In this paper, the alternating group explicit (AGt-j iterative method is applied to cylindrical problems involving

regular domains on a non-uniform grid. The procedure uses the fractional splitting strategy which is applied

alternately at each half (intermediate) time step on a tridiagonal system of difference equations. The method

is shown to be more accurate than the corresponding AGE scheme solved earlier by the authors using an uniform

grid system but with a reduced stability range

Muda, Zaiton

Sahimi, Ahmad Salleh

Universiti Putra Malaysia Institutional Repository

Pertanika 12(3),413-419 (1989).An Iterative Explicit Method for Parabolic Problems withCylindrical Symmetry-Increased Accuracy on Non-Uniform GridMOHD. SALLEH SAHIMI AND ZAiTON Ml DA*Department of MathematicsUniversiti Kebangsaan MalaysiaBangi, Selangor Darul Ehsan, MalaysiaKey words : AGE method, cylindrical problem, non-unifonn grid.ABSTRAKDi dalam makalah ini, kaedah berlelar tidak tersirat kumpulan berselang-seli (ITKS) digunakan untukmenyelesaikan masalah silinder yang melibatkan domain sekata pada sistem grid yang tidak seragam.Tatacara ini menggunakan strategi pecah-belah berperingkat secara berselang-seli pada setiap setengah parasmasa. Kaedah ini diterapkan ke atas sistem tiga pepenjuru persamaan beza. Ternyata kaedah ini lebih jitudaripada kaedah irKS yang sepadan yang digunakan bagi masalah yang sama tetapi pada sistem gridseragarn. Walau bagaimanapun, julat kestabilannya terhad.ABSTRACTIn this paper, the alternating group explicit (AGt-j iterative method is applied to cylindrical problems involvingregular domains on a non-unifonn grid. The procedure uses the fractional splitting strategy which is appliedalternately at each half (intermediate) time step on a tridiagonal system of difference equations. The methodis shown to be more accurate than the corresponding AGE scheme solved earlier by the authors using an uniformgrid system but with a reduced stability range.in one-space(1.4)1. INTRODUCTIONConsider the following equationdimension given by,(1.1 )together with the initial-boundary conditions,U(r,O)=f(r),O~r~l (1.2)au- (0, t) = 0, u(l, t) = 0, 0 ~ t ~ T.arIn Sahimi and Muda (1988), the cylindri-cal equation (1.1) was approximated by stan-dard finite difference analogues on. the usualuniformly-spaced network whose mesh pointswere r = i6.r, t. = j6.t. The truncation error,howev~r, always contains low order deri-.,vatives au and (r U. To overcome this, theor or 2transformation procedure of Mitchell andPearce (1963),1 .,x = -- r- (1.3)4can be used. The "accuracy difficulty" in theneighbourhood of the axis may be avoided tosome extent by considering