Cubic Spline Iterative Method for Poisson’s Equation in

Cylindrical Polar Coordinates

R. K. Mohanty1, Rajive Kumar

2, Vijay Dahiya

2

1 Department of Mathematics

Faculty of Mathematical Sciences

University of Delhi, Delhi-110 007, INDIA

E-mail: rmohanty@maths.du.ac.in

2Department of Mathematics

Deenbandhu Chhotu Ram University of Science & Technology

Murthal 131039, INDIA

E-mail: rajeev_kansal@yahoo.com; vijay_15dahiya@yahoo.com

Abstract: In this article, using non-polynomial cubic spline approximation in x- and finite

difference in y- direction, we discuss a numerical approximation of 2 4( )O k h+ for the solutions

of diffusion-convection equation, where k>0 and h>0 are grid sizes in y- and x- coordinates,

respectively. We also extend our technique to polar coordinate system and obtained high order

numerical scheme for Poisson’s equation in cylindrical polar coordinates. Iterative method of the

proposed method is discussed and numerical examples are given in support of the theoretical

results.

Keywords: Cubic spline approximation; Diffusion-convection equations; Poisson.s equation;

Singular coefficients; Cylindrical polar coordinates; Maximum absolute errors.

AMS (2010) Classifications: 65N06

1. Introduction

We consider the two-dimensional elliptic equation of the form

2 2

2 2= ( ) ( , ), ( , )

u u uD x g x y x y

x y x

∂ ∂ ∂+ + ∈Ω

∂ ∂ ∂ (1)

The Dirichlet boundary conditions are given by

( , ) ( , ), ( , )u x y g x y x y= ∈ Γ (2)

where ( , ) 0 , 1x y x yΩ = < < is the solution domain and Γ is its boundary. For g(x,y)= 0 and

( )D x β= the above equation represent diffusion convention equation. For 1( )x

D x −= the above

equations represent the Poisson’s equation with singular coefficients in rectangular coordinates.

Similarly, for 1( )x

D x −= and replacing the variables x, y by r, z, we obtain a Poisson’s equation

in cylindrical polar coordinates. We shall assume that the boundary conditions are given with

sufficient smoothness to maintain the order of accuracy of the difference scheme and spline

functions under consideration.

In this paper, we are interested to discuss a new approximation based on cubic spline

polynomial for the solution of elliptic equation (1). In many practical problems, coefficients of

the second derivatives term are small compared to the coefficients of the first derivatives term.

These problems are called singular perturbation problems. Singularly perturbed elliptic boundary

value problems are mathematically models of diffusion-convections process or related physical

phenomenon. The diffusion term is the term involving the second order derivative and

convective term is that involving the first order derivative. During last three decades several

numerical schemes for the solution of elliptic partial differential equations have been developed

by many researchers. First Lynch and Rice [1] have discussed high accuracy finite difference

approximations to the solutions of elliptic partial differential equations. Boisvert [2] have

discussed a class of high order accurate discretization for the elliptic boundary value problems.

Yavneh [3] have reported the analysis of fourth order compact scheme for convection diffusion

equation. A fourth order difference method for elliptic equations with non linear first derivative

terms has been discussed by Jain et al [4, 5]. In 1997, Mohanty [6] have derived order h4

difference method for a class of 2D elliptic boundary value problems with singular coefficients.

A new discretization method of order four for the numerical solution of 2-D non linear elliptic

partial differential equations has been studied by Mohanty et al [7-9]. The use of cubic spline

polynomial and its approximation plays an important role for the formation of stable numerical

methods. In the past many authors (see [10-12]) have studied and analysed the use of cubic

splines in the solution of linear two point boundary value problems. In 1983, Jain and Aziz [13]

have developed a new method based on cubic spline approximations for the solution of two point

nonlinear boundary value problems. Later, Al-Said [14-15] has discussed spline methods for

solving a system of second order boundary value problems. Khan [16] has introduced parametric

cubic spline approach for solving second order ordinary differential equations. Mohanty et al

[17-18] have also reported a fourth-order accurate cubic spline alternating group explicit method

for non-linear singular two-point boundary value problems. Recently, Rashidinia et al [19] have

proposed a new cubic spline technique for two point boundary value problems. Most recently,

