Application Of Alternating Group Explicit Method For Parabolic Equations

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1 WSEAS RANSACIONS o INFORMAION SCIENCE ad APPLICAIONS Qghua Feg Applcato Of Alteatg oup Explct Method Fo Paabolc Equatos Qghua Feg School of Scece Shadog uvesty of techology Zhagzhou Road # Zbo Shadog 09 Cha fqhua@sacom Abstact: - Based o the cocept of decomposto two alteatg goup explct methods ae costucted fo D covecto-dffuso equato wth vaable coeffcet ad D dffuso equatos espectvely Both the two methods have the popety of ucodtoal stabllty ad tsc paallelsm Numecal esults show the two methods ae of hgh accuacy Key-Wods: - alteatg goup method; paallel computg; explct scheme; paabolc equato; fte dffeece Itoducto Paabolc equatos ae wdely used descbg may physcal pheomea such as flud flowg ve ad atmosphee polluto ad so o Reseaches o fte dffeece methods fo them ae gettg moe ad moe popula May fte dffeece methods have bee peseted so fa [-] whch ae soted by explct ad mplct methods geeal As we all kow explct methods ae easy fo computg but ae commoly shot stablty ad accuacy Most of mplct methods ae of good stablty whle ae ot sutable fo paallel computg hus the task of pesetg fte dffeece methods wth good stablty ad popety of paallelsm s of mpotat theoetc ad pactcal meag D J Evas peseted a AE method [] ogally he AE method s used computg by applyg the specal combato of seveal asymmety schemes to a goup of gd pots he the umecal solutos at the goup of pots ca be obtaed depedetly ad the computato the whole doma ca be dvded to may sub-domas Futhemoe by alteatg use of asymmety schemes at adheet gd pots ad dffeet tme levels the AE method ca lead to the popety of ucodtoal stablty he AE method s soo appled to covecto-dffuso equatos [6] he AE method s wdely caed fo t s smple fo computg ucodtoally stable ad sutable fo paallelsm Ude the elghtemet of the AE method Baol Zhag ad S Zhu gave alteatg block explct-mplct methods [-9] whle Rohallah avakol deved a class of doma-splt method fo dffuso equatos [0 ] Seveal AE methods ae gve fo two-pot lea ad o-lea bouday value poblems [-] We otce most of the AE methods ae amed at costat coeffcet equato ad D poblems Reseaches o vaable coeffcet equatos ad D poblems have bee scacely peseted Results about the exstece ad uqueess of theoetc soluto fo paabolc equatos ca be foud [-] We ogaze the pape as follows: Fst we peset a class of alteatg goup explct method fo D covecto-dffuso equatos wth vaat coeffcet secto ad gve stablty aalyss fo t secto he we apply the costucto of the method to D dffuso equatos secto also the stablty aalyss s fshed I ode to vefy the effectveess of the two methods we peset umecal esults compag wth othe kow AE method [9] ad Cak-Ncolso scheme secto Some coclusos ae gve at the ed of the pape he Paallel Alteatg oup Explct (AE) Method I ths secto we cosde the followg covecto -dffuso equato u u u + a( x) = b( x) 0 x 0 t a( x) a 0 > 0 b( x) > 0 t x x () wth tal ad bouday value u (0 t ) = f ( x ) u ( t ) = f ( x ) () u ( x 0 ) = g ( t ) Let h ad τ be the gd step sze the x ad t dectos espectvely h = m x = h( = 0 m) t = τ( = 0 τ ) he gd pot ( x t ) s deoted by ( ) ad the umecal soluto s u whle the exact soluto s u( x t ) Let = τ h I ode to get the soluto of ( + )-th tme level whle the soluto of -th tme level kow we peset eght basc schemes usg the secod kd of saul yev asymmety schemes whch wll be used the costucto of the alteatg goup method I smple let ah = p b = q + + ( p q ) u + ( + q ) u + = q u + ( 6 q ) u + (p + q ) u () + ISSN: Issue Volume 6 July 009

