An Algorithm for the Navier-Stokes Problem

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1 A Algorhm for he Naver-Sokes Problem F.Z. NOURI ad.aoura * * DEPARTEENT DE ATHEATIQUES FACUTE DES SCIENCES UNIVERSITE BADJI OHTAR BP 12, ANNABA (AGERIA). RÉSUÉ. Cee éude es la couao des ravaux [7],[8] e [9] qu so basés sur l éude fae par Glowsk Al [3] e [] as que Berard Al (vor [1] e [2]). Ic ous proposos u Algorhme pour résoudre u problème o-léare ssu de la mécaque des fludes. Das [7] ous avos éudé le probléme de Sokes e adapa la echque de Glowsk, grace à laquelle, o peu découpler la presso de la vesse lors de la résoluo du probléme de Sokes. Das ce raval, ous éedos ore éude e moros que cee echque peu êre ulsée das la résoluo d u problème o-léare comme les équaos de Naver Sokes. Des ess umérques cofrme l érê de la dscrésao. ABSTRACT. Ths sudy s a couao of he oe doe [7],[8] ad [9] whch are based o he work, frs derved by Glowsk Al [3] ad [] ad also Berard Al [1] ad [2]. Here, we propose a Algorhm o solve a olear problem arsg from flud mechacs. I [7], we have suded Sokes problem by adapg Glowsk echque. Ths echque s userful as decouples he pressure from he velocy durg he resoluo of he Sokes problem. I hs paper, we exed our sudy o show ha hs echque ca be used solvg a olear problem such as he Naver Sokes equaos. Numercal expermes cofrm he eres of hs dscresao. OTS-CÉS Algorhmes Numérques, Approxmaos Specrales, Problèmes de Sokes Naver- Sokes. EYWORDS Numercal Algorhms, Specral Approxmaos, Sokes Naver Sokes Problems. Volume , pages 289 à 301 -

2 V % F A algorhm for he Naver-Sokes problem 1. Iroduco Over he las decade, here has bee cosderable eres sudyg axsymmerc problems ad her approxmaos. These problems have bee suded by may auhors, amely, Glowsk Al [3],[], Berard Al [1],[2] ad Nour Al [7],[8],[9] ad refereces here. The umercal dscresaos relyg o he curre, he whrpool ad he pressure formulao has frs bee performed for fe eleme mehods see [10], he much smpler case of he Sokes problem, ad exeded o he case of specral mehods [2] ad [7], where specral aalogues of Nédelec fe elemes [10] are used. Relyg o he work [7] ad [8], we propose a dscresao of he Naver Sokes equao he suao where he doma s ax-symmerc. We prese some umercal expermes whch cofrm he opmaly of he dscresao. The aalyss of Naver-Sokes equaos for compressble vscous flows a coeced bouded doma ca be reduced o aalysg he correspodg Sokes problem. Frs, we cosder he Sokes problem he rdmesoal doma ha we suppose vara by roao aroud he axs of he cylder. The, we use he cyldrcal coordaes!#" ad $% The doma s deoed by ')( *+,# -.0/ ;=<' $<>@?A 5 where GIH# s he eror par of he boudary of he axs BC ED F ad we deoe by / 5 Frs, we roduce he fucoal spaces J@ B0 egrable such ha N O PQ SRUT wh he orm VXWZY[P\ O^] `_a N O PQ bretdce [ Y gfh J@ % ad he correspodg weghed Sobolev spaces ( S Jj Xlkm H h p Jj? % - uméro spécal TA TA'05

3 o 5 % D o 291 Volume q 5 )( S -r UD o Hs / 5?8% mu ( b q 5 vu Xwyx 6 { ƒ( S u H x H o S J ID X o?% <'}Z w<? The, he Sokes problem ca be wre as A s H Š H Š Y Š H Š o H o Š o + H QŒ I ;Ž 9 A s H Š o H Š o H oš o + H ö Œ o ;Ž 9 H Š Š H o Š o ID Ž 9 Š ID Š o IDI y9 ; For he kow o Q d 5 Q Š Š Š o here exss q 5 E ˆ % The equao for he dvergece ca be wre as H 2 Š Š o Š q Š o UD ;Ž 9 Equao [2] show ha he vecor s of dvergece free. From heorem 3.1 chaper 1 of [5], here exss a fuco verfyg Š Š o ž Š 2 H ž % šq 5 [1] 8 [2] By akg 0 we have ad H ; H A he we se Š AU Š o H y- Now, le us deoe [2] by Ž Š whch ca also be wre as H Š Š Š o UD ad by he vecor H o C H y- % We also roduce he operaor p IH! [3] [] gve by [5] - uméro spécal TA TA'05

