EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 3, 2021, 905-914
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Group Splitting with SOR/AOR Methods for Solving
Boundary Value Problems: A Computational

Comparison
Abdulkafi Mohammed Saeed,1,∗, Najah Mohammad AL-harbi2
1 Department of Mathematics, College of Science, Qassim University, Buraydah,
Kingdom of Saudi Arabia
2 Department of Mathematics, College of Science and Arts in Uglat Asugour,
Qassim University, Buraydah, Kingdom of Saudi Arabia

Abstract. Many researchers are working on the explicit group methods as the alternative methods
for solving several boundary value problems. These methods have been shown to be much faster
than the other point iterative methods in solving the elliptic partial differential equations (EPDEs),
which is due to the formers’ overall lower computational complexities. This paper is concerned
with the application of a suitable Explicit Group (EG) iterative method for solving EPDEs. This
study will compare several iterative methods such that S5-point-SOR,4 Point-EGSOR, 5S-point-
AOR, and 4 Point-EGAOR. Numerical experiments were carried out to confirm our results by
using MATLAB software. The results reveal that 4 Point-EGAOR is the most superior method
among these methods.
2020 Mathematics Subject Classifications: NO AMS CODE
Key Words and Phrases: Explicit Group (EG) method, Poisson equation, Successive Over-
Relaxation (SOR), Accelerated Over-Relaxation (AOR

1. Introduction

Many physical phenomena in engineering, fluid dynamics, and static field problems,
particularly in the electromagnetic field and the incompressible potential flow field, are
described by partial differential equations (PDEs) [2, 5, 9, 13]. It has been known that
using finite difference schemes to discretize PDEs typically results in a broad and sparse
system of linear equations. Several studies were proposed on different iterative methods
for solving any linear system of equations to speed up the convergence rate due to the
wide range of linear systems. Therefore, Yousif and Evans [13] has also pioneered 4-point
block iterative methods for solving large linear structures using the Explicit Group (EG)
iterative method. The iterative point method proposed by Hadjidimos [5], together with

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.4031

Email addresses: abdulkafi.ahmed@qu.edu.sa (A.M. Saeed), njah.alharbi@qu.edu.sa (N. AL-harbi)

http://www.ejpam.com 905 © 2021 EJPAM All rights reserved.



A.M. Saeed, N. AL-harbi / Eur. J. Pure Appl. Math, 14 (3) (2021), 905-914 906

two accelerated parameters called the Accelerated Over-Relaxation (AOR) method, has
been developed. In this work, the formulation of efficient explicit group by combining the
EG iterative method with the AOR method will be presented for the elliptic partial differ-
ential equations solution. The new applications of the EG AOR method will be conducted
by MATLAB software .
We shall begin with the presentation of several preliminary relevant theorems and lem-
mas, which are needed for the proof of the convergence properties of the solution of the
mentioned iterative methods. The spectral radius of a matrix is denoted byρ(.), which is
defined as the largest of the moduli of the eigenvalues of the iteration matrix, which play
an important role to study these convergence properties.

Theorem 1. [11] If A = M −N is a regular splitting of the matrix A and A−1 ≥ 0, then

ρ(M−1N) = ρ(A−1N)
1 + ρ(A−1N) < 1.

Thus, an iterative method with coefficient matrix M−1N is convergent for any initial vector
x(0).

An accurate analysis of convergence properties of the SOR method is possible if the
matrix A is consistently ordered in the following sense (see [10]).

Definition 1. A matrix A is a generalized (q,r)-consistently ordered matrix (a GCO(q,r)-
matrix) if ∆ = det(αqE + α−rF − kD) is independent of α for all α 6= 0 and for all k.
Here, D = diag A and E and F are strictly lower and strictly upper triangular matrices,
respectively, such that: A = D − E − F .

Definition 2. [10] A matrix A of the form (3.2) is said to be generally consistently
ordered (π,q,r) or simply GCO (π,q,r), where q and r are positive integers, if for the
partitioning π of A the diagonal submatrices A(ii), i = 1, 2, .., p(≥ 2), are non-singular,
and the eigenvalues of

BJ(α) = αrL+ α−qU (1)
are indpendent of α, for all α 6= 0, where L and U are strict blocks lower and upper
triangular parts of A, respectively.
For any matrix C = (cij)in 6⊂ni,ni

π,p , let |C| denote the block matrix in 6⊂ni,ni
π,p with entries∣∣∣C(ij)

∣∣∣. Given the matrix
BJ = L+ U, (2)

then µ̄ denotes the spectral radius of the matrix

|BJ | = |L+ U | , (3)

such that µ := ρ(|BJ |).

