The Solution of Advection Diffusion Equation by the Finite Elements Method

Transcription

1 Iraioal Joural of Basic & Applid Scics IJBAS-IJES Vol: o: 88 T Soluio of Advcio Diffusio Equaio by Fii Els Mod Hasa BULUT, Tolga AKTURK ad Yusuf UCAR Dpar of Maaics, Fira Uivrsiy, 9, Elazig-TURKEY Dpar of Maaics, Iou Uivrsiy, 448, Malaya-TURKEY bulu@fira.du.r, olgaakurkk@gail.co, yusuf.ucar@iou.du.r Absrac-- I is sudy, w av rid o fid urical soluios of Advcio diffusio quaio (ADE) by Galrki od (GM), Adoia dcoposiio od (ADM) ad Hooopy prurbaio od (HPM); ad, w av ford a abl a coais urical rsuls for is quaio by drawig grapic of a quaio usig Origi 8. Fially, w av ad a copariso bw GM, ADM ad HPM for ADE. Id Tr-- Liar Advcio diffusio quaio, Hooopy prurbaio od, Galrki od, Adoia dcoposiio od.. ITRODUCTIO ADE dscribs [,,,6] ay quaiis suc as ass, a, rgy, vlociy, voriciy, c. T soluios of is quaio odl so of poa suc as a rasfr i a draiig fil, war rasfr i soils, sprad of polluas i rivrs ad sras, coaia disprsio i sallow laks, flow i porous dia, disprsio of dissolvd sals i groudwar, ral polluio i rivr syss, c. T slow progrss as b ad owards aalyical soluios of ADE w iiial ad boudary codiios ar coplicad. Bsids ay of aalyical soluios av o uc asy us. So a gra dal of ffors av b giv o dvlopig ffici ad sabl urical ciqus. Various urical ciqus av b proposd o illuia pysical poa dscribd by ADE i ay disciplis. T difficulis arisig i urical soluios of ADE rsul fro doia advcio, wic is for rlaivly ig pclc ubr.. AALYSIS OF THE METHODS. Galrki Mod (GM) For coic of wig fucio qual o approiaio fucio, wigd-rsidual od is i br kow as Galrki od [,4,4,8,,,]. T algbraic quaios of Galrki approiaio ar (..) wr j A f ij i i i i A A ddy ad ij j j j. (..) f f A ddy W o a A ij is o syric.. Hooopy Prurbaio Mod (HPM) To illusra basic idas of is od, w cosidr followig oliar diffrial quaio,, A u f r r, (..) wi boudary codiios B u, u /, r, (..) wr A is a gral diffrial opraor, B is a boudary opraor, f r is a kow aalyical fucio ad is boudary of doai. Grally spakig, opraor A ca b dividd io wo pars, aly L ad, wr L is liar, wil is oliar. Eq.(..) ca b rwri as followig L u ( u) f r. (..) By ooopy ciqu, w cosruc a ooopy, :, V r p or R wic saisfis; H( v, p) ( p)[ L( v) L( u )] p[ A( v) f ( r)] (..4) H v, p L v L u pl u p v f r, wr, (..5) p is a bddig parar, u is a iiial approiaio of Eq.(..). Obviously, fro Eq.(..4) ad Eq.(..5), w will av (..6) H( v,) L( v) L( u ) Fbruary IJES

2 Iraioal Joural of Basic & Applid Scics IJBAS-IJES Vol: o: 89 ad (..7) H( v,) A( v) f ( r) cagig procss of p fro zro o uiy is jus a of V ( r, p ) fro u () r o ur (). L( v) L( u ) I opology, is is calld dforaio, ad ad A( v) f ( r) ar calld ooopy. Accordig o HPM, w ca firs us bddig parar p as a "sall parar", ad assu a soluio of Eq.(..4) ad Eq.(..5) ca b wri as a powr sris i p ; (..8) V V pv p V p V. Sig p rsuls i approia soluio of Eq.(..) ad Eq.(..); (..9) u liv V V V V p T covrgc of sris i Eq.(..9) as b provd by H i is papr [7]. Tis ciqu ca av full advaag of radiioal prurbaio ciqus. T sris i Eq.(..9) is covrg ra dpds o o-liar opraor Av () ( followig opiios ar suggsd by H [7]: () T scod drivaiv of v () wi rspc o v us b sall bcaus parar ay b rlaivly larg, i.., p. () T or of a o so a sris covrgs. (..) L ( / v) us b sallr. APPLICATIO OF METHODS TO ADE. Applicaio of GM I is scio, u u u, < <, >. W cosidr ADE by bgiig codiio of 5 u,, s v (..) [] (v: issio spd, s: issio spac). A ac soluio of is probl is U, p 5. s s Aloug r ar approia soluios of is probl, cssary boudary codiios wr ak fro ac soluio of probl. I is scio, approia rsuls of Eq.(..) wr obaid by GM usig Quadraic B-spli fucios. T approac soluio a corrspods o ac soluio of probl, (..) U i rs of B-spli fucios, U Q j j j will b obsrvd. Hr, j ar i dpd ukow parars. To Quadraic B-spli fucios, if ab, irval is a < < < < pars b by sparaig lgs i, irval. W rasiio followig B-splis (..4) (..5) (..6) Q, is usd, Q Q ar obaid. O zro, i rs of quadric bas fucios, U, is wri as (..7), l, all or splis ar, U Q j j j approac If Eq.(..4) Eq.(..6) Quadraic B-spli fucios ad Eq.(..7) fucio ar usd, o poi of, Fbruary IJES