a rectangular netwhich is uniformly spaced in the t-directiongiven by tj =j6.t and unequally spaced in the x-direction indicated by Xi = i26.x. The latter isconsistent with equal spacing in the r-direction.The transformation of (1.3) converts (1.1) tooU = aU + x 02U , 0 ~ x. ~ 1....at ax ox" 4The higher time derivatives no longer containlow derivatives of U with respect to x. The AGEalgorithm can now be applied to the resultingimplicit replacement of (1.4).*Present address: Department of Mathematics, 'niversiti Pertanian Malaysia.MOHD. SALLEH SAHIMI AND ZAITON MUDAf, = U"J_I' i =2,3, ... ,m -1;f m= u mj _1- J.(- 2m 2- 2m + 2J. + 1)- ,[ 4 i(2i + 1)] ,i = 1,2, ... , m - 1J.fl='u I -1- -(1 + 2J.) u o ;.J 4 .J-I[4m(2m + 1)] U"'+ljwith U . and u l' corresponding to the left and0:.1 m+ ,jright boundary values respectively.Without loss of generality, let us assumethat we have an odd number of internal meshpoints. We can then perform the followingspli tting of A:A = G] + G2l::.!.:.;;..;.:;14....... 3 2 3'-------+----------- f -:------- !--------------·······1Gj····,j·!:;:::;:.:··(mxm)G =1A •• = 1__ M= 1:. and J. = A!:.D.X , n Llx'4 (m + 1)= U i. i-Ii=I,2, ... ,m, j=I,2, ... ,n (2.1)while for points on the axis, we have1( ") 2U . = - 4 - 5J. + n - u . I + -J.0·1 4 o.J- 3J.(J. - 2)u'.i_' + 12(2J. -1)u"j_1for j = 1,2, ... , n (2.2)wherethe mesh ratio.The matrix representation of (2.1) thustakes the form2. PROBLEM FORMUlATIONFollowing Mitchell and Pearce (1963) an opti-mum four point implicit finite difference ap-proximation (the MP approximation) to (1.4)is given by[1 + 2J.C" -J. -1)[ (4i 2-1)fJu;j- I .+ J. ( - 2i' - 2i + n + 1)[ 4 i(2i + 1)] u ;+I. j- I+ J. (- 2 i"+ 2 i + n + 1) [ 4i ( 2 i-I)] u ,_1 jU1U2andc ! a• Z ,- I[ 4 i( 2i - 1)] , i = 2,3, ... , mUm-lUm ..• j ...orAu= fJwherea, = 1+ nC" - J. - 1)( 4i" - 1) - I , i = 1,2, ... , mc; = J.( - 2i" + 2i + n + 1)fm-lfm(2.3)G~= a'••••••••..••? ·••~••··.[··:~::::i·.·I. _.Following Evans and Sahimi (1987), thefollowing iterative AGE convergent scheme wasderived,(G,+; r))k+-;)=(; r-G")~k)+~(G 2+; r)~(k+I)=(G2-(I-w);r)t1+(2_W);Jk+;) (2.4)for any 0 ~ w ~ 2 and ; > 0 being a fixed ac-celeration parameter along each intermediate414 PERTANlKA VOL. 12 NO.3, 1989AN ITERATIVE EXPLICIT METHOD FOR PARABOLIC PROBLEMS(half-time) level or iterate. w =°leads to thePeaceman-Rachford (PR) scheme and w = 1gives us the variant due to Douglas and Rach-ford (DR).Both stable schemes have truncation errorsof the order T pR = O((~r)2 + (M)2) and T DR =°((~r) 2) + M) respectively.We have- b 2' ]1 • ,-a". + r2 -,• -1(G + rI) =1where, (2.5)(mxm)-~~~~-_._~~;~-'------_._._-_.