Lemma 1. [10] Let |BJ | of (4.3) be a GCO (q,r)-matrix and p = q + r. Then for any
real nonnegative constant α, β, and γ with γ 6= 0 satisfying: αrβqµ ≺ γp, the matrix
A′′ := γI − α |L| − β |U | is such that An−1 ≥ 0.



A.M. Saeed, N. AL-harbi / Eur. J. Pure Appl. Math, 14 (3) (2021), 905-914 907

Lemma 2. [8] Suppose A = I–L–U is a GCO(π,q,r), where –L and –U are strictly lower
and upper triangular matrices, respectively. Let B`w be the block iteration matrix of the
SOR method given by (2.2). If 0 < w < 2, then the block SOR method converges, i.e.
ρ(B`w) < 1.

Theorem 2. [8] Suppose A = I–L–U is a GCO(π,q,r), where –L and –U are strictly lower
and upper triangular matrices, respectively. Let B`w and B̃`w be the iteration matrices of
the SOR method given by (2.2) and (2.6) respectively. If 0 < ω < 2, then

(i) ρ(B̃`w) < ρ(B`w) if ρ(B`w) < 1

(ii) ρ(B̃`w) = ρ(B`w) if ρ(B`w) = 1

(iii) ρ(B̃`w) > ρ(B`w) if ρ(B`w) > 1 .

Theorem 3. [3] Let the matrix A = D–L–U be a pcyclic consistently ordered one with
non-singular diagonal submatrices Ajj, j = 1, 2, ..p. If all the eigenvalues of the pth power
of the associated Jacobi matrix T

(
T = I −D−1

A A,DA = diag (A11, A22, . . . , APP )
)

are
real and nonnegative and 0 ≤ (ρ(T )) < 1 then with ω̂p defined by (ρω)P = PP (P − 1)1−P (ω − 1),
when ρ = ρ(T ), it is ρ (Lω) > ρ

(
Lω̂p

)
= (p− 1) (ω̂p − 1) for all ω 6= ω̂p.

This paper is organized as follows: In section 2, we describe the five-point finite dif-
ference (SOR) and EG(SOR). In Section 3, we provide an overview of the iterative five-
point differential formula (AOR) and EG(AOR) for the resolution of Poisson’s Equation.
The numerical findings are presented in Section 4 to demonstrate the EGAOR method’s
efficiency. The final observations and conclusions are given in Section 5.

2. Explict Group SOR Method

The SOR (successive over-relaxation) method is a standard iterative method for solving
linear systems of equations, particularly large sparse systems arising from partial dif-
ferential equations. In the SOR method, the parameter ω must be determined where
a suitable value of ω can lead to dramatic convergence improvements. Therefore, the
method of SOR became popular and was chosen as a method in computer codes to solve
major practical problems such as weather prediction and diffusion of the nuclear reactor
[1, 4, 7, 12].
Consider the elliptic equation that is linear and self-adjoint,

a
∂2U

∂x2 + b
∂2U

∂x∂y
+ c

∂2U

∂y2 + d
∂U

∂x
+ e

∂U

∂y
+ fU + g = 0,

where a, b, c, d, e, f and g may be a function of the independent variables x and y of the
dependent variables U . Elliptic equations describe problems in a closed region. The
Poisson and Laplace equations are examples of the elliptic equations. Only boundary



A.M. Saeed, N. AL-harbi / Eur. J. Pure Appl. Math, 14 (3) (2021), 905-914 908

conditions are considered because the dependent variable does not depend on time. In
this work, we will consider the Poisson equation.

∇2u = ∂2u

∂x2 + ∂2u

∂y2 = f(x, y), (x, y) ∈ Ω (4)

with specific Dirichlet boundary conditions

u(x, y) = g(x, y), (x, y) ∈ ∂ Ω.

