The Modified Bi-quintic B-Splines for Solving the Two-Dimensional Unsteady Burgers' Equation

Transcription

1 Europea Iteratoal Joural of Scece ad Tecolog Vol. No. Te Modfed -qutc -Sples for Solvg te Two-Desoal Ustead urgers' Equato S. KUTLUAY a ad N. M. YAGMURLU b a Departet of Mateatcs Đöü Uverst 448 Malata/TURKEY e-al: selcuk.kutlua@ou.edu.tr Tel: Fa: b Departet of Mateatcs Đöü Uverst 448 Malata/TURKEY e-al: urat.agurlu@ou.edu.tr Tel: Fa: Abstract I te preset paper te odfed b-qutc -sple base fuctos are proposed ad successfull appled to te two-desoal ustead urgers' equato usg te Galerk etod to obta ts uercal solutos. Te accurac of te uercal scee s eaed b te error ors L ad L.Te obtaed uercal results ave bee copared wt te eact oes. Kewords: Galerk Fte Eleet Metod -qutc -sples Two-desoal -sples Modfed b-qutc -sples Two-desoal Ustead urgers' Equato MSC classfcato: 97N4 6N3 6D7 74S. Itroducto Ts paper deals wt te two-desoal ustead urgers' equato gve b ut uu uu ν u u D t > wt te tal codto u / e u v 3

2 Europea Iteratoal Joural of Scece ad Tecolog Vol. No. ad te boudar codtos u t g t > / t e t v u t g t > / t e t v u t t > / t e t v u t t t > t / v e te rego D [ ] []. Te eact soluto of ts proble s gve as [] u t e t / ν 7 were ν / R s a arbtrar uber ad R s Reolds' uber. Te fte eleet etod as bee wdel appled to pscs sold ad flud ecacs egeerg edce ad so o []-[]. Tere ave bee soe studes o uercal solutos of te twodesoal ustead urgers' equato. urgers' equato as bee foud to descrbe varous kds of peoea suc as a ateatcal odel of turbulece ad approate teor of flow troug a sock wave travelg a vscous flud []. Soe researcers ave obtaed aaltcal ad uercal solutos of two-desoal urgers' equato. El-Saed ad Kaa [3] ave proposed Adoa's decoposto etod to approate te uercal ad aaltcal solutos of te sste of two-desoal urgers's equatos wt tal codtos. Zu et al. [4] ave proposed dscrete Adoa decoposto etod to fd uercal solutos of two-desoal urgers' equatos. Fletcer [] as obtaed eact ad uercal solutos of two-desoal urgers' equato b usg Cole-Hopf trasforato. Kater et al. [] ave elaborated spectral collocato etod based o dfferetated Cebsev poloals to obta uercal solutos for soe dfferet kds of olear partal dfferetal equatos cludg two-desoal urgers'. Wu ad Zag [6] ave cosdered te uercal soluto of te two-desoal urgers equato ubouded doas. Radwa [7] as tested ad copared te uercal solutos obtaed fro ger-order accurate fte dfferece scees for te two-desoal urgers' equato avg oderate to severe teral gradets ad also derved fourt-order accurate two-pot copact scee ad fourt order accurate Du Fort Frakel scee. Gurefe ad Mısırlı [8] ave frst obtaed eact solutos of te Rccat equato wt te elp 4

3 Europea Iteratoal Joural of Scece ad Tecolog Vol. No. of G / G -epaso etod ad te ave foud soe varable separato solutos wt two arbtrar fuctos of two-desoal urgers sste. Velvell ad rde [9] ave copared te parallel perforace ad accurac of lattce oltza ad tradtoal fte dfferece etods for solvg te ustead two-desoal urger's equato. Srvastava et el. [] ave obtaed uercal solutos of two-desoal coupled urgers equatos b usg Crak-Ncolso fte dfferece etod. oules ad Eck [] ave obtaed uercal approato of te two-desoal urgers' equato based o a tecque cosstg of approatg te copoets of te eact soluto b a trucated Fourer seres wt te depedet coeffcets. Ia et al. [] ave appled a drect algebrac etod to cople solutos of t-order Korteweg-de Vres equato ad two-desoal urgers equato ad ave obtaed soe cople solutos of te. Aller et al. [3] ave solved twodesoal urgers equato b usg Pror Reducto etod wc s a teratve procedure based o repeatedl proved bases at eac terato. Te redefg process was successfull appled to cubc -sple fuctos to obta odfed b-cubc -sple fuctos [4]. I ts paper we ave proposed te odfed b-qutc -sples ad successfull appled te to te two-desoal ustead urgers' equato order to obta ts uercal soluto.. Te odfed b-qutc -sples A approato u t to te eact soluto u t of two-desoal ustead urgers' equato s sougt ters of te -sples as u t αj t j j 8 were α t 's are te apltudes of b-qutc -sples j gve b j j j 9 ad s defed as

