Explicit Group Methods in the Solution of the 2-D Convection-Diffusion Equations

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1 Proceedigs of the World Cogress o Egieerig 00 Vol III WCE 00 Jue 0 - July 00 Lodo U.K. Explicit Group Methods i the Solutio of the -D Covectio-Diffusio Equatios a Kah Bee orhashidah Hj. M. Ali ad Choi-Hog Lai Abstract I this paper we preset the four poits Explicit Group (EG) ad schemes for solvig the two dimesioal covectio-diffusio equatio with iitial ad Dirichlet boudary coditios. he EG method is derived from the cetred differece approximatio whilst EDG is derived from the rotated differece operator expressed i coordiates rotated 5 0 with respect to the stadard mesh. hese ew formulatios are show to be ucoditioally stable ad the robustess of these ew formulatios over the existig poit Crak-icolso scheme demostrated through umerical experimets. Idex erms Explicit Group (EG) Explicit Decoupled Group (EDG) Covectio-Diffusio Crak-icolso Rotated Crak-icolso. I. IRODUCIO Cosider the two dimesioal covectio-diffusio equatio: U U U U U a x a y b x b y () t x y x y with iitial ad boudary coditios: uxy ( 0) f( xy ) u(0 y) t g( y) t u( X y) t g( y) t () ux (0) t h() xt uxyt ( ) h(). xt Here a x a y b x b y are positive costats o a rectagular grid with grid spacig Δx i x-directio ad Δy i y-directio with x i x 0 iδx y j y 0 jδy ad t Δt (for all i 0 x j 0 y 0 ) X x 0 xδx Y y 0 yδy. Equatio () ca be approximated at ay poit (x i y j t ) i various ways. Oe commoly used itegratio method is the Crak-icolso formula: Mauscript received March his research was supported by the Uiversiti Sais Malaysia Research Uiversity Grat (00/PMAHS/707). a Kah Bee is a doctoral cadidate at the School of Mathematical Scieces Uiversiti Sais Malaysia 00 Peag Malaysia (correspodig author avery7000@gmail.com ). orhashidah Hj. M. Ali is a Associate Professor at the School of Mathematical Scieces Uiversiti Sais Malaysia 00 Peag Malaysia. She is curretly spedig her sabbatical leave at the School of Computig ad Mathematical Scieces Uiversity of Greewich Lodo SE0 9LS UK. ( shidah@cs.usm.my). Choi-Hog Lai is a Professor of umerical Mathematics at the School of Computig ad Mathematical Scieces Uiversity of Greewich Lodo SE0 9LS UK. ( C.H.Lai@gre.ac.uk ). u u a u u u u u u Δt Δx Δx a u u u u u u Δy Δy i j i j x i j i j i j i j i j i j y i j i j i j i j i j i j b u u u u x Δx Δx i j i j i j i j b u u u u Δy Δy y i j i j i j i j Let the Courat umbers (Cx Cy) ad diffusio umbers (Sx Sy) be defied as Sx axδt / Δx Sy ayδt / Δy () Cx bxδt / Δx Cy byδt / Δy. hus () ca be simplified as Sx Cx Sx Cx ( Sx Sy) u u u Sy Cy Sy Cy u i j u i j Sx Cx Sx Cx ( Sx Sy) u u u Sy Cy Sy Cy u i j u i j i j i j i j i j i j i j with the computatioal molecule as i Fig.. Aother itegratio method derived from the Crak-icolso formula ca be obtaied by rotatig the x-y axis clockwise by 5. Usig aylor series expasio the rotated Crak-icolso formula for () ca be show to be of the followig form []: Sx Sy Sx Cx Cy Sx Cx Cy u u u Sy Cx Cy Sy Cx Cy u u Sx Sy