([) ._______ • " • • ,(2.6)-------: ------: -------: ----------: -----------a. = (_,1a., + ~)(.!.a". ,+ ~)-b,,.c,,. " j1 2 -I 2 _I + _I _I + . 1( ) ( )1 = 1, 2, ... , - (m - 1)• 1 • 1 • 2and a, = "2a2'" + r "2a2' + r - b 2,.,C 2,3. COMPUTATIONAL ASPECT OF THE AGE SCHEMEUsing (2.4) - (2.6), the u-vales at each of the half-iterates can be computed as follows:(1) at the ( k + ~}h iterateJ k+~l= (s dk) - b Jk) + f) ISi I I 1 2 I /;) I,'+~l= (A,d,'!, + B,d,') + C,u"'!, + D,J"!2 + E)/a V2 j._J J 1-2,4, ... ,m-1 (3.1),':,; = (A, d,'!, + B, u,,') + C, d i'!, + D, d"!2 + E,)/a V2whereB j = SjSi+I' B i = -C i + J Sj{o if i = m -1D =, b, b,+, otherwise'. {O if i = m -1D =, - b,., S, otherwisewiths,• 1 1r + -a., s. = r --a., i = 1,2, ... ,ln.2" 2'E=fs -b fI I I +Iii+ IPERTANlKA VOL. 12 0.3,1989 415MOHD. SALLEH SAHIMI AND ZAITO MUDAPi = 5i+ I q i - b i C i + I' P i = C i+ {5i - q.)Qi = b i (5i+ I - q i +), Q i =5iq i + I - C i + I b i(3.2)[11k'-b Is1 15I,. 2Fig. 1 Computational molecules at level ( k + ~)416 PERTANlKA VOL. ]2 NO.3, 1989a {I dJ/2"tIt"i~~(5r-'""Z().'-'0-<Dco<D>t>-......'-JTABLE 1The absolute errors of the numerical solutions to the cylindrical problemq = 0.7, t = 0.175, 6x = 0.0025, 6t = 0.00175, i = 0.7r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 O.R 0.9~ Number0.0000 0.0025 0.0100 0.0225 0.0400 0.0625 (L0900 O. L225 0..1600 0.2025 ofiterations'" THOMAS-MP 9.1 x 10" 9.2 X lOA 9.::\ X 10" 9.4 x 10'" 9.5 x 10" 9.::\ X 10" 8.7 X 10" 7.6 X 10" 5.7 X 10" 3.1 X 10"'C -0C E PR 9.6 x lOA 9.9 X lOA 9.6 X 10" 9.6 X 10" 9.6 X 10" 9.4 X 10" 8.8 X 10. 1 7.6 X 10--' 5.7 X 10" 3.1 X lOA 2o ...z<8 AGE-MPc '20;:> DR 3.5 x 10'" 3.4 x 10'" 3.4 x 10':< 3.1 x 10'" 2.7 x 10'" 2.2 x 10'" 1.7 x 10'" 1.1 x 10'" 7.0 x 10'" 3.0 x 10.4 6THOMAS-IMP 3.9 x 10'" 3.8 x 10'" 3.6 x 10'" 3.4 x 10'" 3.0 x 10'" 2.6 x 10" 2.1 x 10'" 1.6 x 10'" 1.1 x Hr' 5.5 x 10" -PR .3.9 x 10'" 3.8 x 10" 3.7 x 10'" 3.4 x 10'" 3.0 x 10'" 2.6 x 10'" 2.1 x 10'" 1.6 x 10'" I. 1 x 10'" 5.5 X lOA 3AGE-IMPDR 8.6 x 10':< 8.6 x 10'" 8.2 x 10'" 7.6 x \0'" 6.8 X HY' 5.9 x I'" 4.8 x 10'" ::\.6 x 10-" 2.4 x 10'" l.l x 10'" 6"'C0 THOMAS-CN 2.0 x 10'" 2.0 x 10'" \.9 x 10:" 1.8 x \0'" 1.6 x 10':< \.4 X 10" 1.1 x 10" 8.9 X 10'" 6.0 X 10" 3.\ X 10'" -E~ PR 2.0 x 10'" 2.0 x 10" 1.9 x 10'" 1.8 x 10'" 1.6 x 10'" \.4 x 10'" 1.1 x 10'" 8.9 x 10.4 6.0 x \ O~' 3.1 X 10" 3'2AGE·CN;:><= DR 7.0 x 10'" 6.9 x 10:< 6.6 X Hr' 6.1 x 10'" 5.5 x 10':< 4.7 x W-" 3.8 X 10" 2.9 X 10·:1 1.9 X IO~ 9.6 X 10" 60THOMAS-DGE 2.8 x 10·:1 2.8 x 10' 2.2 x 10':< 1.7 x 10·:1 1.3 x 10'" 9.4 X 10" 6.4 x 10' 4.0 X 10" 2.1 X 10" 7.6 x 10'-'-PR 2.8 x 10':< 2.8 x 10" 2.2 X 10" 1.7 x 10':< 1.3 X 10" 9.::\ x 10" 6.3 X 10' 4.0 X 10" 2.1 X /0" 7.5 x 10'" 2AGE-DGEDR 2.0 x ](Y' \.9 x 10':< 2.3 x 10'" 2.5 x 10':< 2.5 x \0'" 2.3 x lO''' 2.0 x 10':< 1.6 x 10'" 1.1 x 10'" 5.7 X 10" 6EXACT SOLUTION 0.36341707 0.35818100 0.34269894 0.31763846 0.28407656 0.243448\1 0.19747738 0.