4 Europea Iteratoal Joural of Scece ad Tecolog Vol. No. 6. ] [ 6 6 ] [ 6 ] [ 6 ] [ 6 ] [ 6 ] [ oterwse []. j s ca easl be foud b replacg wt j ad wt. Fg. F depcts a rego were so tat t s dvded to fte eleets b te teger kots j ad a sgle bqutc -sple 33 wc peaks o te pot 33 ad also covers a total of 36 square eleets. We te etre set of b-qutc sples j eac of wc peaks o a kot j 6 j are added to ts fgure a total uber of 36 sples cover eac fte eleet [6]

5 Europea Iteratoal Joural of Scece ad Tecolog Vol. No. Fgure : Te b-qutc -sple 33 cetered o 33 coverg 36 fte eleets of sde. To fd a approate soluto of te for 8 to te proble gve b Eqs. -6 b te Galerk etod -sples ad j j eed odfg to be zero o te boudar of te proble te ad -drectos. Te odfed b-qutc - sple bass fuctos are defed as [6] Tus te approato 8 ca be epressed ters of te odfed b-qutc -sples as 7

6 Europea Iteratoal Joural of Scece ad Tecolog Vol. No ~ ~ ~ ~ ~ ~ ~ Fgure : Modfed -sple fuctos - No-oogeous boudar codtos are satsfed b te ter / t t ϕ ϕ were t ϕ ad t ϕ are gve as follows: g g t ϕ g g g g t ϕ 6 7

7 Europea Iteratoal Joural of Scece ad Tecolog Vol. No. g g Te odfed qutc -sples are grapcall sow Fg. for ol 4 eleets. 3.Nuercal Eaple ad Results I ts secto we wll tr to obta te uercal solutos of te two-desoal ustead urgers' proble gve b Eqs. -6 b te Galerk fte eleet etod usg te odfed b-qutc - sple base fuctos. To appl te etod to te proble frst of all we eed to costruct ts weak for. 3. Weak For of te Proble To obta te weak for of te proble all ters Eq. are take to te rgt ad sde of te equato ad te ultpled b te wegt fucto Ψ. Te b tegratg te resultg equato over te rego D ad settg t to zero we get D [ u uu uu ν u u ] Ψdd t 8 were k l k ; l. applg te Gree Teore see e.g. Redd [7] to Eq. 8 te weak for of te odel proble global coordate sste s obtaed as D [ uψ uu Ψ uu Ψ ν u Ψ νu Ψ ] dd. t Te applcato of te fte eleet etod usuall apples a trasforato of global coordates to local coordates as follows: 9 ca be rewrtte as: ξ lad 9 η. Applg suc trasforatos Eq. 9

8 Europea Iteratoal Joural of Scece ad Tecolog Vol. No. D [ u Ψ u u Ψ u u Ψ νu Ψ νu Ψ ] dξdη. t ξ η ξ ξ η η I a tpcal eleet equato ters eac value of u e u u Ψd ξ ξdη ad u u Ψd η ξdη are te o-lear ters. I tese wll be take as a real costat e λ depedg o te eleet posto. 3. Galerk Fte Eleet Solutos of te Proble To obta te approate soluto of te odel proble b te Galerk fte eleet etod usg te odfed b-qutc -sples te approate soluto ξ η t s costructed for eac eleet usg te weak for ad a coeffcet atr s obtaed. cobg te coeffcet atrces for eac eleet we fall get te followg algebrac equatos for te proble: ZA g u I Eqs. ad da t G PA t. dt 3