Sx Cx Cy Sx Cx Cy ui j u u Sy Cx Cy Sy Cx Cy u u i j i j i j i j i j i j i j i j i j It is clearly see that the applicatio of either () or (6) at each time step will result i a large ad sparse liear system A u B u (7) A ad B are square osigular matrices while u ad u are specific colum matrices. he solutio of (7) ca be obtaied by direct or iterative methods. Sice the equatio is large ad sparse iterative method is more suitable to be used i solvig this type of problem either i poit or block formulatios. () (5) (6) ISB: ISS: (Prit); ISS: (Olie) WCE 00

2 Proceedigs of the World Cogress o Egieerig 00 Vol III WCE 00 Jue 0 - July 00 Lodo U.K. Sx Cx Sx Cx Sy Cy Sy Cy ( Sx Sy) u u u u u i j Sx Cx Sx Cx Sy Cy Sy Cy ( Sx Sy) ui j ui j ui j u u i j i j Sx Cx Sx Cx Sy Cy Sy Cy ( Sx Sy) u u u u u i j Sx Cx Sx Cx Sy Cy Sy Cy ( Sx Sy) ui j ui j u u u i j i j i j i j i j i j i j j j j j j i i i i i i j i j i j. Fig. : he Crak-icolso scheme with atural orderig he Explicit Group (EG) ad Explicit Decoupled Group (EDG) schemes ca be costructed based o (5) ad (6) respectively. he origial EG scheme was formulated by Yousif ad Evas [] i solvig the two dimesioal elliptic equatio by costructig ew groupig of the mesh poits ito smaller size groups of poits the gais i executio timigs of the four poit EG method over the -lie smoother rages from 5%-6%. Usig the idea of smaller size groupigs o rotated grids Abdullah [] developed the four poits EDG which was show to be more efficiet computatioally tha the EG method. Yousif ad Evas [5] later exteded the method to six ad ie poits groupigs ad showed that they ca be easily parallelised o a MIMD multiprocessor. Sectios II ad III describe the formulatio the EG ad EDG methods respectively for the two dimesioal covectio-diffusio equatio. he trucatio error ad cosistecy aalysis are preseted i Sectio IV followed by the stability aalysis i Sectio V. umerical experimets ad results are preseted i Sectio VI. he cocludig remark is give i Sectio VII. II. EXPLICI GROUP (EG) o formulate the EG scheme we apply () to ay group of four poits o the solutio domai at each time step. hus at ay particular time level () this will result i a (x) system of the form: Sx Cx Sy Cy Sx Sy 0 Sx Cx Sy Cy ui j rh Sx Sy 0 u i j rh Sy Cy Sx Cx u i j rh 0 Sx Sy ui j rh Sy Cy Sx Cx 0 Sx Sy Sx Cx Sy Cy u u rh Sx Cx Sy Cy u u rh rh Sx Cx Sy Cy u u rh Sx Cx Sy Cy u u i j i j i j j i i j i j i j i j i j i j i j i j () (9) (0) Equatio () ca be iverted to obtai the four-poit EG equatio: ui j q q q q rh u i j q5 q q q 6 rh u i j cost q7 q q q 5 rh ui j q q9 q q rh () Sx Cx Sx Cx Sy Cy Sy Cy cost ( Sx Sy) Sx Cx Sx Cx Sy Cy Sy Cy q ( Sx Sy) ( Sx Sy) ( Sx Sy) Sx Cx Sx Cx Sx Cx Sx Cx Sy Cy Sy Cy q ( Sx Sy) Sx Cx Sy Cy q ( Sx Sy) Sy Cy Sx Cx Sx Cx Sy Cy Sy Cy Sy Cy q ( Sx Sy) Sx Cx Sx Cx Sx Cx Sx Cx Sy Cy Sy Cy q5 ( Sx Sy) Sx Cx Sy Cy q6 ( Sx Sy) Sx Cx Sy Cy q7 ( Sx Sy) Sy Cy Sx Cx Sx Cx Sy Cy Sy Cy Sy Cy q ( Sx Sy) Sx Cx Sy Cy q9 ( Sx Sy) he solutios may be obtaied by imposig the Gauss-Seidel iterative scheme to the four-poit EG formula () at each time level. Iteratios are geerated i groups of four poits over the etire spatial domai util the covergece test is satisfied. he coverged solutios are the take as iitial guesses for the iteratios at the ext time level. III. EXPLICI DECOUPLED GROUP (EDG) Similar to the EG method we apply (6) to ay group of four poits i the solutio domai at each time step to obtai the followig (x) system of equatios: Sx Sy Sx Cx Cy 0 0 Sy Cx Cy Sx Sy ui j rh 0 0 u i j rh Sx Sy Sx Cx Cy u i j rh 0 0 ui j rh Sy Cx Cy Sx Sy 0 0 with () rh bui j dui j eui j i j rh bui j cui j dui j i j rh cui j dui j eui j i j rh bui j cui j eui j i j () ISB: ISS: (Prit); ISS: (Olie) WCE 00

3 Proceedigs of the World Cogress o Egieerig 00 Vol III WCE 00 Jue 0 - July 00 Lodo U.K. au bu cu du eu au bu cu du eu au bu cu du eu i j i j i j i j i j i j i j i j i j i j i j i j i j i j i j i j i j i j i j aui j bui j cui j dui j eui j () he system () leads to a decoupled system of x equatios i explicit form: Sx Sy Sx Cx Cy ui j rh Sy Cx Cy Sx Sy u i j rh (5) ad Sx Sy Sx Cx Cy ui j rh Sy Cx Cy Sx Sy u i j rh (6) Referrig to Fig. (a) it is observed that the iterative evaluatio of (5) at ay time level ivolves poits of type oly while the evaluatio of (6) ivolves poits of type oly (see Fig. (b)). hus the iteratios may be chose to ivolve oly oe type of poits. Suppose we choose to iterate o poits of type. Hece the EDG scheme correspods to geeratio of iteratios o these poits usig the group formula (5) util a covergece test is satisfied. After covergece is achieved the solutios at the poits of type are evaluated directly oce usig the Crak-icolso formula (5) before proceedig to the ext time level. i-j i-j- ij ij ij ij- ij ij Fig. (a) Computatioal Molecule of (5) i-j i-j ij ij- ij ij ij ij- Fig. (b) Computatioal Molecule of (6) ime level IV. RUCAIO ERROR AD COSISECY he local trucatio for the Crak-icolso scheme may be obtaied by usig the aylor series expasio about the poit (x i y j t / ): Δt u Δt u u C a x a y t t x t x i j 0.5 i j 0.5 u u ax u bx u bx b... y Δx t x t y x 6 x i j 0.5 i j 0.5 i j 0.5 i j 0.5 a y u by u Δy y 6 y i j 0.5 i j 0.5 Δx Δt ax u u b x t x t x i j 0.5 i j 0.5 Δy Δt ay u u b... y t y t y i j 0.5 i j 0.5 i.e. C O( t ) O( x ) O( y ) (7) Let h x y k t the local trucatio error for this scheme is the k u C t k u u u u ax a... y b x b y t x t x t x t y i j 0.5 i j 0.5 i j 0.5 i j 0.5 a u a u b u b u 0.5 x y x y h x y 6 x 6 y i j 0.5 i j 0.5 i j 0.5 i j hk ax u ay u u u b... x b y t x t y t x t y ij 0.5 ij 0.5 ij 0.5 ij 0.5 i.e. C O(k ) O(h ). () As x y t 0 the trucatio error C teds to zero. Hece as the grid spacigs x y t 0 i the limit sese the Crak icolso formula (5) is equivalet to the covectio-diffusio equatio ad thus is cosistet. EG is also cosistet ad its trucatio error is similar with the Crak-icolso scheme sice it is derived from the same formula. Assumig that a a x a y the trucatio error for the rotated Crak-icolso scheme becomes: RC k u t k u u u u a a b x b... y t x t x t x t y ij 0.5 ij 0.5 ij 0.5 ij 0.5 h a u a u a u