\4809604 0.09735\72 0.04731180 -2':::1t"i~~tTl><"tII"""n::j~tJ:toob;<j~ocn"tI;;0o(}:lI"""t"i~en,j:>........00-01"'1~Z~<or-""Z()yo<.0(Xl<.DTABLE 2The absolute errors of the numerical solutions to the cylindrical problemq = 1.0, t = 0.25, 6x = 0.0025, M = 0.0025, r = 0.3r 0.0 0.1 0.2 O.~ 0.4 0.5 0.6 0.7 0.8 0.9~ Nlimber0.0000 0.0025 0.0100 0.0225 0.0400 0.0625 0.0900 0.1225 0.1600 0.2025 ofileraliolls-0 THOMAS-MP 1.6 x 10'" 1.6 x 10'" 1.3 x 10':< 1.2 x 10'" 1.1 x 10'" 1.0 x 10'" 9.7 X 10" 8.1 X 10" 5.9 X 10-' 3.1 X 10" -.;::v:: E PR 3.0 x 10'" 3.2 X 10" 1.8 x 10-:1 1.5 x 10'" 1.3 x 10" 1.2 x 10'" 1.0 X 10-:' 8.7 x 10" 6.3 X 10'" ~.3 X 10' 4o ....z-S AGE-MP<: 'cDR 6.2 x 10' 6.9 x 10" 1.8 x 10" 1.0 x 10" 6.4 x 10'" 4.1 x 10'" 2.7 x 10'" 1.7 x 10'" LOx 10" 5.2 X 10" 7O;:JTHOMAS-IMP 4.3 x lO·" 4.~ x 10'" 4.1 x 10'" 3.8 X 10" 3.4 x 10'" 2.9 x lO·" 2.4 x 10" 1.8 x 10'" 1.2 x 10'" 6.0 x 10" -PR 4.2 x 10'" 4.2 x 10'" 4.0 x 10'" 3.8 x 10'" ~.4 x lO·" 2.9 X lO" 2.4 x 10'" 1.8 x 10-:' 1.2 x 10'" 5.9 X 10" 3AGE-IMPDR 8.1 x lO·' 8.2 x 10'" 7.8 x 10'" 7.3 x 10'" 6.5 x 10'" 5.6 x 10-:' 4.6 X 10" 3.4 X 10'" 2.~ x 10" 1.1 x 10'" 6~....2.8 X 10"v THOMAS-CN 1.9 x lO·" 1.9 x 10'" 1.8 x 10'" 1.6 x 10'" 1.5 x 10'" I.~ X 10':< 1.0 x 10·:1 8.2 X 10'" 5.5 X 10" -E....-S PR 1.9 x lO·" 1.9 x 10" 1.8 x 10'" 1.7 X 10" 1.5 x 10'" 1.3 X 10" 1.1 X 10" 8.5 X 10. 1 5.7 X I()' 2.9 X 10' 8'c:J AGE-CN<: DR 6.2 x 10'" 6.2 x 10" 5.9 x 10'" 5.5 x 10'" 4.9 x 10'" 4.2 x 10'" 3.4 x 10'" 2.6 x 10'" 1.7 x 10'" 8.5 X 10" 60THOMAS-DGE 2.3 x lO·" 2.3 x 10" 1.9 x 10'" 1.5 x lO·" 1.2 x 10'" 9.3 x 10" 6.7 X 10'" 4.5 x 10~ 2.6 X 10'" \.I X lO··'-PR 2.2 x 10'" 2.2 x 10'" 1.8 x 10'" 1.4 x J0-" 1.1 x 10'" 8.7 X 10" 6.2 X lO-' 4.1 X 10'" 2.4 x 10" 9.9 x 10'" ~AGE-DGEDR 2.1 x 10'" 2.J x 10" 2.3 x 10'" 2.4 x 10'" 2.3 x 10'" 2.1 x 10'" 1.8 X 10" J.4 X \{r' 9.5 x 10" 4.S X 10. 1 6EXACT SOLUTION 0.2~550905 0.23211586 0.2220S286 0.20584265 0.IS409~17 0.15776428 0.127973~S 0.95972263 0.6~087S77 0.30659974 -~o:r::I:'~t""1"'1:r::(fl~~;J>ZoNC:-loZs:c:o;J>Fig 2 continued9CDAN ITERATIVE EXPLICIT METHOD FOR PARABOLIC PROBLEMSThe MP approximation of (2.1) was solvedusing both the AGE scheme and the Thomaselimination procedure. The solutions werecompared with the results obtained from theapplication of the two schemes on the fullyimplicit (IMP), the Crank-Nicolson (CN) andthe Douglas-equivalent (DGE) approximationsto (1.1) (Sahimi and Muda (1988)) for differ-ent mesh ratios. Tables 1 and 2 