3 Iraioal Joural of Basic & Applid Scics IJBAS-IJES Vol: o: 9 poial valus of priary drivaios o o j parars ar wri as; U ad accordig B ' ij i j Q Q d U U ' wr,,,, ow, o for oial igral for of ADE, if i is uliplid wi oial fucio ad is igral is ak fro spac, i is obaid as, U U U d (..8) T oial igral for o fii l obaid as U U U d (..9),, i is Eq.(.. 9) oial igral for is arragd as followig, (..) U d U d U d W parial igraio is applid o U d r, U U U d U (..) is obaid. Accordig o GM, i Eq.(..) quaio, U approia soluio is ak Q i is slcd ad if isad of U, i rs of i,, ' ' ' ' j QiQ j QiQ j j Q Q i j j Qi Q j j j (..) is obaid. Suprscrips pri ad do do drivaiv wi rspc o spac ad i, rspcivly. Fro is, if A ij i j Q Q d C ' ' ij i j D Q Q d ' ij QQ i j Eq. (..) quaio sys is wri i ari for as, A B C D (..) Fro is, aris of A B C D ar obaid. T gral lis of A, B, C, D a ar obaid by uifyig of A, B, C ad aris ar as follows, A:,6,66,6, B :,,,, 6 C :,, 6,, D :,,,, D l Fbruary IJES

4 By usig uifid aris i (..), A B C D (..4) is obaid. I Eq.(..4) quaio, isad of, if forward fii diffrc approac blow is wri ad isad of, if Crak-icolso fii diffrc approac is wri, Iraioal Joural of Basic & Applid Scics IJBAS-IJES Vol: o: 9 valus, j U, U,, j,, j parars U, U, U U U,,, ar obaid. I is sys, is. As a coplry codiio, if quaio ukow A B C D A B C D (..5) quaio sys is foud. To calcula parars, firsly iiial vcor sould b calculad. Vcor will b calculad by usig U, iiial ad boudary codiios giv wi probls. ' U j j j drivd boudary codiio is usd, ' ' U U is obaid. If parar is abolisd, a sys cosisig of ukow ubrd is obaid. By is daa a is obaid, wi lp of forra ad aaica progras, for ADE, approia soluios, aalyic soluios ad absolu rror daa abls obaid wi Quadraic B- Spli fucios by usig GM, ar giv o abls blow. To calcula iiial parars, follows, Hr, usig U, Q j j j U approac is wri as j is parars a will b drid. So, by Fbruary IJES

5 X Iraioal Joural of Basic & Applid Scics IJBAS-IJES Vol: o: 9.5, IF IT IS ACCEPTED THAT T ABLE.. THE APPROXIMATE AD AALYTIC SOLUTIO THAT IS OBTAIED, = =5 = =. =. =.. Applicaio of HPM Aalyic Soluio Appr. Soluio Absolu Error Appr. Soluio Absolu Error Appr. Soluio Absolu Error W w cosidr iiial codiio as (..) (..) u u u,, 5 u,, s v for is quaio a ooopy as follows ca b ford; Fig... Two disioal visio of approia soluio of ADE by u, aalyic soluio w.5 (..) p Y U p Y Y Y. Hr, Y '' Y ' Y,, Y Y Y W a gral pasio of Eq.(..) is ford, ad p,. (..4) Y U pu py py quaio is obaid. T soluio of Eq.(..) ca b adld as, (a) Approia soluio (b) Aalyic soluio Fig... Tr disioal visio of approia soluio of ADE by u, aalyic soluio w Y Y py p Y p Y p Y P Y, (..5) Fbruary IJES