Mohanty and Dahiya [20], and Mohanty et al [21] have used cubic spline polynomials and

developed high order stable numerical methods for the solution of one space dimensional

parabolic and hyperbolic partial differential equations. To the authors knowledge no high order

method using cubic spline polynomial for the solution of 2D elliptic differential equations (1) has

been discussed in the literature so far. In this paper, using nine point compact cell (see Fig.1), we

discuss a new compact cubic spline finite difference method of accuracy two in y- and four in x-

coordinates for the solution of elliptic differential equation (1). In next section, we discuss the

derivation of the proposed cubic spline method. It has been experienced in the past that the cubic

spline solutions for the Poisson’s equation in polar coordinates usually deteriorate in the vicinity

of the singularity. We overcome this difficulty by modifying the method in such a way that the

solution retains its order and accuracy everywhere in the vicinity of the singularity. In section 3,

we discuss an iterative method. In section 4, we compare the computed results with the results

obtained by the method discussed in [8]. Concluding remarks are given in section 5.

(, ) (, ) (, )

(, ) (, ) (, )

ℎ

(, ) (, ) (, )

Fig.1: 9-point Computational Network

2. The approximation based on cubic spline polynomial

We consider our region of interest, a rectangular domain Ω = 0, 1 × 0, 1 . We choose grid

spacing ℎ > 0 and > 0 in the directions - and - respectively, so that the mesh points

(, ) denoted by (,) are defined as = ℎ and = ; = 0,1, … , + 1; =0,1, … , + 1, where and are positive integers such that (N+1)h = 1 and ( + 1) = 1.

The mesh ratio parameter is denoted by ( / )k hλ = . The notations , ,andl m l mu U are used for the

discrete approximation and the exact value of ( , )u x y at the grid point ( , )l mx y , respectively.

At the grid point ( , )l mx y , we denote

a b

ab a b

l m

WW

x y

+∂=

∂ ∂, W = U, D and g (3)

Let ( )mS x be the cubic spline interpolating polynomial of the function ( , )mu x y between the

grid point 1( , )l mx y− and ( , )l mx y , and is given by

3 3 2 2

1 11, , 1, 1, , ,

( ) ( )( ) ,

6 6 6 6

l l l lm l m l m l m l m l m l m

x x x x x x x xh hS x M M U M U M

h h h h

− −− − −

− − − − = + + − + −

1 ; 1,2,..., 1 ; 0,1,2,..., 1l lx x x l N m M− ≤ ≤ = + = + (4)

which satisfies at mth-line parallel to x-axis the following properties:

(i) ( )mS x coincides with a polynomial of degree three on each 1[ , ],l lx x−

1, 2,..., 1; 1, 2,... ,l N m M= + =

(ii)

2( ) [0,1],mS x C∈ and

(iii) ,( ) , 0,1, 2,..., 1; 1, 2,...,m l l mS x U l N m M= = + = .

The derivatives of cubic spline function ( )mS x are given by

2 2

, 1,11, , , 1,

( ) ( )( )

2 2 6

l m l ml lm l m l m l m l m

U Ux x x x hS x M M M M

h h h

−−− −

−− − −′ = + + − − , (5)

11, ,

( ) ( )( ) l l

m l m l m

x x x xS x M M

h h

−−

− −′′ = + (6)

where

, ,, ,,

( ) , 0,1,2,..., 1; 1,2,...,l m m l xx yy x l ml m l ml ml

M S x U U U f l N j Jx

α′′= = = − − = + = , (7)

, 1,

, 1, , 1,( ) 2 ,

6

l m l m

l m m l x l m l m l ll m

U U hm S x U M M x x x

h

−

− −

−′ = = = + + ≤ ≤ , (8)

and replacing h by ‘–h’, we get

1, ,

, 1, , 1,( ) 2 ,

6

l m l m

l m m l x l m l m l ll m

U U hm S x U M M x x x

h

+

+ −

−′ = = = − + ≤ ≤ , (9)

Combining (8) and (9), we obtain

1, 1,

, 1, 1,,( )

2 12

l m l m

l m m l x l m l ml m

U U hm S x U M M

h

+ −

+ −

−′ = = = − − , (10)

Further, from (8), we have

1, ,

1, 1 . 1,1,( ) 2

6

l m l m

l m m l x l m l ml m

U U hm S x U M M

h

+

+ + ++

−′ = = = + + , (11)

and from (9), we have

, 1,

1, 1 , 1,1,( ) 2

6

l m l m

l m m l x l m l ml m

U U hm S x U M M

h

−

− − −−

−′ = = = − + , (12)