9 Europea Iteratoal Joural of Scece ad Tecolog Vol. No. were A s t t [ A t A t... A 3 3 t] ; j k l l j ad j l l k were j k l l j ad j l l k were j l j. Te uercal scee s pleeted b dvdg te rego D [ ] [] to frstl 6 / 4 ad secodl / eleets. Te obtaed uercal results are copared wt te eact oes ad tabulated wt te error ors L ad L. I all uercal calculatos R s take as. Te dvso of te rego to 4 46 eleets If te soluto doa D of te proble s equall dvded to eleets 6 squares avg te sdes of / 4 are obtaed. If te approate soluto ξ η t s costructed for eac eleet a total uber of global eleet paraeters over te rego D are obtaed depedg o te local eleet paraeters α t ad β t j. It s clear tat we eed to fd te global eleet paraeters A t 49 to obta te approate soluto ξ η t for eac eleet. To do ts te approate soluto ξ η t for eac u j u u eleet s wrtte te weak for ad te a coeffcet atr s obtaed for eac eleet. 3

10 Europea Iteratoal Joural of Scece ad Tecolog Vol. No. A t 49 are solvg te algebrac equatos ad eleet paraeters obtaed at tes t. ad t.. Te obtaed eleet paraeters are put ter places te eleet equatos ad te te approate soluto u ξ η t for eac eleet at tes t. ad t. s foud. Te dvso of te rego to eleets If te soluto doa D of te proble s dvded to equvalet eleets squares avg te sdes of / are obtaed. solvg algebrac equatos ad we obta te global eleet paraeters A t 69 at tes t. ad t. respectvel. Te obtaed eleet paraeters are put ter places te eleet equatos ad te te approate soluto u ξ η t for eac eleet at tes t. ad t. are foud. Te obtaed uercal results b te preset etod usg te odfed b-qutc -sples ave bee dsplaed ad also copared wt ts eact oes Tables - 4. It s obvousl see fro te tables tat te uercal results are good agreeet wt te eact oes. Fgs. 3 ad 4 sow grapcall ow closel te approate soluto u t atces wt te eact soluto u t at tes t. ad t. respectvel. Sce te uercal soluto s ver close to te eact soluto ter graps are dscratel slar to eac oter.... u u u a Eact soluto 3

11 Europea Iteratoal Joural of Scece ad Tecolog Vol. No. b 4 4 eleets c eleets Fgure 3: a Eact soluto b Nuercal soluto for 446 eleets ad c Nuercal soluto for eleet at t... u. u u c eleets a Eact soluto b 4 4 eleets Fgure 4: a Eact soluto b Nuercal soluto for 446 eleets ad c Nuercal soluto for eleet at t. wt t.. I order to easure ow good te uercal solutos obtaed b te Galerk Fte Eleet Metod wt te odfed b-qutc -sple bass fuctos te L ad L error ors defed as follows L u u p u u p u 33

12 Europea Iteratoal Joural of Scece ad Tecolog Vol. No. L u u a < < p u u are coputed ad tabulated Table or3. Here p s te uber of er pots u ad u are te eact ad uercal solutos at te pot. respectvel. Te L ad L error ors are coputed b takg te values u ad u at pots obtaed b dvdg p te rego D [ ] [] to 4 equal eleets te ad drectos. As clearl see fro te table te uercal soluto becoes better as te uber of eleets creases. Table : Nuercal ad eact solutos of te proble for 4 46 ad eleets at t.. Nuercal Soluto u 4 4 Eact Soluto u

13 Europea Iteratoal Joural of Scece ad Tecolog Vol. No. Table : Nuercal ad eact solutos of te proble for 4 46 eleets at t. wt t.. Nuercal Soluto u 4 4 Eact Soluto u Table 3: Nuercal ad eact solutos of te proble for 4 46 ad eleets at t.. Nuercal Soluto u 4 4 Eact Soluto u

14 Europea Iteratoal Joural of Scece ad Tecolog Vol. No Table 4: Nuercal ad eact solutos of te proble for 4 46 ad eleets at t. wt t.. Nuercal Soluto u 4 4 Eact Soluto u