2 WSEAS RANSACIONS o INFORMAION SCIENCE ad APPLICAIONS Qghua Feg (p q ) u + (+ 6 q ) u + ( p q ) u = ( q ) u + ( p + q ) u () ( p q ) u + (+ 6 q ) u + ( p q ) u = ( p + q ) u + ( q ) u () + (+ q ) u + ( p q ) u + + = ( p + q ) u + ( 6 q ) u + qu (6) + q u + (+ 6 q ) u + ( p q ) u = ( p + q ) u + ( q ) u () + (+ q ) u + ( p q ) u + + = ( p + q ) u + ( 6 q ) u + ( p + q ) u (8) + (+ q ) u + ( p q ) u = ( p + q ) u + ( 6 q ) u + ( p + q ) u (9) + (p q ) u + (+ 6 q ) u q u = ( q ) u + (p + q ) u (0) Based o ()-(0) we peset fou basc computg goups as follows: goup: fou gd pots ae volved ad ()-(6) ae used at each gd pot espectvely goup: fou gd pots ae volved ad ()-(0) ae used at each gd pot espectvely L goup: fou gd pots ae volved ()-(8) ae used at each gd pot espectvely ad R goup: fou gd pots ae volved ad (9)-(0) ae used at each gd pot espectvely he pupose of the pape s to get the soluto of the (+)-th ad the (+)-th tme level wth the soluto of the -th tme level kow Let m- = p hee p s a tege the the alteatg goup method wll be peseted as followg: Fst at the (+)-th tme level we wll have p goups () () () (6) ae used each goup Secod at the (+)-th tme level we wll have (p+) pot goups () ad (8) ae used to + + solve u u (9) ad (0) ae used to solve + + u m- u m- whle the est (p-) e gd pots ae dvded to (p-) goups ad () (8) (9) (0) ae used each goup hus the alteatg goup method s establshed by alteatg use of the schemes ()-(0) the two tme levels We otce the computato each goup ca be fshed depedetly Let U = ( u u um ) the we ca deote the alteatg goup explct method as below + ( I + ) U = ( I ) U + F () + + ( + ) = ( ) + I U I U F Hee F = (( p + q ) u 000( p + q ) u ) 0 m m m = (( p + q ) u 000( p + q ) u ) + + F 0 m m m = p ( m ) ( m) = p = = ( p+ ) ( m ) ( m) p q( ) + p( ) + q( ) + = p( ) q( ) 6q + + ( ) = p( ) + q( ) p ( ) + q ( ) + = 0 0 6q( ) + p( ) + q( ) + = p( ) q( ) q + + ( ) + qm pm qm p = qm 6q m 6q q = p q q ( + ) p q( ) + p( ) + q( ) + = q( ) 6q + ( ) = p( ) + q( ) + 0 = = ISSN: Issue Volume 6 July 009

3 WSEAS RANSACIONS o INFORMAION SCIENCE ad APPLICAIONS Qghua Feg 0 p q 0 0 6q( ) + q( ) + = p( ) 6 q( ) 6 q + + ( ) + 6 ( ) + ( ) + = he alteatg use of the asymmety schemes ()-(0) ca lead to patly couteactg of tucato eo ad the ca cease the umecal accuacy O the othe had goupg computato ca be obvously obtaed hus computg the whole doma ca be dvded to may sub-domas So the method has the obvous popety of paallelsm Aalyss Of Stablty Lemma (Kellogg) [8] Assume θ>0 ad ( + ) s o egatve defte eal matx the ( θi I + ) exsts ad the fallowg equaltes hold ( θ I + ) θ () ( θ I )( θ I + ) heoem he alteatg goup method () s of absolute stablty Poof: Fom the costucto of the matces ( + ) ad ( + ) we ca see they ae o egatve defte eal matces he we have ( I )( I + ) I I I I ( I )( I + ) Let s a eve umbe + the we have U = U Hee s gowth matx = ( I + ) ( I )( I + ) ( I ) Let = ( I + ) ( I + ) = ( I )( I + ) ( I )( I + ) By Lemma we have ρ( ) = ρ( ) heefoe the alteatg goup method () s of absolute stablty Applcato Of AE Method Fo D Dffuso Equatos Cosdeg the tal bouday value poblem of D dffuso equatos: u u u = + t x y 0 x 0 y 0 t u( x y 0) = f ( x y) u(0 y t) = g( y t) u( y t) = g ( y t) u( x 0 t) = f( x t) u( x t) = f ( x t) () he doma : [0 ] [0 ] [0 ] wll be dvded to (m m N) meshes wth spatal step sze h = x = y = m x y decto ad the tme step sze τ = = t d pots ae deoted by (x yj t) o ( j ) x = h( = 0 m) y = jh( j = 0 m) t = τ ( = 0 τ ) j he umecal soluto of (0)-() s deoted by u whle the exact soluto u x y t ) = τ h j ( j We peset 6 basc asymmety schemesby use of the seced class of saul yev schemes as follows (Fgue -6): ( + ) u j u + j u j + = u j + u + ( ) u + u + u () j j j + + j u + ( + ) u u u j + j + j + j+ = u + u + ( ) u + u () + j j + j + j+ u + ( + ) u u u j + j + j + j+ = u + ( ) u + u + u (6) u+ j + ( + ) u+ j u+ j+ = u+ j + j + j + j + j+ + ( ) u+ j + u+ j + u+ j+ + u+ j () ISSN: Issue Volume 6 July 009