We consider the following approximations:

( ) 2 2

, 1 , , 1, ,2 /( ) ( )yy l m l m l m yyl m l m

U U U U k U O k+ −= − + = + , (13.1)

( ) 2 2

1, 1 1, 1, 11, 1,2 /( ) ( )yy l m l m l m yyl m l m

U U U U k U O k+ + + + −+ += − + = + , (13.2)

( ) 2 2

1, 1 1, 1, 11, 1,2 /( ) ( )yy l m l m l m yyl m l m

U U U U k U O k− + − − −− −= − + = + , (13.3)

( )2

4

1, 1, 30, ,/(2 ) ( )

6x l m l m xl m l m

hU U U h U U O h+ −= − = + + , (14.1)

( )2

3

1, , 1, 301, 1,3 4 /(2 ) ( )

3x l m l m l m xl m l m

hU U U U h U U O h+ −+ +

= − + = − + , (14.2)

( )2

3

1, , 1, 301, 1,3 4 /(2 ) ( )

3x l m l m l m xl m l m

hU U U U h U U O h− +− −

= − + − = − + , (14.3)

Since the derivative values of ( )mS x defined by (7), (10), (11) and (12) are not known at

each grid point ( , )l mx y , we use the following approximations for the derivatives of ( )mS x .

Let

, ,,,l m yy l x l ml ml m

M U DU g= − + + , (15.1)

1, 1 1,1,1,l m yy l x l ml ml mM U D U g+ + +++

= − + + , (15.2)

1, 1 1,1,1,l m yy l x l ml ml m

M U D U g− − −−−= − + + , (15.3)

1, ,

, 1,1,2

6

l m l m

x l m l ml m

U U hU M M

h

+

++

− = + + , (16.1)

, 1,

, 1,1,2

6

l m l m

x l m l ml m

U U hU M M

h

−

−−

− = − + , (16.2)

1, 1 1,1,l m l x l ml mF D U g+ + ++

= + , (16.3)

1, 1 1,1,l m l x l ml mF D U g− − −−

= + (16.4)

1, 1,, , 1, 1,

ˆ12 12

x x l m l m yy yyl m l m l m l m

h hU U F F U U+ − + −

= − − + − (16.5)

, ,,ˆ ˆl m l x l ml m

F DU g= + (16.6)

Then at each grid point ( , )l mx y , a cubic spline finite difference method of Numerov type

with accuracy of 2 4( )O k h+ for the solution of differential equation (1) may be written as

2 2

2 2

, 1 11, 1, ,1, 1, ,

ˆ10 1012 12

x l m yy yy yy l x l x l xl m l m l ml m l m l m

k kU U U U D U D U DUλ δ + −+ −+ −

+ + + = + +

2

1, 1, ,1012

l m l m l m

kg g g+ − + + + + ,l mT . (17)

where the local truncation error 4 2 4

, ( )l mT O k k h= + . Note that the method (17) is of

2 4( )O k h+ for the numerical solution of equation (1). However the method (17) fail to

compute at 1l = , when D(x) and/or g(x,y) contains the singular terms like 1x

, 2

1

x, etc. For

example, if D(x) = 1x, this implies

1

11 ll x

D−− = , which cannot be evaluated at l=1 (since x0=0).

We modify the method (17) in such a manner, so that the method retains its order and

accuracy everywhere in the vicinity of the singularity.

We use the following approximation:

2 3

1, , 10 202( )h

l m l mD D hD D O h+ = + + + , (18.1)

2 3

1, , 10 202( )h

l m l mD D hD D O h− = − + − (18.2)

2 3

1, , 10 202( )h

l m l mg g hg g O h+ = + + + , (18.3)

2 3

1, , 10 202( )h

l m l mg g hg g O h− = − + − , (18.4)

Substituting the approximations (18.1)-(18.4) into (17) and neglecting higher order

terms and local truncation error, we get

( )( ) ( )2

22 2 2 2 2 2 2 2

, , 00 20 , 10 00 ,(12 ) 12 12 22

x y l m x l m x x l m x l m

hu u D h D u h D D u

λδ δ λ δ µ δ λ δ+ + = + + −

( ) ( )2 2 2 2 20010 , , 00 20 10 00 00 102 12

2y l m y x x l m

hDh D u u k g h g D g D gδ δ µ δ − + + + + − (19)

where ( )1 12 2

12x l l l

u u uµ+ −

= + and ( )1 12 2

x l l lu u uδ

+ −= − . Note that, the cubic spline method

(19) is of 2 4( )O k h+ for the numerical solution of elliptic equation equation (1), which is

also free from the terms 1

1

lx ±, hence can be computed for l=1(1)N, m= 1(1)M.

