Research Article Cubic Spline Iterative Method for Poisson s Equation in Cylindrical Polar Coordinates

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1 International Scolarly Researc Network ISRN Matematical Pysics Volume 202, Article ID 2456, pages doi:0.5402/202/2456 Researc Article Cubic Spline Iterative Metod for Poisson s Equation in Cylindrical Polar Coordinates R. K. Moanty, Rajive Kumar, 2 and Vijay Daiya 2 Department of Matematics, Faculty of Matematical Sciences, University of Deli, Deli 0 007, India 2 Department of Matematics, Deenbandu Cotu Ram University of Science & Tecnology, Murtal 09, India Correspondence sould be addressed to R. K. Moanty, rmoanty@mats.du.ac.in Received 4 October 20; Accepted 6 November 20 Academic Editors: J.-C. Wallet and H. Zou Copyrigt q 202 R. K. Moanty et al. Tis is an open access article distributed under te Creative Commons Attribution License, wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly cited. Using nonpolynomial cubic spline approximation in x- and finite difference in y-direction, we discuss a numerical approximation of O k 2 4 for te solutions of diffusion-convection equation, were k>0and>0aregridsizesiny-andx-coordinates, respectively. We also extend our tecnique to polar coordinate system and obtain ig-order numerical sceme for Poisson s equation in cylindrical polar coordinates. Iterative metod of te proposed metod is discussed, and numerical examples are given in support of te teoretical results.. Introduction We consider te two-dimensional elliptic equation of te form 2 u x 2 u D x u 2 y2 x g( x, y ), ( x, y ) Ω.. Te Diriclet boundary conditions are given by u ( x, y ) g ( x, y ), ( x, y ) Γ,.2 were Ω { x, y 0 <x,y<} is te solution domain and Γ is its boundary. For g x, y 0 and D x β, te above equation represents diffusion-convention equation. For D x /x,

2 2 ISRN Matematical Pysics te above equations represent te Poisson s equation wit singular coefficients in rectangular coordinates. Similarly, for D x /x and replacing te variables x, y by r, z, we obtain a Poisson s equation in cylindrical polar coordinates. We will assume tat te boundary conditions are given wit sufficient smootness to maintain te order of accuracy of te difference sceme and spline functions under consideration. In tis paper, we are interested in discussing a new approximation based on cubic spline polynomial for te solution of elliptic equation.. In many practical problems, coefficients of te second derivatives term are small compared to te coefficients of te first derivatives term. Tese problems are called singular perturbation problems. Singularly perturbed elliptic boundary value problems are matematically models of diffusion-convections process or related pysical penomenon. Te diffusion term is te term involving te secondorder derivative, and convective term is tat involving te first-order derivative. During last tree decades, several numerical scemes for te solution of elliptic partial differential equations ave been developed by many researcers. First-Lync and Rice ave discussed ig-accuracy finite difference approximations to te solutions of elliptic partial differential equations. Boisvert 2 as discussed a class of ig-order accurate discretization for te elliptic boundary value problems. Yavne as reported te analysis of fourt-order compact sceme for convection diffusion equation. A fourt-order difference metod for elliptic equations wit non-linear first derivative terms as been discussed by Jain et al. 4, 5. In 997, Moanty 6 as derived order 4 difference metod for a class of 2D elliptic boundary value problems wit singular coefficients. A new discretization metod of order four for te numerical solution of 2D non-linear elliptic partial differential equations as been studied by Moanty et al Te use of cubic spline polynomial and its approximation plays an important role for te formation of stable numerical metods. In te past, many autors see 0 2 ave studied and analysed te use of cubic spline approximations in te solution of linear two-point boundary value problems. In 98, Jain and Aziz ave developed a new metod based on cubic spline approximations for te solution of two-point nonlinear boundary value problems. Later, Al-Said 4, 5 as discussed cubic spline metods for solving te system of second-order boundary value problems. Kan 6 as introduced parametric cubic spline approac for solving second order ordinary differential equations. Moanty et al. 7, 8 ave also reported a fourt-order accurate cubic spline alternating group explicit metod for nonlinear singular two-point boundary value problems. Recently, Rasidinia et al. 9 ave proposed a new cubic spline tecnique for two-point boundary value problems. Most recently, Moanty and Daiya 20 and Moanty et al. 2 ave used cubic spline polynomials and developed ig order stable numerical metods for te solution of one space dimensional parabolic and yperbolic partial differential equations. To te autors knowledge, no ig-order metod using cubic spline polynomial for te solution of 2D elliptic differential equations. as been discussed in te literature so far. In tis paper, using nine-point compact cell see Figure, we discuss a new compact cubic spline finite difference metod of accuracy two in y- and four in x-coordinates for te solution of elliptic differential equation.. In te next section, we discuss te derivation of te proposed cubic spline metod. It as been experienced in te past tat te cubic spline solutions for te Poisson s equation in polar coordinates usually deteriorate in te vicinity of te singularity. We overcome tis difficulty by modifying te metod in suc a way tat te solution retains its order and accuracy everywere in te vicinity of te singularity. In Section, we discuss an iterative metod. In Section 4, we compare te computed results wit te results obtained by te metod discussed in 8. Concluding remarks are given in Section 5.