4 WSEAS RANSACIONS o INFORMAION SCIENCE ad APPLICAIONS Qghua Feg u + ( + ) u u u j j+ + j+ j+ = u + ( ) u + u + u (8) j+ j+ j + j+ u + ( + ) u u u j + j+ j+ + j+ + u + j+ = u+ j + u j+ + ( ) u + j+ (9) u u + ( + ) u u j + j+ + j+ + j+ u = u + u + ( ) u (0) + + j+ + j + j+ + j+ u u + ( + ) u u j + j+ + j+ + j+ = u + u + ( ) u + u () + j + j+ + j+ + j+ u + ( + ) u u u j+ j+ + j+ j+ = u + ( ) u + u + u () j+ j+ + j+ j+ = u + ( ) u + u + u () + j+ + j+ + j+ + j+ u + ( + ) u u = u j+ j+ + j+ j+ + u j+ + ( ) u j+ + u+ j+ + u j+ (6) u u + ( + ) u u j+ j+ + j+ + j+ = u + j+ + u j+ + ( ) u + j+ + u + j+ () u u + (+ ) u u j+ + j+ + j+ + j+ = u + ( ) u + u + u (8) + j+ + j+ + j+ + j+ u u + ( + ) u = u +! j+ + j+ + j+ + j+ + u + j+ + ( ) u + j+ + u + j+ + u + j+ (9) u + j+ u j+ + ( + ) u + j+ u + j+ + u + j+ = u j+ + ( ) u + j+ + u+ j+ () u u + (+ ) u u j+ + j+ + j+ + j+ u = ( ) u + u + u () + + j+ + j+ + j+ + j+ u u + (+ ) u u j+ + j+ + j+ + j+ ISSN: Issue Volume 6 July 009

5 WSEAS RANSACIONS o INFORMAION SCIENCE ad APPLICAIONS Qghua Feg Lx 6 Ly Fgue 8 the goupg at + tme level We peset seveal basc computg goups as below: j j j+ j+ j+ ( 6 pot goup ) Let u =(u u u u ) u = ( u u u u ) k = 0 j+ k j+ k + j+ k + j+ k + j+ k j ( j j+ j+ j+ ) F = F F F F F = ( u + u u j j j + j u u + u ) + j + j + j F = ( u 00 u ) j+ j+ + j+ F = ( u 00 u ) j+ j+ + j+ F = ( u + u u j+ j+ j+ + j+ u u + u ) + j+ + j+ + j+ he we debote the 6 pot goup as follows: Let m = s + s s a tege he we costuct the alteatg goup schemes at the two adheet tme levels as Fgue -8: Ry 6 Fgue the goupg at + tme level H Rx + j j j ( I + A ) u = ( I B ) u + F (0) Let 0 C C C = = = C = C = [ 0] 0 the t follows A A A A A A = A A A A A 6 6 B B B = B B B B B B 6 6 C C A = C O A = C C O C A = I + A C O B = I + A = A O C ISSN: Issue Volume 6 July 009