3. Iterative Method

Now consider the convection-diffusion equation

+

= !

, 0<x,y <1 (20)

where ! > 0 is a constant and magnitude of ! determines the ratio of convection to diffusion

term. Substituting "() = !and&(, ) = 0 in the difference scheme (19) and simplifying, we

obtain a nine point cubic spline difference scheme of 2 4( )O k h+ accuracy for the solution of the

convection-diffusion equation (20) given by

0 , 1 1, 2 1, 3 , 1 4 , 1l m l m l m l m l mu u u u uα α α α α+ − + −+ + + + 5 1, 1 6 1, 1 7 1, 1 8 1, 1 0l m l m l m l mu u u uα α α α+ + + − − + − −+ + + + = (21)

where2

hR

β= is called the cell Reynold number and the coefficients jα , j = 0,1,2,…,8 are

defined by

22

0 320 24 (1 )Rα λ= + + ,

22

1 312 (1 ) 2(1 )RR Rα λ= − − + + − ,

22

2 312 (1 ) 2(1 )RR Rα λ= − + + + + ,

3 4 10α α= = − , 5 6 (1 )Rα α= = − − , 7 8 (1 )Rα α= = − + .

The scheme (21) may be written in matrix form

'( = ) (22)

where ' is a square matrix of order × , ( is a solution vector and ) is the right hand side

vector consisting of boundary values.

The coefficient matrix ' has a block tri-diagonal structure' = *+,−.,/,−0, with the sub

matrices –L, D and -U given by

-L 8 4 6[ , , ]tri α α α= = −0 , D= 2 0 1[ , , ]tri α α α (23)

We focus on line stationary iterative methods for solving the linear system (23). The

coefficient matrix ' can be written as ' = / − . − 0, where / is block tri-diagonal matrix of

A, −. is strictly block lower triangular part and −0 is strictly block upper triangular part of

matrix ' . The iteration matrices of the block Jacobi and block Gauss-Seidel methods are

described by

12 = /(. + 0) and 134 = (/ − .)0 (24)

The matrix ' has block tri-diagonal form and hence is block consistently ordered (see Varga

[22] ).

It can be verified that 0α >0 and jα <0 for 5 = 1,2, . . ,8 provided 1R < . One can also verify that

8

0

1

i

i

α α=

=∑ (25)

which implies that ' is weakly diagonally dominant. Since ' is reducible, we conclude that it is

an 9-matrix (see Varga [22]).

Applying the Jacobi iterative method to the scheme (21), we get the iterative scheme

1, 1 1, 1, 1

2( ) 2 ( ) ( )(1 ) 12 (1 ) 2(1 ) (1 )

3l m l m l m

k k kRR u R R u R uλ

+ + + + −

− + − + − − + −

, 1 , , 1 1, 1

2( ) 2 ( 1) ( ) ( )10 [20 24 (1 )] 10 (1 )

3l m l m l m l m

k k k kRu u u R uλ

+ − − +

++ − + + + + +

( )1, 1, 1

22 ( ) ( )12 1 2 1 (1 ) 0

3 l m l m

k kRR R u R uλ

− − −

+ + + − + + + =

. (26)

The propagating factors for the Jacobi iterative methods is given by

2

2

22

15cos( ) 6 1 (1 )(1 cos )

35 6 1

3

j

Rk p R R k

Rp

ξ π π = + + + − + − + +

2

26 1 (1 )(1 cos ) cos3

Rp R R k hπ π

× − + − − −

(27)

where (M+1)h=1, (N+1)k = 1. Consequently, the spectral radii : of the Jacobi and Gauss-Seidel

matrices are related by

:(134) = :(12); (28)

Hence, the associated iteration

((<) = 1((<) + = (29)

converges for any initial guess, where G is either Jacobi or Gauss-Seidel iteration matrix. The

Jacobi and Gauss-Seidel splitting are regular for both the line and point versions, and hence they

converge for any initial guess.