6 Iraioal Joural of Basic & Applid Scics IJBAS-IJES Vol: o: 9 ' '' '' '' 4 '' '' 4 Y Y py p Y p Y p Y P Y, (..6) 4 Y Y py p Y p Y p Y 4 P Y, (..7) ' ' ' ' ' 4 ' ' 4 Y Y py p Y p Y p Y P Y, (..8) By wriig Eq.(..5) Eq.(..8) i Eq.(..4), Y U pu py py 4 ' '' '' 4 '' Y py p Y p Y p Y 4 U pu p Y py p Y p Y p Y4 p Y py p Y p Y p Y ' ' ' ' 4 ' 4 4 ' '' '' 4 '' 4 4 Y py p Y p Y p Y U pu py py p Y p Y p Y py py p Y p Y p Y ' ' ' ' 4 ' 4 ar obaid. If is quaio is r-ford accordig o rs i sa ordr of p, p Y U (..9) (..) : p : Y Y Y (..) (..) 4 4 p : Y Y Y (..) p : Y U Y Y p : Y Y Y : p Y U Y Y Y U Y Y Y U Y Y d Y 5 : p Y Y Y Y Y Y Y Y Y d (..5) 5 Y Y Y (..6) : p Y Y Y Y Y Y Y Y Y d 5 Y (..7) 4 : 4 4 p Y Y Y Y Y Y 4 Y Y Y d 5 Y firs four rs of Eq.(..5) sris ar obaid. Wil Eq.(..4) Eq.(..8) ar p, approia soluio of Eq.(..) w wri i is plac i Eq. (..5) is obaid. Wil soluio of Eq.(..9) Eq. (..) ar as follows, p : Y U Y U Y (..4) Fbruary IJES

7 Iraioal Joural of Basic & Applid Scics IJBAS-IJES Vol: o: U, liy Y Y Y Y Y p is obaid. So, closd for of aalyic soluio of Eq.(..) is, u, p 5. s s Accordig o obaid soluios, D-D grapics of giv ADE by HPM w.5 ar giv blow. (a)approia soluio (b) Aalyic soluio Fig..6. (HPM) r disioal visio of approia soluio ad aalyic soluio of ADE by,. Applicaio of ADM u aalyic soluio w.5 Cosidr ADE giv wi Eq.(..) ad Eq.(..) quaios. Tis quaio ca b wri as follows i for of a opraor, (a)approiasoluio (b) Aalyic soluio (..) L Lu Lu Fig..5. (HPM) Two disioal visio of approia soluio ad aalyic soluio of ADE by, u aalyic soluio Fbruary IJES

8 Hr, L, L igral opraor ad If L (..) ad. Iraioal Joural of Basic & Applid Scics IJBAS-IJES Vol: o: 95 L L d. is applid o bo sids of Eq.(..), is obaid. So, L L u L L u L u. Hr, L is a u(, ) u(,) L Lu Lu (..) is obaid. For Eq.(..), a rcurrc cocio ca b wri as follows, 5 u, u, s v,, k, uk, L uk, Luk, d (..4) Fro Eq.(..4) rcurrc cocio a is obaid,,, u L L u L u L u L u d 5,, u L L u L u L u L u d ,, u L L u L u L u L u d ,, u L L u L u L u L u d T firs four rs of dcoposiio sris ar obaid. u, u, u, u, u ar wri i W obaid rs 4 Eq.(..6), approia soluio of Eq.(..) ADE is,,, u u u u u u u Fbruary IJES

9 ca b obaid as. So closd for aalyical soluio of Eq.(..) Iraioal Joural of Basic & Applid Scics IJBAS-IJES Vol: o: 96 u, p 5. s s Accordig o obaid soluios, D-D grapics of giv ADE by ADM w.5 ar giv blow. (a) Approia soluio (b) Aalyic soluio Fig... (HPM) Tr disioal visio of approia soluio ad aalyic soluio of ADE by, u aalyic soluio w.5 (a) Approia soluio (b) Aalyic Soluio Fig..9. (ADM) Two disioal visio of approia soluio ad aalyic soluio of ADE by, u aalyic soluio w.5 4. COCLUSIO I is sudy, GM, HPM ad ADM av b applid succssfully o ADE. I is s a w approia soluio of ADE is ford by GM for.5 ad., i is s a rror is oo slig. Howvr, if approia soluio is ad for. or., approia soluio is vry clos o aalyic soluio. Bsids, for.5, w Tabl.. is aid, i is s a approia soluio obaid by GM for grad valus of ad sall valus of is uc closr o aalyic soluio. T wo disioal ad r disioal grapics of soluio fucios obaid by usig Maaica progra for ADM ad HPM, ad Forra, Maaica ad Origi 8 progras for GM av b draw. I is sudy i is s a soluios obaid by ac r ods ak vry clos rsuls o is aalyic soluio. I is obsrvd a soluio grapics ar alos sa w Maaica progra is usd for urical ad aalyic soluios obaid by applyig HPM ad ADM o ADE. REFERECES [] Adoia G., 986. oliar Socasic Opraor Equaios, Acadic Prss, Sa Digo. [] A. Doğa.,, urical soluio of RLW quaio usig liar fii ls wii Galrki s od, Applid Maaics Modll, 6, [] A.H.Hasaov.,. Varyasyol Probllr v Solu Elalar Yöi, Liraür Yayıcılık [4] Ala J.Davis., T Fii El Mod A Firs Approac,Clardo Prss Oford [5] Crruaul, Y., 988. Covrgc of Adoia s od, Kybrs 8,(), - 8. Fbruary IJES

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