6 WSEAS RANSACIONS o INFORMAION SCIENCE ad APPLICAIONS Qghua Feg B C O = O C B = B I ( Lx goup) Accodg to the pcple of ow fst ad colum secod (6) () (0) () () () (8) (9) ae used to get the soluto of the eght gd pots as show fgue -8 Let j = j j+ j+ j+ v ( v v v v ) v = ( v v ) k = j+ k j+ k j+ k j = j j+ j+ j+ w ( w w w w ) w = ( u + u u + u ) + j j 0 j j j w = ( u u ) + j+ 0 j+ j+ w = ( u u ) + j+ 0 j+ j+ w = ( u + u u + u ) + j+ 0 j+ j+ j+ j+ the we deote the Lx goup as follows: + j j j ( I + A ) v = ( I B ) v + w () A A A = B O A A B = O B I + C C B = C C C C B = C I C + C C A = C I + C A = A I + C C A = C C O O A = C O ( Ly goup) Accodg to the pcple of ow fst ad colum secod () (6) () () () (8) () (9) ae used to get the soluto of the eght gd pots as show fgue -8 ( Rx goup) Accodg to the pcple of ow fst ad colum secod () () (8) (9) () () (6) () ae used to get the soluto of the eght gd pots as show fgue -8 ( Ry goup) Accodg to the pcple of colum fst ad ow secod () (8) () (9) (6) (0) () () ae used to get the soluto of the eght gd pots as show fgue -8 ( goup) () () (8) (9) ae used to get the soluto of the fou gd pots ()()( )( ) ( H goup) () () (8) (9) ae used to get the soluto of the fou gd pots ( m m )( m m ) ( m m )( m m ) he costucto of Ly goup Ry goup Rx goup s smla to Lx goup Accodg to the goups show Fgue -8 f we let ( 9 m6 m) u = u u u u u j = ( j j 9 j m6 j m j ) = 9 6 u u u u u u j m m j = m j m j m j+ m j+ m j+ m j+ m j+ m j+ u ( u u u u u u u u ) m m m 9 m m6 m m m u = ( u u u u u ) m = m m + m + m + m + m + m + m u ( u u u u u u u u ) = 9 m6 m m = m m m m m m m m u ( u u u u ) he we deote the alteatg goup explct method as follows: ( I + H) u+ = ( I H) u () ( I + H ) u + = ( I H) u + Let ( m ) = a ( m ) + = b ( m ) + 8 = c the H H H = H H a a A A H = A A c c A A H = A A b b A A A = B A = A A A 8 8 = A A = A A = B A = B B = A P M M P M H = M P M M P Q a a ISSN: Issue Volume 6 July 009

7 WSEAS RANSACIONS o INFORMAION SCIENCE ad APPLICAIONS Qghua Feg P E E P E P = E P E E P F F P c c Q E E Q E Q = E Q E E Q F F Q b b P P O O P P O O P = O O P P O O P P 6 6 P O P = I + C O P = O P O I + C C O P = C O O C P = O C P = A P = A E E E = E E 6 6 F F F = F F 8 8 O C E = O O F = [ O C ] 8 8 O C E = O C O O F = 8 O O 8 8 Q = P Q = A M M = M M O M M = O O 6 6 C O M = O C c c M O C = O O 8 8 = c b = [ O ] 8 6 = [ O ] = C C = C C 8 Fom the costucto of the matces H ad H we ca see they ae o egatve defte eal matces he we have ( I H ) + ( I H )( I + H ) ( I H ) + ( I H )( I + H ) Let s a eve umbe the we have u = u Hee = ( I + H ) ( I H)( I + H) ( I H ) s the gowth matx Let H = ( I + H ) ( I + H ) = ( I H ) he by Lemma we have ρ( ) = ρ( H) H he we have the followg theoem: heoem he alteatg goup explct method deoted by () s ucodtoally stable umecal Examples Example : Cosde the tal-bouday poblem of covecto -dffuso equato as below u u u + a = b 0 x 0 t t x x () u (0 t ) = 0 u ( t ) = u ( x 0 ) = 0 he exact soluto [6] s a x / b e u ( x t ) = + a b e ( ) π π + e s ( π x ) e ( π ) + ( a b ) = a ( x ) / b [( ) b ( a / b )] t Let A E ad P E deote absolute eo ad elevat eo espectvely We compae the umecal esults of ISSN: Issue Volume 6 July 009

8 WSEAS RANSACIONS o INFORMAION SCIENCE ad APPLICAIONS Qghua Feg () wth the Cak-Ncholso scheme able ad able able : Patly esults of compasos m = τ = 0 t = 000 τ a = 0 b = 0 able : Compasos wth [9] wth = 0 m = 9 h = 9 τ = 6 t = 90 τ y = 6h A E A E (C-N) P E P E (C-N) A E P E x 6h h 8h able : Patly esults of compasos m = 9 τ = 0 t = 000 τ a = b = A E A E (C-N) P E P E (C-N) he esults able - show the method () s of ealy the same accuacy as the mplct C-N scheme Example : Cosdeg the followg poblem: u u u = + t x y 0 x 0 y 0 t u( x y 0) = s( π x) s( π y) u(0 y t) = 0 u( y t) = 0 u( x 0 t) = 0 u( x t) = 0 he exact soluto [9] of the poblem s π t u( x y t) = e s( πx)s( πy) We compae the alteatg goup method () wth the method [9] ad the exact soluto I able - we peset pat of the compasos wth = 0 m = 9 h = 9 τ = 6 t = 90 τ y = 6h A E P E [ 9 ] [ 9 ] Exact 86 0 A E P E A E P E able : Compasos wth [9] wth = 0 m = 9 h = 9 τ = 6 t = 90 τ y = 6h x 0h h h [ 9 ] [ 9 ] Exact he esults able - show the peset method () s of hghe accuacy tha the method [9] I Fgue 9 we peset the compaso betwee the umecal esults of the peset method ad the exact soluto wth = 0 m = h = τ = 6 t = 90 τ y = 6h ISSN: Issue Volume 6 July 009

