Linearization Technique and its Application. to Numerical Solution of Bidimensional Nonlinear. Convection Diffusion Equation

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1 Appled Mahemacal Sceces Vol o HIKARI Ld hp://d.do.org/0.988/ams Learzao Techqe ad s Applcao o Nmercal Solo of Bdmesoal Nolear Coveco Dffso Eqao Marco Dosee de Campos ederal Uvers of of Mao Grosso Ise of Eac ad Earh Sceces Barra do Garças MT Brazl Esaer Claro Romão Deparme of Basc ad Evromeal Sceces Uvers of São Palo Lorea SP Brazl Lz elpe Medes de Mora Sae Uvers of Campas Egeerg acl Campas SP Brazl Coprgh 04 Marco Dosee de Campos Esaer Claro Romão ad Lz elpe Medes de Mora. Ths s a ope access arcle dsrbed der he Creave Commos Arbo Lcese whch perms resrced se dsrbo ad reprodco a medm provded he orgal work s properl ced. Absrac I hs work a mercal scheme sg Crak-Ncolso scheme ad he hgh-order e Dfferece Mehod were sed respecvel for he emporal ad spaal dscrezao of he olear wo-dmesoal coveco-dffso eqao. Nmercal applcaos are mplemeed ad valaed makg se of he L error orm ad he mercal resls were compared wh he eac solo obaed va lerare revew. Kewords: Coveco-dffso eqao learzao echqe hgh-order fe dfferece mehod

2 744 Marco Dosee de Campos e al. Irodco The wo-dmesoal parabolc dffereal eqaos appeared ma scefc felds of egeerg ad sceces sch as ero dffso hea rasfer ad fld flow problems []. A aracve problem s coveco-dffso eqao for whch varos mercal mehods have bee sggesed b a mber of researchers o solve hem. Applcao of hs eqao ma be see compaoal fld damcs hdrodamc rblece shockwave heor wave processes hermoelasc medm raspor ad dsperso of pollas rvers ad sedme raspor for modelg coveco-dffso of qaes sch as mass eerg vorc hea amog ohers []. I rece ears he solo of wo-dmesoal olear coveco-dffso eqao has araced a lo of aeo ad ma ahors have sed varos mercal echqes for he solo of hs eqao. Nmercal ormlao I hs work we propose a solo b he hgh-order fe dfferece mehod for he olear wo-dmesoal coveco-dffso eqao whch s gve b () where () s he veloc feld he -drecos ad s he kemac vscos. Here de o effcec ad smple mplemeao o carr o he me dscrezao of he eqao () wll se he Crak-Ncolso mehod [4] as follows: () The ma prpose of hs work s o se he Newo s mehod for he learzao of he erms ad he followg maer (a) (b) Noe ha hs echqe does o reqre a erave learzao a each me sep makg qcker he compao of.

3 Learzao echqe ad s applcao o mercal solo 745 Sbsg he Eqs. (a-b) he Eq. () we have: 5 0 where. I he applcaos ad below he al vale of he eac solo s ake as he al codo ad bodar codos are also specfed b he above eqao ad chage wh me. However for he applcao he codos wll be specfed he same case. Cosderg odes wh or dsace from he bodar sg he Ceral Dfferece Mehod wh O( ) we have: where. Now cosderg he eral odes sg he Ceral Dfferece Mehod wh O( 4 ) we have:

4 746 Marco Dosee de Campos e al where Nmercal Applcaos I he followg applcaos he mercal solo o olear wo-dmesoal coveco-dffso eqao o a compaoal doma 0 L 0 L wh > 0. I he applcaos ad he mercal solo s compared o he eac solo fod he lerare sg he L orm defed as: / / Nos e Nos ε where Nos s he oal mber of odes he mesh ad a m e ) ( ) ( where (m) s he resl from he mercal solo ad (a) s he resl from he mercal solo ad he resl from he aalcal solo. Now sce he applcao has o aalcal solo sed a coarse mesh ad a more refed o prese he mercal oscllaos he solo each case s possble o oe ha he oscllaos decreased as he refeme. Applcao : I hs applcao he eac solo of he Eq. () s specfed b he followg eqao ν ep ) ( [5] ad compared wh he mercal solo. The vales of e v were chose so ha he veloc profle assme vales bewee 0 ad. Ths he b g. () o he rgh for eample were ake for ν 0. ad