4. Numerical Results

If we replace the partial derivatives in equation (1) by the central difference approximations at

the grid point ( , )l m , we obtain a central difference scheme (CDS) which is of 2 2( )O k h+ . We now

solve the following two benchmark problems whose exact solutions are known. The right-hand side

hogrneous function and boundary conditions may be obtained by using the exact solution as a test

procedure. We use block Gauss-Seidel iterative method (See [22-25]) to solve the proposed scheme

(19). In all cases, we have considered (0) =u 0 as the initial guess and the iterations were stopped

when the absolute error tolerance ( 1) ( ) 1210k k+ −− ≤u u was achieved. In all cases, we have calculated

maximum absolute errors ( l∞ -norm) for different grid sizes. All computations were performed using

double precision arithmetic.

Example 1 (Convection-diffusion equation)

The problem is to solve (20) in the solution region 0 < , < 1 whose exact solution is

?(, ) = @AB CDEF CDEGH I2@JA K,LℎM + K,LℎM(1 − )N , where M; = O; + P

Q and 0β > .

The maximum absolute errors for ? are tabulated in Table 1.1 and Table 1.2.

Example 2 (Poisson’s equation in polar cylindrical coordinates)

2 2

2 2

1( , )

u u uf r z

r z r r

∂ ∂ ∂+ + =

∂ ∂ ∂, 0<r, z<1 (30)

The exact solutions are given by u(r, z) = r2 sinhr coshz. The maximum absolute errors for ?

are tabulated in Table 2.1 and Table 2.2.

5. Concluding remarks

Available numerical methods based on spline approximations for the numerical solution of

2D Poisson’s equation are of 2 2( )O k h+ accurate, which require nine grid points. In this article,

using the same number of grid points , we have discussed a new stable compact nine point cubic

spline finite difference method of 2 4( )O k h+ accuracy for the solution of Poisson’s equation in

polar cylindrical coordinates. For a fixed parameter 2

k

hγ = , the proposed method behaves like a

fourth order method, which is exhibited from the computed results.

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Table 1.1

Example 1: The maximum absolute errors

(h, k) Proposed

2 4( )O k h+ - method 2 2( )O k h+ - method

! = 10 ! = 50 ! = 10 ! = 50

( )1 120 10

,

0.2645E-02

0.1982E-01

0.9212E-02

0.2240E+00

( )1 140 20

,

0.6560E-03

0.1823E-02

0.2279E-02

0.6511E-01

( )1 120 40

,

0.2167E-03

0.1875E-01

0.1125E-01

0.2253E+00

( )1 180 40

,

0.1635E-03

0.1511E-03

0.5682E-03

0.1416E-01

( )1 140 80

,

0.4352E-04

0.1604E-02

0.2755E-02

0.6534E-01

Table 1.2

Example 1: The maximum absolute errors 2( 64)k

h=

h

Proposed 2 4( )O k h+ - method 4 4( )O k h+ - method discussed in [8]

!=5 !=10 !=15 !=5 !=10 !=15

1

16

0.1808E-01

0.1636E-01

0.1494E-01

0.4408E-01

0.4212E-01

0.4018E-01

132

0.1129E-02

0.1026E-02

0.9396E-03

0.2692E-02

0.2554E-02

0.2482E-02

1

64

0.7054E-04

0.6419E-04

0.5881E-04

0.1645E-03

0.1566E-03

0.1512E-03

Table 2.1

Example 2: The maximum absolute errors

(h, k) Proposed

2 4( )O k h+ - method 2 2( )O k h+ - method

( )1 120 10

,

0.3986E-03

0.1133E-02

( )1 140 20

,

0.1089E-03

0.3003E-03

( )1 120 40

,

0.4117E-03

0.1132E-02

( )1 180 40

,

0.2893E-04

0.7836E-04

( )1 140 80

,

0.1116E-03

0.2984E-03

Table 2.2

Example 2: The maximum absolute errors 2( 64)k

h=

h

Proposed 2 4( )O k h+ - method 4 4( )O k h+ - method discussed

in [8]

1

16

0.2711E-02

0.4842E-02

132

0.1654E-03

0.2988E-03

1

64

0.1016E-04

0.1811E-04