x x y y ij 0.5 ij 0.5 ij 0.5 bx u bx u by u by u 6 x x y 6 y x y i j 0.5 i j 0.5 i j 0.5 i j 0.5 hk a u u a u a t x t x y t y ij 0.5 ij 0.5 ij 0.5 u u u u bx b x b... y b y t x t x y t y t x y ij 0.5 ij 0.5 ij 0.5 ij 0.5 i.e. R-C O(k ) O(h ). (9) Similarly the rotated Crak-icolso equatio (6) is cosistet ad the cosistecy of EDG is also maitaied sice it is based o the same formula. ime level Fig. Grid geeratio at time level ad (mesh size 9) V. SABILIY AALYSIS Explicit Group (EG) Equatio () ca be writte explicitly i differece form as u u A - B. Here ISB: ISS: (Prit); ISS: (Olie) WCE 00

4 Proceedigs of the World Cogress o Egieerig 00 Vol III WCE 00 Jue 0 - July 00 Lodo U.K. R R R R R A R R R R R G G G G G G5 R R 5 G G G G G G G a c e 0 R b a 0 e G G d 0 a c 0 d b a 0 b 0 0 G c G b 0 0 c d d G G5 e e 0 0 a Sx Sy Sx Cx b Sx Cx c Sy f Sx Sy. Cy d A ( Sx Sy 0.5Sx 0.5Cx 0.5Sx 0.5Cx 0.5Sy 0.5Cy 0.5Sy 0.5 Cy ). S S S S S B S S S S S H H H5 H H H S S H 5 H H H H H f c e 0 H S b f 0 e H d 0 f c H 0 d b f 0 b 0 0 H c H b 0 0 c d d H H 5 e e 0 0 B ( Sx Sy 0.5Sx 0.5Cx 0.5Sx 0.5Cx 0.5Sy 0.5Cy 0.5Sy 0.5 Cy ) Sy Cy e A B A B B A for all Cx Cy Sx Sy 0. herefore the EG iterative method is ucoditioally stable. Equatio () may also be expressed explicitly as u u A - B. he matrix A is of the form: R R R R R A R R R R R G G G G G G G G G R R G G G G G G G G5 G a e G R d a G5 G G5 G d G c 0 G b 0 G e 0 G5 0 0 with Sx Sy a Sx Cx Cy b Sx Cx Cy c Sy Cx Cy d e Sx Sy f. Sy Cx Cy Sx Sy Sx Cx Cy Sx Cx Cy A Sx Cx Cy Sx Cx Cy. S S S S S B S S S S S H H H H H H H H H S S H H H H H H H H5 H S H5 H H5 H f e H d f 0 0 H c H b d H e 0 H5 0 0 ISB: ISS: (Prit); ISS: (Olie) WCE 00

5 Proceedigs of the World Cogress o Egieerig 00 Vol III WCE 00 Jue 0 - July 00 Lodo U.K. B Sx Sy Sx C x C y Sx C x Cy Sx Cx C y Sx C x C y Sice the amplificatio matrix A-B A B A B B A for all Cx Cy Sx Sy 0. herefore the EDG iterative scheme is ucoditioally stable. Fig. : Experimetal Results of Example VI. UMERICAL EXPERIMES he experimets were carried out o a PC with Itel (R) Corel(M) Duo CPU GHz.9 GB of RAM ruig Widows XP Pro usig C compiler i Cygwi. hroughout the whole experimets the absolute error test was used with tolerace equals to 0-0. Oe average error was obtaied at each time step. he depicted i ables I-III deotes the maximum of all the average errors for the particular mesh size. ables I II ad III preset the umerical results of the four methods the classical Crak-icolso rotated poit Crak-icolso EG ad EDG i solvig Examples ad respectively for the umber of time step 00 ad t 0.0. Example (Diffusio problem) We cosider the followig example (axay bxby0): U U U t x y 0 x 0 y 0 t. he iitial ad boudary coditios are defied so that they satisfy the exact solutio []: U ( x y t ) ( x 0.5 ) ( y 0.5 ) exp t > 0. t t t (0) Fig. 5: Experimetal Results of Example Example We will cosider a covectio domiat problem. Let ax ay 0. bx by.0 the the exact solutio of the problem above is deoted as below []: U (x y t) ( x t 0.5) ( y t 0.5) t > 0. 