15 Europea Iteratoal Joural of Scece ad Tecolog Vol. No Cocluso I ts paper te odfed b-qutc -sples were proposed ad successfull appled to twodesoal ustead urgers' equato order to obta ts uercal solutos. Te accurac of te etod was eaed b te error ors L ad L. Te obtaed results were foud to be good agreeet wt te eact oes. Te etod ca be effcetl appled to ts tpe of olear twodesoal partal dfferetal equatos arsg pscs ad egeerg.. Referece [] A. H. Kater R. S. Tesa ad M. M. Hassa A Cebsev spectral collocato etod for solvg urgers'-tpe equatos J. Coput. Appl. Mat. : [] M. A. att Fudaetal Fte Eleet Aalss ad Applcatos: wt Mateatca ad Matlab Coputatos Jo Wle & Sos Ic.New York. [3] J.N. Redd A troducto to Nolear Fte Eleet Aalss Oford Uverst Press New York 8. 37

16 Europea Iteratoal Joural of Scece ad Tecolog Vol. No. [4] E. G. Topso Itroducto To Te Fte Eleet Metod Teor Prograg ad 38 Applcatos Jo Wle Sos Ic. USA. [] J. Wu X. Zag Fte Eleet Metod b Usg Quartc -Sples Nuercal Metods for Partal Dfferetal Equatos [6] L.R.T. Garder G.A. Garder A two desoal b-cubc -sple fte eleet: used a stud of MHD-duct flow Coput. Metods Appl. Mec. Egrg [7] O.C. Zekewcz Te fte eleet etod McGraw-Hll Lodo 977. [8] S.Kutlua A. Ese I.Dağ Nuercal solutos of te urgers equato b te least-squares quadratc -sple fte eleet etod J Coput Appl Mat [9] H.atea Soe recet researcer o te oto of fluds Mo. Weater Rev [] J.M. urger A ateatcal odel llustratg te teor of turbulece Advaces Appled Mecacs I Acadec Press New York [] J.D. Cole O A Quas Lear Parabolc Equao Occurg Aerodacs Quart. Appl. Mat [] P. Jaet R. ooret Nuercal soluto of copressble flow b fte eleet etod wc flows te free boudar ad te terfaces J. Coput. Ps [3] S. M. El-Saed ad Doga Kaa O te uercal soluto of te sste of two-desoal urgers equatos b te decoposto etod Appl. Mat. ad Coput [4] H. Zu H. Su ad M. Dg Nuercal solutos of two-desoal urgers' equatos b dscrete Adoa decoposto etod Coput. ad Mat. Appl [] C. A. J. Fletcer Coputatoal Tecques for Flud Dacs Sprger Seres Coputatoal Pscs Verlag erl 99. [6] Xaoa Wu ad Jwe Zag Artfcal boudar etod for two-desoal urgers' equato Coputers ad Mateatcs wt Applcatos [7] S. F. Radva Coparso oaof Te Hger-Order Accurate Scees For Solvg Two- Desoal Ustead urgers' Equato J. Coput. Appl. Mat [8] Y. Gurefe ad E. Msrl New varable separato solutos of two-desoal urgers sste Appl. Mat. ad Coput [9] A.C. Velvell ad K.M. rde Parallel perforace ad accurac of lattce oltza ad tradtoal fte dfferece etods for solvg te ustead two-desoal urger's equato Psca A [] V.K. Srvastava M. Tasr U. ardwaj ad Y. Saasraju Crak-Ncolso Scee for

17 Europea Iteratoal Joural of Scece ad Tecolog Vol. No. Nuercal Solutos of Two-desoal Coupled urgers' Equatos It. J. of Coput. Metods Egrg. Sc. ad Mec [] A.N. oules ad I.J. Eck A Spectral Approato of te Two Desoal urgers' Equato Ida J. Pure Appl. Mat [] I.E. Ia Y. Ugurlu ad S. Dura Cople Solutos for Nt-Order Korteweg-de Vres Kdv Equato ad -desoal urgers equato Turks Joural of Scece ad Tecolog [3]C. Aller A. Hadou D. Rckelck ad N. Verdo A pror reducto etod for solvg te two-desoal urgers' equatos Appl. Mat. ad Coput [4] K.N.S. Kas Vswaada S.R. Koeru Fte eleet etod for oe-desoal ad twodesoal te depedet probles wt -sples Coput. Metods Appl. Mec. Egrg [] P.M. Preter Sples ad varatoal etods Wles New York 97. [6] N.M. Yagurlu "Nuercal Solutos of -Desoal Partal Dfferetal Equatos wt - sple Fte Eleet Metods"PD. Tess Đöü Uverst Malata Turke. [7] J.N. Redd A troducto to te fte eleet etod McGraw-Hll Iteratoal Edtos trd ed. New York 6. 39