4 [ Y [ Y V Y Y / A algorhm for he Naver-Sokes problem ad he operaor We ca he wre Remark 1 If Ž ID, we have we ca he show ha such ha for ay, we have 2 EH 2 H ž [6] [7] m. If we defe he operaor ; H Q - (For deals see proposo [7]). = H o by [8] [9] 2. Couous Problem The queso s ow wha are he ecessary boudary codos o esure he exsece ad uqueess of he curre fuco defed by Š AU Š o H y- ad he deduce s regulary from he regulary of he velocy Š he soluo of he Sokes problem, see [7] for deals. We cosder Ž ( S Jj wh he orm [ VXª [ \ «š P q OZ] V V \WZY[P\ O^] ] Y V V W Y[P\ Y Ž O^]Q± [11] Noe ha he boudary operaor s defed ad couous from [ Y oo X% Noe also ha { [ cocdes wh 65y² he pos away from % I he same way, we defe he ormal boudary operaor whch s couous from Ž oo { [ (The dual of by Ž S J@? [10] - uméro spécal TA TA'05

5 Ê H H Y 9 Ë [ Y 293 Volume f#s Ž ad ³ # 2^% 9 ³ Q N O Ž ^% ³ X2 µ >N O h% k³ C µ!% %! [ b s he scalar produc of dualy bewee ad { o he sde whch s parrallel o I ad o he wo sdes of % fh I has bee show [7] ha Proposo 1 Q Ž such ha Ž From heorem 3.1 of [5], ºp> ge where ¾y ¾0À ¼, we have ¼½ J ^% [12] wh he measure UD ad ^% 9s µ¹id»ºp> such ha j% such ha y. If we se ¼ verfyg H 2¼ J ad ¼ H^ IDI y9 s he ageal dervave gve by The fuco ¼ all he oes ha ouch he axs B k-%. By egrag by pars, we % [13] Hh % [1] ÂÁ ÃyÄÅ Hs Á D o eachidpf coeced compoe of where o. Hece #% e us ow defe he boudary codos for he curre fuco. For, we cosder U Ž such ha Ž ÆD. e us deoe by he fuco assocaed o, From proposo 1, we have ÈÇ (a cosa) o % wever ouch he axs BC UD F ÇUD, hece ad É ID. We se he followg resul Corollary 2 e Ž such ha Ž ÂD ad h% É ÂD assocaed o ^ he s he soluo of he followg problem 2 Q -% Q Ë C PQ d Ê q 5 2^% Q0Ë X2 µ To sudy hs problem (see [5] ad [7]), we roduced he space. If s he curre fuco ^µf šq 5 [15] - uméro spécal TA TA'05

6 % ] Ë Ë 9 Š 29 - A algorhm for he Naver-Sokes problem 2 ( S; J@ ad we defe he boudary operaor from f# Xµf! / p J@ 2 ) oo {? [16] [ Y as Ì ^% Ë Í N O h% Ë X2 µ m N O 2 2 µ [17] Hece we ca wre where ¾y ¾0Ð kj% 9 Proposo 3 We suppose ha ' ˆ ¾ q 5 q 5 exss a curre fuco /, where s defed by ( J P/I0x 6 { H Remark 2 pž 9ÏÎ \ É ID [18] h% CÉ ED ¾0Ð x H / Ž p D verfyg o > J X ad h% É D ½D Hs o ad ¾0 ¾0Ð D$Ò 2}ZdÓ Ò Ô? %, he here ÑD o For he proof of hs proposo, we use Hardy equaly (See [7] ad refereces here). Remark 3 I [8], we showed how we ca geeralse hs resul. 3. Curre-Whrpool fuco formulao I hs seco, we roduce a ew ukow called whrpool fuco Õ whch gves Š Q Õ [19] - uméro spécal TA TA'05