5 Learzao echqe ad s applcao o mercal solo 747 gre : Mesh wh 0./00 cosderg o he lef ν 0. wh ε 7.78E-04 ad o he rgh ν 0.0 wh ε 6.E-0 boh 0.. gre : Mesh wh 0./00 ad 0./00 cosderg o he lef ν 0. wh ε 4.56E-04 ad o he rgh ν 0.0 wh ε.6e-0 boh 0.. I gres ad s oed ha he mercal resls for some refed meshes was obaed a precso of a leas wo decmal places whch s cosdered sable for egeerg. Applcao : or he secod eample we cosder he followg aalcal solo ( ) ah [6]. The al vale of he eac solo s ake as he al codo ad bodar codos are also specfed b he above eqao ad chage wh me. I ma be oed he aalss of gs. ()-(4) for large vales for ν he forces domae dffsve profle ad he solo evolves oward a fla srface. As he ν decreases he covecve forces ake over ad he solo develops o a vscos shock ha ravels o he rgh.

6 748 Marco Dosee de Campos e al. gre : Mesh wh 0./00 cosderg o he lef ν 0.05 wh ε.74e-0 ad o he rgh ν 0.0 wh ε 07E-0 boh 0.. gre 4: Mesh wh 0./00 ad 0./00 cosderg o he lef ν 0. wh ε.7e-0 ad o he rgh ν 0.05 wh ε.7e-0 boh 0.. Applcao : Here was cosdered he compaoal doma 0 ad > 0 ad he followg al codo (0) 0 as well he bodar codos (0) (0) (0.) 0 ad (.0 ) 0. A qalave aalss of he resls of he wo me sas for a mesh adoped ad ν 0. (g. 5) shows ha he have o mercal oscllaos ad frher he mercal po are he erval 0 o. However for 0.05 obaed a mamm vale for.4857 oall o of erval 0 o. I order o redce mercal oscllaos prese gre 6 (lef) was appled sep a refeme ( 0.00) sep me.

7 Learzao echqe ad s applcao o mercal solo 749 gre 5: profle for ν 0. ad o he lef ad for o he rgh wh 0.0 ad 0.05 he applcao. gre 6: v profle for ν 0.0 ad wh 0.0 ad 0.05 he lef ad 0.00 he rgh he applcao. A qalave aalss of he resls of he wo me sas for a mesh adoped ad ν 0. (g. 5) shows ha he have o mercal oscllaos ad frher he mercal po are he erval 0 o. However for 0.05 obaed a mamm vale for.4857 oall o of erval 0 o. I order o redce mercal oscllaos prese gre 6 (lef) was appled sep a refeme ( 0.00) sep me. 4 Coclsos I hs paper we propose a hgh-order e Dfferece Mehod for olear wo-dmesoal coveco-dffso eqao. Nmercal eamples show ha he mehod ca be sed o smlae he mercal solo of he eqao. B

8 750 Marco Dosee de Campos e al. observg he dealed comparso of mercal ad aalcal resls s covced ha he proposed scheme s ver smple sable ad accrae for he solos of he eqao. Especall regardg he learzao of olear erms of he eqao hs work was preseed a ver effce echqe wh large me ervals whe compared wh hose sed b oher ahors. Moreover hs echqe does o reqre a pe of erave process wh each me sep whch represes a maor savg o he compaoal me. Ackowledgemes The prese work was sppored b he Mao Grosso Research odao apema Brazl (Proc. 9470/00) ad he Naoal Cocl of Scefc Developme ad Techolog CNPq Brazl (Proc. 5008/0-5). Refereces [] B. Gsafsso Hgh-Order Dfferece Mehods for Tme Depede PDE Sprger 008. [] H. Loma T.H. Pllam ad D.W. Zgg dameals of Compaoal ld Damcs Sprger 999. [] D.W. Peacema dameals of Nmercal Reservor Smlaos Elsever Scece Amserdam 977. [4] T.J. Chg Compaoal ld Damcs. Cambrdge Uvers Press 00. [5] J. L Y. Che ad G. L Hgh-Order Compac ADI Mehods for Parabolc Eqaos Comp. Mah. Appl. 5 (006) [6] J. Zhag ad G. Ya Lace Bolzma mehod for oe ad wo-dmesoal Brgers eqao. Phsca A 87 (008) Receved: December 7 0