3 ISRN Matematical Pysics (x l,y m+ ) (x l,y m+ ) (x l+,y m+ ) k (x l,y m ) (x l,y m ) (x l+,y m ) (x l,y m ) (x l,y m ) (x l+,y m ) Figure : 9-point computational network. 2. Te Approximation Based on Cubic Spline Polynomial We consider our region of interest, a rectangular domain Ω 0, 0,. We coose grid spacing >0 and k > 0 in te directions x- and y- respectively, so tat te mes points x l,y m denoted by l, m are defined by x l l and y m mk; l 0,,...,N ; m 0,,..., M, were N and M are positive integers suc tat N and M k. Te mes ratio parameter is denoted by λ k/. Te notations u l,m and U l,m are used for te discrete approximation and te exact value of u x, y at te grid point x l,y m, respectively. Similarly, we write D x l D l and D x l± D l±. At te grid point x l,y m, we denote a b W W ab, W U, D and g. 2. x la y b m Let S m x be te cubic spline interpolating polynomial of te function u x, y m between te grid point x l,y m and x l,y m and is given by S m x x l x M l,m x x ( ) l (xl ) M l,m U l,m M x l,m ( ) ( ) U l,m 2 x 6 M xl l,m, x l x x l ; l, 2,...,N ; m 0,, 2,...,M, 2.2 wic satisfies at mt-line parallel to x-axis te following properties: i S m x coincides wit a polynomial of degree tree on eac x l,x l,l, 2,...,N andm, 2,...M, ii S m x C 2 0,, iii S m x l U l,m, l 0,, 2,...,N andm, 2,...,M.

4 4 ISRN Matematical Pysics Te derivatives of cubic spline function S m x are given by S m x x l x 2 M l,m x x l 2 M l,m U l,m U l,m 2 2 S m x x l x M l,m x x l M l,m, 6 M l,m M l,m, 2. were M l,m S m x l U xxl,m U yyl,m D l U xl,m g l,m, l 0,, 2,...,N, j, 2,...,J, 2.4 m l,m S m x l U xl,m U l,m U l,m and replacing by, we get 6 M l,m 2M l,m, x l x x l, 2.5 m l,m S m x l U xl,m U l,m U l,m Combining 2.5 and 2.6,weobtain 6 M l,m 2M l,m, x l x x l, 2.6 m l,m S m x l U xl,m U l,m U l,m 2 2 M l,m M l,m, 2.7 Furter, from 2.5, we ave m l,m S m x l U xl,m U l,m U l,m 6 M l,m 2M l,m, 2.8 and from 2.6, m l,m S m x l U xl,m U l,m U l,m 6 M l,m 2M l,m, 2.9 We consider te following approximations: U yyl,m U l,m 2U l,m U l,m k 2 U yyl,m U l,m 2U l,m U l,m k 2 U yyl,m U l,m 2U l,m U l,m k 2 ( U yyl,m O k 2), ( U yyl,m O k 2), ( U yyl,m O k 2),