9 WSEAS RANSACIONS o INFORMAION SCIENCE ad APPLICAIONS Qghua Feg Fgue 9 Compaso betwee umecal esults ad exact esults wth = 0 m = h = τ = 6 t = 90 τ y = 6h Fom Fgue 9 we ca see the umecal esults s of hgh accuacy 6 Coclusos I ths pape based o the cocept of doma decomposto we peset a class of paallel alteatg goup explct method fo covecto-dffuso equatos whch s vefed to be ucodtoally stable Fom the esults of able ad able we ca see that the umecal soluto fo the method s of ealy the same accuacy as the mplct Cak-Ncholso scheme Futhemoe we costcut aothe alteatg goup explct method fo D dffuso equatos he method s also ucodtoally stable he esults able - show the method s supeo to the method [9] Both of the two methods ae sutable fo paallel computg ad the computg the whole doma ca be dvded to may depedet sub domas So the two alteatg goup methods ae effectve methods solvg lage system of equatos Refeeces: [] Damelys Zabala Aua L Lopez De Ramos Effect of the Fte Dffeece Soluto Scheme a Fee Bouday Covectve Mass asfe Model WSEAS asactos o Mathematcs Vol 6 No 6 00 pp 69-0 [] Ramods Vlums Ads Buks Cosevatve Aveagg ad Fte Dffeece Methods fo aset Heat Coducto D Fuse WSEAS asactos o Heat ad Mass asfe Vol No 008 [] Mastoaks N E A Exteded Cak-Ncholso Method ad ts Applcatos the Soluto of Patal Dffeetal Equatos: -D ad -D Coducto Equatos WSEAS asactos o Mathematcs Vol 6 No 00 pp - [] M Styes ad L obska A fte dffeece aalyss of a steamle dffuso method o a Shshk mesh Numecal Algothms pp -60 [] D J Evas A R B Abdullah oup Explct Method fo Paabolc Equatos [J] Ite J Comput Math (98) -0 [6] D J Evas A R B Abdullah A New Explct Method fo Dffuso-Covecto Equato Comput Math Appl (98) - [] W Yua L J She Y L Zhou Ucodtoal stablty of paallel alteatg dffeece schemes fo sem lea paabolc systems Appl Math Comput (00) 6-8 [8] Z B l L J fu Y xue oup mplct method fo the olea heat coducto equatos ad umecal expemets Chese Joual of Comp Phys (9) (99) 8- [9] Zhag Baol Su Xum Alteatg Block Explct-Implct Method fo the wo- Dmes -oal Dffuso Equato Ite J Compute Math 99 8: - [0] R avakol P Davam New stable goup explct fte dffeece method fo soluto of dffuso equato Appl Math Comput 8 (006) 9-86 [] Rohallah avakol Pavz Davam D paallel ad stable goup explct fte dffeece method fo soluto of dffuso equato Appl Math Comput 8(006)8-9 [] R K Mohaty D J Evas Hghly accuate two paamete CAE paallel algothms fo olea sgula two pot bouday poblems Ite J of Comp Math 8 (00) - [] R K Mohaty N Khosla A thd-ode accuate vable-mesh AE teatve method fo the umecal soluto of two-pot o-lea sgula bouday poblems Ite J of Comp Math 8 (00) 6- [] S V Meleshko Methods fo Costuctg Exact Solutos of Patal Dffeetal Equatos Spge 00 [] C Sweezy adet Nom Iequaltes fo Weak Solutos to Paabolc Equatos o ISSN: Issue Volume 6 July 009

10 WSEAS RANSACIONS o INFORMAION SCIENCE ad APPLICAIONS Qghua Feg Bouded Domas wth ad wthout Weghts WSEAS asactos o SystemVol No 00 pp 96-0 [6] H Cheg he tal value ad bouday value poblem fo D Nave-Stokes Math Sca (998) - []S Ng Istataeous shkg of suppots fo o-lea eacto-covecto equato J P D E (999)9-9 [8] B Kellogg A alteatg Decto Method fo Opeato Equatos J Soc Idust Appl Math (SIAM) (96) 88-8 ISSN: Issue Volume 6 July 009