4 bnTza Yµ°W9 qrY l hAew]^f?bQ—uawf³]¡bkYtkacbd]^˜wYom=o'`7Y X+a l Y2ewœ

(116) bC= ²e1?@2SjnA _zBC£zD j™bkYhzjp]^jo-egfzj™bktd_zobnY l ! V    #$ % $&    1 4  a V  9   +    ^     . . â%â Örë'óí«ô. . \. 1 4 a. . 9. . +.

(117)     $!    

(118) 

(119) . N. o o o o o o o o o o o o o. 0. 1. o o. o o o o o o o. 

(120) ]^‰g_ztdYS!8\rhRYobntn_zX*ecœ. V . o o o o o o o o. œegt. b>1 B . awf l \. 1. \. . . é%âæ.

(121) N . 

(122) 

(123) !"

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(126) ^ 4 91 BIB + a ²TzY-tdY \ : \  \ —WmM]^Y` l ]^fz‰+bnTzYŸœew`^`^eN²]7fW‰+dt Y l _zo'bn]^egfœÜawobnegt ! 1 4`Ba 9  4 a 9   ^    4             Ba 

(127) ^ ?   ^   1 1 1 SUT=_Wj-—ˆ]7fKbnTzYŸ`^]^Xi]^b 3   —WbnTzYŸjnhAY-o' bktnac`%Btka + l ]^_zjewœ%bkTWBIY+ acfzfW]^TW]^`7acbd]7egfÇom=o'`7Y2bkYf l j bke(! \ ^ ^ \ B 

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(129)     $!    

(130) 

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(132) ewœzbkTWY‚jnY‰gXY-f?b©o-awf:£AY‚a lzl Y l bke awf l ¬©SUTr]7j

(133) X] l hAew]^fMb-— \r—A]7jya+tnY-ac`

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