At this point, (xi, yj) the Poisson equation (4) may in several ways be approximated.
Assume u(xi, yj) is a function of independent variables x and y. The grid lines are equally
spaced in both directions, x, and y. then, we can set h = ∆x = ∆y where the grid
point P is (xi, yj) with xi = i∆x,i = 1, 2, ..., n − 1, and yi = i∆x,j = 1, 2, ..., n − 1. The
basic five-point formula is obtained by discretizing Eq. (4) and using the finite difference
approximation.

ui+1,j + ui−1,j + ui,j+1 + ui,j−1 − 4ui,j = h2fij . (5)
If we use the standard five-point approximation scheme, SOR iterative method can be
written as equations (5)

u
(k+1)
ij = ω

4 (u(k)
i+1,j + u

(k)
i,j+1 + u

(k+1)
i−1,j + u

(k+1)
i,j−1 − h

2fij) + (1− ω)u(k)
ij . (6)

Each of equations (5), and (6) has local order truncation errors O(h2). In the (EG)
method, the solution domain is divided into groups of four-points. The mesh points are
grouped in blocks of four points, and the standard five-point equation is implemented at
each of these points, and this will result in a (4 × 4) system of equations,

4 −1 0 −1
−1 4 −1 0
0 −1 4 −1
−1 0 −1 4




uij
ui+1,j
ui+1,j+1
ui,j+1

 =


ui−1,j + ui,j−1 − h2fij

ui+2,j + ui+1,j−1 − h2fi+1j
ui+2,j+1 + ui+1,j+2 − h2fi+1j+1
ui−1,j+1 + ui,j+2 − h2fi,j+1

 . (7)

Equation (7) can be easily inverted to produce a four-point explicit group equation:
uij
ui+1,j
ui+1,j+1
ui,j+1

 = 1
24


7 2 1 2
2 7 2 1
1 2 7 2
2 1 2 7




ui−1,j + ui,j−1 − h2fij
ui+2,j + ui+1,j−1 − h2fi+1j

ui+2,j+1 + ui+1,j+2 − h2fi+1j+1
ui−1,j+1 + ui,j+2 − h2fi,j+1


whose each explicit equations can be written in the form

uij = 1
24[7r1 + 2(r2 + r4) + r3],

ui+1,j = 1
24[7r2 + 2(r1 + r3) + r4],

ui+1,j+1 = 1
24[7r3 + 2(r2 + r4) + r1],

ui,j+1 = 1
24[7r4 + 2(r1 + r3) + r2]

, (8)



A.M. Saeed, N. AL-harbi / Eur. J. Pure Appl. Math, 14 (3) (2021), 905-914 909

where

r1 = ui−1,j + ui,j−1 − h2fij r2 = ui+2,j + ui+1,j−1 − h2fi+1,j

r3 = ui+2,j+1 + ui+1,j+2 − h2fi+1,j+1 r4 = ui−1,j+1 + ui,j+2 − h2fi,j+1.

The approach uses the collection of equations to iteratively evaluate the solution in blocks
of four points around the entire solution domain (4). Until convergence is achieved, the
process is continuous.

3. Explicit Group AOR Method

The AOR method, Au = b, is a well-known iterative method for solving linear systems
of equations with two real parameters r and ω. When the two parameters involved in the
(AOR) method take on some particular values, it can be thought of as an extrapolation
of the (SOR) method [4, 5]. A matrix A can be decomposed as

A = D − L− U, (9)

D is a diagonal matrix, and L and U are strictly lower and upper triangular matrices.
The AOR iterative method can be written as:

u(k+1) = Lr,ωu
(k) + ω(D − rL)−1b (10)

where
Lr,ω = (I − rD−1L)−1[(1− ω)I + (ω − r)D−1L+ ωD−1U ].

According to equation (6) in section 2, the AOR iterative scheme for the standard five-
point can be written as we have to determine the expressions consist in the lower triangular
matrix . The values of these expressions have been computed. From equation (6), we have
already computed u(k+1)

i−1,j and u(k+1)
i,j−1 . So, we have to change u(k+1)

i−1,j and u(k+1)
i,j−1 to u(k)

i−1,j and

u
(k)
i,j−1. Then, add r(u(k+1)

i−1,j −u
(k)
i−1,j)

4 , and r(u(k+1)
i,j−1−u

(k)
i,j−1)

4 [6]. The coefficient is 1
4 , corresponds

to the equation (6).

u
(k+1)
ij = r(

u
(k+1)
i−1,j − u

(k)
i−1,j + u

(k+1)
i,j−1 − u

(k)
i,j−1)

4 ) + ω(
u

(k)
i−1,j + u

(k)
i+1,j + u

(k)
i,j−1 + u

(k)
i,j+1 − h2fij)

4 )

+ (1− ω)u(k)
ij .

