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

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1 Pertanika 12(), (1989) An Iterative Explicit Method for Parabolic Problems with Cylindrical SymmetryIncreased Accuracy on NonUniform Grid MOHD. SALLEH SAHIMI AND ZAITON MUDA* Department of Mathematics Universiti Kebangsaan Malaysia Bangi, Selangor Darul Ehsan, Malaysia Key words : AGE method, cylindrical problem, nonuniform grid. ABSTRAK Di dalam makalah ini, kaedah berlelar tidak tersirat kumpulan berselangseli (TTKS) digunakan untuk menyelesaikan masalah silinder yang melibatkan domain sekata pada sistem grid yang tidak seragam. Tatacara ini menggunakan strategi pecahbelah berperingkat secara berselangseli pada setiap setengah paras masa. Kaedah ini diterapkan ke atas sistem tiga pepenjuru persamaan beza. Temyata kaedah ini lebih jitu daripada kaedah IYKS yang sepadan yang digunakan bagi masalah yang sama tetapi pada sistem grid seragam. Walau bagaimanapun, julat kestabilannya terhad. ABSTRACT In this paper, the alternating group explicit (AGE) iterative method is applied to cylindrical problems involving regular domains on a nonuniform 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. L I N TRODUCTION, transformation procedure of Mitchell and Consider the following equation in onespace p (\Qf>%\ dimension given by, x=r (1.) U = jhj + J_ U. (1 1} 4 dt r' 2 r dr can be used. The "accuracy difficulty" in the neighbourhood of the axis may be avoided to together with the initialboundary conditions, SQme extem by considering a rectangular net U( r 0) = f(r) 0 < r < 1 U2) which is uniformly spaced in the tdirection given by t = jat and unequally spaced in the x (0, t) = 0, U( 1, t) = 0, 0 < t < T. direction indicated by x. = rax. The latter is " r consistent with equal spacing in the rdirection. In Sahimi and Muda (1988), the cylindri The transformation of (1.) converts (1.1) to cal equation (1.1) was approximated by standard finite difference analogues on.the usual ^ = + x 7, 0 < x < ' (1.4) uniformlyspaced network whose mesh points " l " x " x were r. iar, t. = jat. The truncation error, The higher time derivatives no longer contain however, always contains low order deri low derivatives of U with respect to x. The AGE ju V 2 j algorithm can now be applied to the resulting vatives and. To overcome this, the irnplicit replacement of (1.4). Present address: Department of Mathematics, Universiti Pertanian Malaysia.

2 MOHD. SALLEH SAHIMI AND ZAITON MUDA 2. OBLEM FORMULATION Following Mitchell and Pearce (19) an optimum four point implicit finite difference approximation (the MP approximation) to (1.4) is given by b L = \ 2i 2i [4i(2i + l)] ',i = 1,2,...,m 1 f,= u,._, (l + 2) u (K] ; f. = u,..,, i =2,,...,m 1; f =u (2m 2 2m + 2?i+ l) 4i(2i l)] u,.,,, = u i= 1, 2,..., m, j1, 2 n (2.1) while for points on the axis, we have [4m(2m + l)] 1 u^+lij with u and u, corresponding to the left and o,j m+l,) right boundary values respectively. Without loss of generality, let us assume that we have an odd number of internal mesh points. We can then perform the following splitting of A: A = G, + G 2 l '' for j = 1, 2,..., n (2.2) Ax=, At= and =, 4 \ m + 1; the mesh ratio. The matrix representation of (2.1) thus takes the form _ Is b 2 k.s J. z a.. b.. u 1 and c a b or Au.= f,c a b 1 ml m1 V, u f ml i i c k, Following Evans and Sahimi (1987), the following iterative AGE convergent scheme was (2.) derived, Ui'l), i = 1,2,..,,m c, =?i(2i"' + 2i +21+ l) [4i(2i 01. i =2,,...m (G 2 + r l)u (k + 1) F^V t U WJ r ; M + (2w) r u 2J (2.4) for any 0 < w < 2 and r > 0 being a fixed acceleration parameter along each intermediate 414 PERTANIKA VOL. 12 NO., 1989