indicate theaccuracy of these methods.It is evident that the AGE-MP(PR) schemecan have comparable accuracy with the mostaccurate of the standard methods. However,for poin ts near the axis, the AGE-MP schemeis clearly seen to be more superior. It is some-what restricted by the stability requirement ofA s:; VlS". Nevertheless, it has the advantagethat being explicit, a parallel algorithm can bedeveloped for computation and although itera-tive in nature, it requires a small number ofiterations for convergence.Fig. 2 Computational molecules at level (k + 1)The computational molecules at each ofthe half-iterates are shown in Figs. 1 and 2 :Solutions are obtained along each timelevel using (2.2) followed by the application ofthe AGE algorithm which is executed explicitly,"utilising equations (3.1) - (3.2) at the(k + ~)and (k + 1)'h iterates in alternate sweeps untila specified convergence criterion is satisfied.4. NUMERICAL RESULTSThe cylindrical problem (1.1) was consideredand the same boundary conditions were used.The initial condition, however, is specificallygiven byU(r,O) = Jo(~r)where JoWr) is the Bessel function of thefirst kind of order 0 and ~ is the first root ofJ)~) = O. The exact solution is U (r,t) =Jo(~r)exp (_~2t) (Mitchell and Pearce (1963)). Theconvergence criterion employed isII X(k+l) - X(k) 1100s:; 10-4 and the accelerationpa~ameter t was chosen to provide the mostrapid convergence.REFERENCESDOUGLAS, J. and H.H. RACHFORD. 1956. On TheNumerical Solution of Heat Conduction Prob-lems in Two or Three Space Variables. Trans.A mer. Math. Soc. 82 : 421-439.EVANS, DJ. and M.S. SAHIMI. 1987. The Alternat-ing Group Explicit Method (AGE) to Solve Para-bolic and Hyperbolic Partial Differential Equa-tions. In A nnual Review ofNumerical Fluid Mechanicand Heat Transfer 2, ed. Chang-Lin Tien, Hemis-phere Pub. Corp.MITCHELL, A.R. and R.P. PEARCE. 1963. ExplicitDifference Methods for Solving the CylindricalHeat Conduction Equation. Maths. Compo 17 :426-432.PEACEMAN, D.W. and RH. R"CHFORD. 1955. TheNumerical Solution of Parabolic and EllipticDifferential Equations. f. Soc. Indust. Appl. Math.3 : 28-41.SAHIMI, M.S.and Z. MUDA. 1988. An Iterative Ex-plicit Method for Parabolic Problems with SpecialGeometries. Technical Report PPK/Mac 88/LT 7.(Received 21 June, 1989)PERTANIKA VOL. 12 NO.3, 1989 419

An Iterative Explicit Method for Parabolic Problems with

Cylindrical Symmetry-Increased Accuracy on Non-Uniform Grid

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An Iterative Explicit Method for Parabolic Problems with Cylindrical Symmetry-Increased Accuracy on Non-Uniform Grid

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