5 ISRN Matematical Pysics 5 U xl,m U l,m U l,m 2 U xl,m U l,m 4U l,m U l,m 2 U xl,m U l,m 4U l,m U l,m 2 U xl,m 2 ( 6 U 0 O 4), U xl,m 2 ( U 0 O ), U xl,m 2 ( U 0 O ). 2.0 Since te derivative values of S m x defined by 2.4, 2.7, 2.8, and 2.9 are not known at eac grid point x l,y m, we use te following approximations for te derivatives of S m x. Let M l,m U yyl,m D l U xl,m g l,m, M l,m U yyl,m D l U xl,m g l,m, M l,m U yyl,m D l U xl,m g l,m, U xl,m U l,m U l,m U xl,m U l,m U l,m 6 6 [M l,m 2M l,m ], [M l,m 2M l,m ], F l,m D l U xl,m g l,m, 2. Û xl,m U xl,m 2 F l,m D l U xl,m g l,m, ] [F l,m F l,m ] [U yyl,m U yyl,m, 2 F l,m D l Û xl,m g l,m. Ten at eac grid point x l,y m, a cubic spline finite difference metod of Numerov type wit accuracy of O k 2 4 for te solution of differential equation. may be written as λ 2 δxu 2 l,m k2 ] [U yyl,m U yyl,m 0U yyl,m 2 ] [D k2 l U xl,m D l U xl,m 0D l Û xl,m k2 [ ] gl,m g l,m 0g l,m Tl,m, were te local truncation error T l,m O k 4 k 2 4. Note tat te metod 2.2 is of O k 2 4 for te numerical solution of.. However, te metod 2.2 fails to compute at l, wen D x and/or g x, y contains te singular terms like /x, /x 2, and so fort. For example, if D x /x, tis implies D l /x l, wic cannot be evaluated at l since x 0 0.

6 6 ISRN Matematical Pysics We modify te metod 2.2 in suc a manner, so tat te metod retains its order and accuracy everywere in te vicinity of te singularity. We use te following approximation: D l D l D 0 2 ( 2 D 20 O ), D l D l D 0 2 ( 2 D 20 O ), g l,m g l,m g 0 2 ( 2 g 20 O ), g l,m g l,m g 0 2 ( 2 g 20 O ). 2. Substituting te approximations 2. into 2.2 and neglecting iger-order terms and local truncation error, we get ( ) 2 δx 2 δyu 2 l,m 2λ 2 δxu 2 l,m λ2 ( ) (2μx 2D 00 2 ) D 20 δ x ul,m λ 2 2( ) 2 D 0 D 00 δ 2 2 xu l,m 2 D 0 δyu 2 l,m D 00 2 (δ 2 y2μ x δ x ) u l,m k 2[ 2g 00 2( g 20 D 0 g 00 D 00 g 0 ) ], 2.4 were μ x u l /2 u l /2 u l /2 and δ x u l u l /2 u l /2. Note tat te cubic spline metod 2.4 is of O k 2 4 for te numerical solution of elliptic equation., wic is also free from te terms /x l± and ence can be computed for l N, m M.. Iterative Metod Now consider te convection-diffusion equation 2 u x 2 u u β, 0 <x,y<,. 2 y2 x were β>0 is a constant, and magnitude of β determines te ratio of convection to diffusion term. Substituting D x β and g x, y 0 into te difference sceme 2.4 and simplifying, we obtain a nine-point cubic spline difference sceme of O k 2 4 accuracy for te solution of te convection-diffusion equation. given by α 0 u l,m α u l,m α 2 u l,m α u l,m α 4 u l,m α 5 u l,m α 6 u l,m α 7 u l,m α 8 u l,m 0,.2

7 ISRN Matematical Pysics 7 were R β/2 is called te cell Reynold number, and te coefficients α j,j 0,, 2,...,8are defined by ( ( ) α λ 2 ), R2 α 2λ 2 R R2 2 R, ( ) α 2 2λ 2 R R2 2 R, α α 4 0,. α 5 α 6 R, α 7 α 8 R. Te sceme.2 may be written in matrix form Au b,.4 were A is a square matrix of order NM NM, u is te solution vector, and b is te rigt and side vector consisting of boundary values. Te coefficient matrix A as a block tridiagonal structure A tri L, D, U, witte submatrices L, D,and U given by L tri α 8,α 4,α 6 U, D tri α 2,α 0,α..5 We focus on line stationary iterative metods for solving te linear system.5. Te coefficient matrix A can be written as A D L U, were D is block tri-diagonal matrix of A, L is strictly block lower triangular part, and U is strictly block upper triangular part of matrix A. Te iteration matrices of te block Jacobi and block Gauss-Seidel metods are described by G J D L U, G GS D L U..6 Te matrix A as block tri-diagonal form and ence is block consistently ordered see Varga 22. It can be verified tat α 0 > 0andα j < 0forj, 2,...,8 provided R <. One can also verify tat α 0 8 α i,.7 i wic implies tat A is weakly diagonally dominant. Since A is reducible, we conclude tat it is also an M-matrix see Varga 22.