0 exp 0 t ( t ) ( t ) () As show i able III ad Fig. 6 EDG scheme requires the least executio timigs compared to the other three methods. I all of the examples the EG method produces almost the same accuracies as the classical Crak-icolso while the EDG method is almost as accurate as the rotated Crak-icolso. EG reduces the executio times up to 50% of the classical Crak-icolso while maitaiig the same degree of accuracies. he executio timigs of EDG are early 65% of the rotated Crak-icolso scheme. he latter was also observed to require lesser computig timigs tha the origial Crak-icolso scheme. Example Cosider the followig example (ax ay bx by ): U U U U U 0 x 0 y 0 t t x y x he exact solutio of the problem above is as follows []: ( x t 0.5 ) ( y t 0.5 ) U ( x y t ) t exp Fig. 6: Experimetal results of Example y t t t > 0. () Similar with Example EG is faster tha the Crak-icolso scheme while EDG is faster tha the rotated Crak-icolso ad the EG schemes. ISB: ISS: (Prit); ISS: (Olie) VII. COCLUSIOS I this paper we have preseted effective ucoditioally stable group explicit iterative algorithms i solvig the two dimesioal covectio-diffusio problem. he methods serve as viable alterative solvers to the problem with the group scheme derived from the rotated fiite differece approximatio requirig the least computig efforts amog the schemes tested. he parallel implemetatio of these group methods are still uder ivestigatio ad will be reported soo. WCE 00

6 Proceedigs of the World Cogress o Egieerig 00 Vol III WCE 00 Jue 0 - July 00 Lodo U.K. REFERECES [] A. R. Abdullah he Four Poit Method: A Fast Poisso Solver Iteratioal Joural of Comp. Math vol. 99 pp []. M. Ali he Desig Ad Aalysis of Some Parallel Algorithms For he Iterative Solutio of Partial Differetial Equatios PhD hesis Uiversiti Kebagsaa Malaysia 99. [] B.J. oye umerical Methods for Solvig the rasport Equatio umerical Modellig: Applicatio to Marie Systems (ed. oye) (Amsterdam: orth Hollad Publishig Compay ). [] W. S. Yousif ad D. J. Evas Explicit Group Over-relaxatio Methods for Solvig Elliptic Partial Differetial Equatios Mathematics ad Computers i Simulatio vol. 96 pp [5] W. S. Yousif ad D. J. Evas Explicit DeCoupled Group Iterative Methods ad heir Parallel Implemetatios Parallel Algorithms ad Applicatios vol pp able I: Experimetal results of Example for each scheme with atural orderig ( 00 t 0.0) Classical C Explicit Group (EG) Size otal iteratio otal iteratio ime (s) ime (s) 00.6E E E E E E E E Size Rotated C otal iteratio otal iteratio ime (s) ime (s) E E E E E E E E Size able II: Experimetal results of Example for each scheme with atural orderig ( 00 t 0.0) Classical C Explicit Group (EG) otal otal iteratio iteratio ime (s) ime (s) 5.099E E E E E E E E 05. Size otal iteratio Rotated C ime (s) otal iteratio ime (s) E E E E E E E E Size able III: Experimetal results of Example for each scheme with atural orderig ( 00 t 0.0) Classical C Explicit Group (EG) otal otal iteratio iteratio ime (s) ime (s) Size otal iteratio Rotated C ime (s) otal iteratio ime (s) ISB: ISS: (Prit); ISS: (Olie) WCE 00