7 5 5 Õ Õ Ý É Õ Þ Þ Ý 295 Volume s appled o Š Š o he wo frs equaos of [1]. If we subsue hs resul [1], we oba Ö ÕG H Œ I Ö H 0Õj+ Œ o [20] The eresg po of hs formulao s o decouple he pressure from he velocy o oba a separae problem for he pressure. We have where Š! ad Õ A% Ö H0 o [21] Hece we ca coclude ha hs formulao s equvale o he oe derved by Glowsk ad Proeau [3] ad [] The reduced Problem The advaage of hs decouplg s o wre [1] as 2 Ø ˆ É UD ¾ Ù É UD ¾ Ú y9ó y9g Problem (P) s exacly a Drchle problem for he bharmoc operaor soluo s he curre fuco such ha Problem 2 ˆ É ID ¾0Ù ved ¾ Ð ca be wre a marx form as where Î s he marx wh he eres 7pÝ ßÞ ÎÛÑUÜ }àšþ y9ó } u defed by, where he 7 Ý Þ } à PÞ } u ßÞ } à šþ } u IßÞ } à šþ } u Ý U } à } u - uméro spécal TA TA'05

8 Š Û Ý Ý á Ý á à Š k Õ Ô Õ % A algorhm for he Naver-Sokes problem where Š Ý!p âå s gve by Å 5 ã äšà0µ#ßã äšà0 Õ 2å à <>Žš < æó <çèp<>æé Here Þ respecvely }âà, are he agrage polyomals assocaed a he odes ã ad äšà, <'Ž@<'æê D ad < ç<'æó% s Š Ý cosruced å by he values of he soluo a he odes ã ä à Ü ad by he erms, ßã ä à Õ à where Õ å ad à are he assocaed weghs. We remark ha Î s a symmerc posve defe marx, herefore he grade cojugae Algorhm ca be used. A sudy of he couous problem was doe by [2] ad [3], ad he specral dscresao was suded [7] ad [9]. Here we would lke o exed hs sudy o he Naver-Sokes equao.. Numercal Sudy I he same way we have appled he above aalyss o a olear problem (see [9] for deals) ad aalogues resuls have bee foud. Usg he above formulao, we ca wre he Naver Sokes equao as ad where Q 2 Ø ˆ A Õv k % Q É UD ¾0Ù É UD ¾0Ú ë H o - H ë H 2 ì 2 X ì Therefore we se a Algorhm o solve problem (P)..1. Algorhm y9g y9g for ay regular fuco -% We propose he followg Algorhm - Approxmae ad B - Approxmae Õ ad BCÕ F0í Å Fyí Å by by dî- ; % - uméro spécal TA TA'05

9 D ñ ñ 297 Volume Õ ad ˆ A Õ are soluos of Õ dî- Õ võ % Õ H y Õ H 0Õ Õ É ED 796 ¾ ïpð É ED ¾0Ú È Hece he olear problem ca be solved as four problems for For he umercal ess, we have ake he followg examples.. Example 1 The frs es s he fuco Example 2 UòXóô s ô õâö 26C The secod es s a sgular sream ; Q. ; P! d Numercal ess were carred ou he IRISA laboraory Rees (Frace)..2. Numercal Resuls Numercal expermes ur ou o be agreeme wh he aalyss doe [7] ad [9]. As show Fgures 1, 2 ad 3, umercal resuls for examples 1 ad 2 were compared o he exac soluos. The error plos are he J -orm. We oced ha 1- The error aalyss for he curre fuco s qualavely he same for boh examples 1 ad 2 (Fgure 3). 2- The choce of he bass does o affec he qualy of our umercal resuls. wever, whe usg he odes o be he roos or he exremums of Chebyshev polyomals, beer resuls were foud. 3- From Fgure 2, where umercal vesgaos for a sgular problem (example 2) was cosdered, he error has a specral behavor, ha s o say, we do o have ay spurous modes ha desruc he soluo. - ay choces for he relaxao parameer were used ad he opmal value was î, he %ùøp% For he choce of parameers for our plos, we have ake he vscosy - uméro spécal TA TA'05