(11)

Unlike SOR, however, the optimal values of r and ω for which the minimum number of
iterations is given are not generally specified. According to [5], the r is typically chosen
to be close to the ω value of the respective SOR; then, a certain ω range will be executed
by the numerical experiment.



A.M. Saeed, N. AL-harbi / Eur. J. Pure Appl. Math, 14 (3) (2021), 905-914 910

For the four-point group AOR method, we can use the following iterative scheme

u
(k+1)
ij = 1

4[ω(7b1 + s2 + b4) + r(7t1 + 2t2)] + (1− ω)u(k)
ij ,

u
(k+1)
i+1,j = 1

24[ω(7b2 + s1 + b3) + r(7c4 + 2t1 + c3)] + (1− ω)u(k)
i+1,j ,

u
(k+1)
i,j+1 = 1

24[ω(7b3 + s1 + b2) + r(7c3 + 2t1 + c4)] + (1− ω)u(k)
i,j+1,

u
(k+1)
i+1,j+1 = 1

24[ω(7b4 + s2 + b1) + r(2t2 + t1)] + (1− ω)u(k)
i,j+1 ,

(12)

where

b1 = u
(k)
i,j−1 + u

(k)
i−1,j − h

2fi,j , b2 = u
(k)
i+1,j−1 + u

(k)
i+2,j − h

2fi+1,j ,

b3 = u
(k)
i−1,j−1 + u

(k)
i,j+2 − h

2fi,j+1, b4 = u
(k)
i+1,j+2 + u

(k)
i+2,j+1 − h

2fi+1,j+1,

c1 = u
(k+1)
i−1,j + u

(k)
i−1,j , c2 = u

(k+1)
i,j−1 + u

(k)
i,j−1, c3 = u

(k+1)
i−1,j+1 + u

(k)
i−1,j+1,

c4 = u
(k+1)
i+1,j−1 + u

(k)
i+1,j−1, s1 = 2(b1 + b4), s2 = 2(b2 + b3),

t1 = c1 + c2, t2 = c3 + c4.

4. Numerical Experiments And Results

In order to compare the standard five-point and four-point group SOR iterative meth-
ods, some numerical experiments have been performed. These methods were implemented
to Poisson equation,

∇2u = ∂2u

∂x2 + ∂2u

∂y2 = (x2 + y2)exy, (13)

with u(x, 0) = u(0, y) = 1, u(x, 1) = ex, u(1, y) = ey, 0 ≤ x, y ≤ 1.
The exact solution for this problem is u(x, y) = exy. In this experimental work, we choose
the value of tolerance ε = 10−5 . The computer processing unit was Intel(R) Core(TM)
i7- 7500U CPU with a memory of 8 Gb and the software used to implement and generate
the results was MATLAB. We have computed the average absolute errors and record the
number of iterations for convergence for different sizes of grids. Due to the MATLAB
program’s use, a larger number of processors were used in this experiment 12 ,46 ,86 ,
106, 146, 186, 226, 350, and 426. Table 1 represents the comparison of S5 and 4-point
EGSOR, and Table 2 represents the comparison of S5 and 4-point EGAOR, where N the
number of squares, ω the SOR parameter, r the second parameter of AOR, k the number
of iterations, e the maximum errors and t represents CPU time. The convergence of
the iteration methods relies on the spectral radius, which is defined as the largest of the
moduli of the iteration matrix’s eigenvalues. It is stated and proven that a linear system
with a smaller spectral radius will have a better convergence rate [6]. Therefore, the
spectral radius of the resulted iteration matrices was calculated through these numerical
experiments to verify that the superior iterative method has the smallest spectral radius.
We can also observe that the resulted spectral radius of all methods used is strictly less



A.M. Saeed, N. AL-harbi / Eur. J. Pure Appl. Math, 14 (3) (2021), 905-914 911

than 1 which coincides with the theoretical parts and previous studies. Through the
figures 1,2 and 3, it becomes clear the progress of the four-point EGAOR iterative method
in reducing time and number of iteration among the other studied methods.

Table 1: Comparison of standard five-point and four-point EGSOR iterative methods.