3 AN ITERATIVE EPLICIT METHOD FOR PARABOLIC OBLEMS (halftime) level or iterate, w = 0 leads to the PeacemanRachford () scheme and w = 1 gives us the variant due to Douglas and Rachford (). Both stable schemes have truncation errors of the order T pr = 0((Ar) 2 + (At) 2 ) and T = 0((Ar) 2 ) + At) respectively. We have Defining *( G c; i) i) a i 1 \ V c +1 b. (G + rl)" 1 " Mb A \ '" /is viz a * (mxm) (2.5) j { G <2) ) * J 2'" 1 "A2""* 1 i..\ri. COMPUTATIONAL ASPECT OF THE AGE SCHEME Using (2.4) (2.), the uvales at each of the halfiterates can be computed as follows: (l) at the k + th iterate I 2) +B i u< i k) u (k i =2,4,...,m 1 (.1) O if i = m 1 [b.b. + 1 otherwise' D = 0 if i = m 1 b j + l s". otherwise with 1 A s" = r + a., s. = r a, i = l,2,...,m. 2 ' ' 2 ' PERTANIKA VOL. 12 NO.,

4 MOHD. SALLEH SAHIMI AND ZAITON MUDA (2) at the (k + 1) th iterate i = l,,...,m2 (.2) ' R, = r 4l d, R =c. + 1d, S. = b.d, S =s d Fig. 1 Computational molecules at level [ k + 41 PERTANIKA VOL, 12 NO., 1989

5 TABLE 1 The absolute errors of the numerical solutions to the cylindrical problem q = 0.7, t = 0.175, Ax , At , f = 0.7 r ().? Method ^ ^ ^ ^ Number of iterations Grid THOMAS^MP 9.1 x 10^ 9.2 x 9. x 10^ 9.4 x 9.5 x 9. x 8.7 x 7. x 5.7 x 10 '.1 _ On Non Uniform AGEMP 9. x.5 x 10^ 9.9 x.4 x 9. x 10^.4 x 9. x.1 x io :1 9. x 2.7 x 10 * 9.4 x 10"' 2.2 x 8.8 x 7. x 1.1 x 5.7 x 7.0 x $ THOMASIMP.9 x.8 x. x 10 s.4 x.0 x 2. x LI x 10 s 5.5 $ K> o On Uniform Gn AGEIMP THOMASCN AGECN.9 x 8. x 2.0 x 2.0 x 7.0 x 10 : *.8 x 8. x 2.0 x 2.0 x.9 x io s.7 x 8.2 x. x.4 x 7. x.1 x 10 : '.0 x.8 x 5.5 x 2. x 5.9 x 1' 1.4 x 1.4 x 4.7 x 4.8 x 1.1 x x. x 8.9 x 8.9 x 2.9 x io LI x.0 x.0 x 10 s io 1 10 " W THOMASDGE 2.8 x 2.8 x 2.2 x 1. x 10 : * 9.4 x.4 x 4.0 x 7. 10"' AGEDGE 2.8 x 2.0 x 2.8 x 2.2 x 2. x 2.5 x 1. x 2.5 x 10 : * 9. x 2. x. x 2.0 x 4.0 x S S EACT SOLUTION