10 298 - A algorhm for he Naver-Sokes problem Error Aalyss for he velocy, example 1 egedre Exremums egedre Zeros Chebyshev Exremums Chebyshev Zeros ERROR N Fgure 1. Error Aalyss for he velovy, Example 1 relaxao parameer î UD %ùøp% A Sudy of he pressure s uder cosderao. 5. Cocluso I hs paper, we proposed a ew echque smlar o he oe by Glowsk Al [5] ad show ha we ca reduce Sokes ad Naver-Sokes problems o a se of saddle po oes ; where he ukows are he curre ad whrpool fucos. Here he pressure s compleely decoupled from he velocy o gve a smple Drchle problem o be solved. - uméro spécal TA TA'05

11 299 Volume ERROR Error Aalyss for he velocy, example 2 egedre Exremums egedre Zeros Chebyshev Exremums Chebyshev Zeros log(error) N log(n) Fgure 2. Error Aalyss for he velocy, Example 2 I s clear ha hs echque ca be very useful olear cases as has bee show above for he Naver Sokes equaos. Opmal resuls were foud. Akowelegeme We are very graeful for Professor Berard Phlppe for provdg he umercal facles a he IRISA.The auhors are also graeful o professor Berard for he fruful dscussos durg her vs our deparme Sepember We also wsh o hak he referees for her commes ad suggesos. - uméro spécal TA TA'05

12 300 - A algorhm for he Naver-Sokes problem Curre Fuco Example 1 Example 2 ERROR log N Fgure 3. Chebyshev Specral Error Aalyss for he curre fuco 6. Bblographe [1] C. Berard ad Y. aday, Specral ehods, Hadbook of Numercal Aalyss, Eded by P.G. Carle ad J.. os, 1997 Elsever Scece ; [2] C. Berard ad. Dauge ad Y. aday, Specral ehods for ax-symmerc Problems, Numercal algorhms ad ess due o ajd Aa, 1999, Edo Elsever Sceces edcales. [3]R.Glowsk, Numercal ehods for olear problems, Sprger Verlag (1982). [] R. Glowsk O.Proeau, Numercal ehods for he Frs Bharmoc equao, SIA SIREV, Vol 21 No 2 (1979). [5] V. Graul, P.A. Ravar, Fe Eleme ehods for Naver-Sokes Equaos, Theory ad Algorhms, Sprger Verlag, New-York (1986). - uméro spécal TA TA'05

13 301 Volume [6] F.Z. Nour,New Fg for a Specral ehod for he soluo of Sokes problem Preseed he Beal Coferece of Numercal Aalyss, Absracs, Dudee NA099. [7] F.Z. Nour A.Bouaghou Sokes problem he case of ax-symmerc Daa, Far Eas I. J. Of Appled mahemacs, vol21, No2, [8] F.Z. Nour ad. Amoura,Glowsk Techque for he Sokes problem, Acceped Revue des Sceces e Techologe, Cosae Uversy (Algera), o appear. [9] F.Z. Nour A. Bouagou, Fe Specral ehod for ax-symmerc Ellpc Problems, Preseed he Beal Coferece of Numercal Aalyss,Dudee 2001.o appear S.A.S. I. Publcaos, Joural of Aalyss ad Applcaos. [10]S. Salmo, Développeme umérque de la formulao ourbllo-vesse-presso pour le problème de Sokes, Thèse Uversé Perre e are Cure, Pars uméro spécal TA TA'05

Key words: Fractional difference equation, oscillatory solutions,

Key words: Fractional difference equation, oscillatory solutions, OSCILLATION PROPERTIES OF SOLUTIONS OF FRACTIONAL DIFFERENCE EQUATIONS Musafa BAYRAM * ad Ayd SECER * Deparme of Compuer Egeerg, Isabul Gelsm Uversy Deparme of Mahemacal Egeerg, Yldz Techcal Uversy * Correspodg