Standard Five-Point Four-Point Group (SOR)
N ω k t ρ(J) E ω k t ρ(J) E

12 1.5604 30 0.0076 0.9595 6.2000e− 06 1.3520 22 0.021156 0.9184 6.2312e− 06
46 1.8696 108 0.0460 0.9976 8.6000e− 06 1.7846 76 0.040845 0.9951 9.6553e− 06
86 1.9287 196 0.1747 0.9993 9.5000e− 06 1.8797 138 0.106233 0.9986 9.3103e− 06
106 1.9419 239 0.3154 0.9996 9.7000e− 06 1.9015 169 0.174041 0.9991 8.1532e− 06
146 1.9576 323 0.4282 0.9998 9.8000e− 06 1.9277 227 0.181077 0.9995 9.1567e− 06
186 1.9666 405 0.7470 0.9999 9.9000e− 06 1.9429 287 0.227092 0.9997 9.7462e− 06
226 1.9725 487 0.9530 0.9999 9.6000e− 06 1.9528 344 0.309206 0.9998 9.6445e− 06
350 1.9822 732 2.8480 0.9999 9.7000e− 06 1.9693 528 0.796697 0.9999 9.5333e− 06
426 1.9853 882 6.4068 0.9999 9.4000e− 06 1.9747 644 2.142877 0.9999 9.3794e− 06

Table 2: Comparison of standard five-point and four-point EGAOR iterative methods.
Standard Five-Point

N ω r k t ρ(J) E
12 1.5604 1.5709 26 0.0032s 0.9595 8.8565e− 06
46 1.8696 1.8749 100 0.0146s 0.9976 9.5039e− 06
86 1.9287 1.9329 181 0.0678s 0.9993 9.8956e− 06
106 1.9419 1.9449 222 0.1072s 0.9996 9.9577e− 06
146 1.9576 1.9592 306 0.2163s .9998 9.9646e− 06
186 1.9666 1.9690 395 0.6039s 0.9998 9.9828e− 06
226 1.9699 1.9742 466 0.67701s 0.9999 9.4652e− 06
350 1.9818 1.9823 724 2.1866s 0.9999 9.8868e− 06
426 1.9845 1.9856 869 4.90743s .9999 9.91628e− 06

Four-Point Group (AOR)
N ω r k t ρ(J) E
12 1.4331 1.4537 17 0.0085s 0.9184 9.7737e− 06
46 1.792− 1.763 1.8270 69 0.0261s 0.9951 9.8840e− 06
86 1.847− 1.853 1.9050 124 0.0920s 0.9986 9.9985e− 06
106 1.880− 1.867 1.9220 152 0.0990s 0.9991 9.8045e− 06
146 1.903− 1.843 1.9430 208 0.1782s 0.9995 9.8531e− 06
186 1.919− 1.917 1.9550 263 0.2521s 0.9997 9.9905e− 06
226 1.931− 1.927 1.9620 318 0.4317s 0.9998 9.9684e− 06
350 1.954− 1952 1.9753 488 0.9133s 0.9999 9.9444e− 06
426 1.935− 1.929 1.9800 587 1.9163s7 0.9999 9.9413e− 06



A.M. Saeed, N. AL-harbi / Eur. J. Pure Appl. Math, 14 (3) (2021), 905-914 912

(a) (b)

Figure 1: Comparison of the CPU time (t) and the number of iterations (k) for Standard
Five-Point SOR and Foure point EGSOR iterative methods.

(a) (b)

Figure 2: (a) approximation solution of EGSOR for N=86,(b) approximation solution
EGAOR for N=86.

(a) (b)

Figure 3: Comparison of the CPU time (t) and the number of iterations (k) for Standard
Five-Point AOR and Foure point EGAOR iterative methods.

5. Conclusions

The results reported in Tables 1 and 2, clearly show that the 4- EGAOR process
outperforms the corresponding 4-EGSOR, point SOR, and AOR methods. The number of



REFERENCES 913

iterations and execution time has also decreased. In all of the cases studied, it is evident
that the 4- EGAOR approach needs the least amount of computational effort, which
corresponds to the pattern of timing results obtained in our experiments. This is due in
large part to the AOR method’s reduction of computational complexity and acceleration
parameters. This research can be continued in the future to look at the use of another
group iterative methods combined with AOR scheme as a smoother with a complexity
reduction approach.

Acknowledgements

The authors are supported by Qassim University and the Ministry of Education of
Saudi Arabia.

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