6 :< 00 TABLE 2 The absolute errors of the numerical solutions to the cylindrical problem q = 1.0, t = 0.25, Ax = , At = , r = 0. r I 0/, 0.? [ 0.5 o.e ^^\ Method x Number of iterations Grid THOMASMP 1. x x 8.1 x 5.9 x 10 (.1 x On Non Uniform AGEMP THOMASIMP.0 x.2 x 4. x.2 x.9 x 4. x x 1.5 x.8 x x 2.9 x x 10 < 8.7 x. x x 5.2 x.0 x ' p E On Unifor AGEIMP THOMASCN AGECN 4.2 x 8.1 x.2 x lor x 8.2 x.2 x 10 < 4.0 x 7.8 x 5.9 x 10 * 10 *.8 x 7. x 5.5 x \(V x 5. x 1. x 1. x 4.2 x 10 ( 10 : ' 4. x 1.1 x.4 x.4 x 8.2 x 8.5 x 2. x 2. x 5.5 x 5.7 x 5.9 x 10 ' x 2.9 x 1 0 «8.5 x > THOMAS^DGE 2. x 2. x 1.5 x 1.2 ]<)! 9. x.7 x 4.5 x 2. x 1.1 AGEDGE 2.2 x 2.2 x 2. x 1.4 x x.2 x 10 ' 4.1 x 1.4 x 9.5 x 10 ' 9.9 x 4.8 x 10 r ' 1 0 ' EACT SOLUTION

7 AN ITERATIVE EPLICIT METHOD FOR PARABOLIC OBLEMS Fig 2 continued M k The MP approximation of (2.1) was solved using both the AGE scheme and the Thomas elimination procedure. The solutions were compared with the results obtained from the application of the two schemes on the fully implicit (IMP), the CrankNicolson (CN) and the Douglasequivalent (DGE) approximations to (1.1) (Sahimi and Muda (1988)) for different mesh ratios. Tables 1 and 2 indicate the accuracy of these methods. It is evident that the AGEMP() scheme can have comparable accuracy with the most accurate of the standard methods. However, for points near the axis, the AGEMP scheme is clearly seen to be more superior. It is somewhat restricted by the stability requirement of < V 15. Nevertheless, it has the advantage that being explicit, a parallel algorithm can be developed for computation and although iterative in nature, it requires a small number of iterations for convergence. Fig 1. 2 Computational molecules at level (k + 1) The computational molecules at each of the halfiterates are shown in Figs. 1 and 2 : Solutions are obtained along each time level using (2.2) followed by the application of the AGE algorithm which is executed explicitly ih utilising equations (.1) (.2) at the(k + l ) and (k + l) th iterates in alternate sweeps until a specified convergence criterion is satisfied. 4. NUMERICAL RESULTS The cylindrical problem (1.1) was considered and the same boundary conditions were used. The initial condition, however, is specifically given by U(r,0) =J o (pr) J o ( Sr) is the Bessel function of the first kind of order 0 and [ is the first root of J o (P) 0. The exact solution is U(r,t) = J () (pr) exp (p 2 t) (Mitchell and Pearce (19)). The convergence criterion employed is 11 x (k+i>_ x ck) ^ < IQ4 an(j th e acceleration parameter r was chosen to provide the most rapid convergence. REFERENCES DOUGLAS, J. and H.H. RACHFORD On The Numerical Solution of Heat Conduction Problems in Two or Three Space Variables. Trans. Amer. Math. Soc. 82 : EVANS, DJ. and M.S. SAHIMI The Alternating Group Explicit Method (AGE) to Solve Parabolic and Hyperbolic Partial Differential Equations. In Annual Review of Numerical Fluid Mechanic and Heat Transfer 2, ed. ChangLin Tien, Hemisphere Pub. Corp. MITCHELL, A.R. and R.P. PEARCE. 19. Explicit Difference Methods for Solving the Cylindrical Heat Conduction Equation. Maths. Comp. 17 : PEACEMAN, D.W. and H.H. RACHFORD The Numerical Solution of Parabolic and Elliptic Differential Equations. / Soc. Indust. AppL Math. : SAHIMI, M.S.and Z, MUDA An Iterative Explicit Method for Parabolic Problems with Special Geometries. Technical Report PPK/Mac 88/LT 7. (Received 21 June, 1989) PERTANIKA VOL. 12 NO.,