8 8 ISRN Matematical Pysics Applying te Jacobi iterative metod to te sceme.2, we get te iterative sceme [ ( ) ] R u k 2λ 2 R R2 2 R u k R u k l,m l,m l,m [ ( )] 0u k 20 24λ 2 R2 u k 0u k R u k l,m l,m l,m l,m ( ) ] [2λ 2 R R2 2 R u k R u k 0. l,m l,m.8 Te propagating factors for te Jacobi iterative metods are given by ( ) ξ j 5 cos πk 6λ 2 R R2 R cos kπ 5 6λ 2 R 2 / ( ) 6λ 2 R R2 R cos kπ cos π,.9 were M, N k. Consequently, te spectral radii ρ of te Jacobi and Gauss- Seidel matrices are related by ρ G GS ρ G J 2..0 Hence, te associated iteration u k Gu k c. converges for any initial guess, were G is eiter Jacobi or Gauss-Seidel iteration matrix. Te Jacobi and Gauss-Seidel splitting are regular for bot te line and point versions, and ence, tey converge for any initial guess. 4. Numerical Results If we replace te partial derivatives in. by te central difference approximations at te grid point l, m, we obtain a central difference sceme CDS wic is of O k 2 2.Wenow solve te following two bencmark problems wose exact solutions are known. Te rigt and side omogeneous function and boundary conditions may be obtained by using te exact solution as a test procedure. We use block Gauss-Seidel iterative metod see to solve te proposed sceme 2.4. In all cases, we ave considered u 0 0 as te initial guess, and te iterations were stopped wen te absolute error tolerance u k u k 0 2 was acieved. In all cases, we ave calculated maximum absolute errors l -norm for different grid sizes. All computations were performed using double-precision aritmetic.

9 ISRN Matematical Pysics 9 Table : Example 4.: Te maximum absolute errors., k 20, 0 40, 20 20, 40 80, 40 40, 80 Proposed O k 2 4 metod O k 2 2 metod β 0 β 50 β 0 β E E E E E E E E E E E E E 0 0.5E E E E E E E 0 Table 2: Example 4.: Te maximum absolute errors k/ Proposed O k 2 4 metod O k 4 4 metod discussed in 8 β 5 β 0 β 5 β 5 β 0 β E E E E E E E E E E E E E E E E E E 0 Example 4. Convection-Diffusion Equation. Te problem is to solve. in te solution region 0 <x,y< wose exact solution is given by u ( x, y ) βx/2 sin πy[ ] e 2e β/2 sin σx sin σ x, 4. sin σ were σ 2 π 2 β 2 /4, β>0. Te maximum absolute errors for u are tabulated in Tables and 2.

10 0 ISRN Matematical Pysics Table : Example 4.2: Te maximum absolute errors., k Proposed O k 2 4 metod O k 2 2 metod 20, E 0 0.E 02 40, 20 20, 40 80, 40 40, E E E E E 0 0.2E E E 0 Table 4: Example 4.2: Te maximum absolute errors k/ Proposed O k 2 4 metod O k 4 4 metod discussed in E E E E E E 04 Example 4.2 Poisson s Equation in Polar Cylindrical Coordinates. Consider te following: 2 u r 2 u 2 z u f r, z, 0 <r,z< r r Te exact solution is given by u r, z r 2 sin r cos z. Te maximum absolute errors for u are tabulated in Tables and Concluding Remarks Te available numerical metods based on spline approximations for te numerical solution of 2D Poisson s equation are of O k 2 2 accurate, wic require nine grid points. In tis paper, using te same number of grid points, we ave discussed a new stable compact nine point cubic spline finite difference metod of O k 2 4 accuracy for te solution of Poisson s equation in polar cylindrical coordinates. For a fixed parameter γ k/ 2, te proposed metod beaves like a fourt-order metod, wic is exibited from te computed results.

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