FINITE DIFFERENCE SCHEMES

Mohd Norfadli Suardi*¹ and Jumat Sulaiman²

1,2 Faculty of Science and Natural Resources, Universiti Malaysia Sabah, Kota Kinabalu, Sabah, MALAYSIA.

2 Preparatory Centre for Science and Technology, Universiti Malaysia Sabah, Kota Kinabalu, Sabah, MALAYSIA.

(E-mail: norfadli1412@gmail.com, jumat@ums.edu.my)

*corresponding author ABSTRACT

Various finite difference schemes have been established to discretize boundary value problems.

Therefore, this paper proposes a numerical solution based on newly established second-order Redlich-Kister Finite Difference (RKFD) discretization schemes to approximate the one- dimensional second-order linear hyperbolic telegraph equation. By doing the discretisation process, the RKFD approximation equation leads to a large-scale and sparse system of RKFD approximation equations, which can be solved iteratively using the Gauss-Seidel (GS) Successive Over-Relaxation (SOR) iterative methods. In order to validate the capability of the SOR iterative method, two model examples have been solved iteratively via the SOR iteration and then compared to the GS iterative method at five different mesh sizes. The comparison results for both iterative methods are illustrated by focusing on three measuring parameters: number of iterations, execution time, and maximum norm. As can be observed from the numerical results, the proposed method approximates the exact solution very well.

Keywords: SOR iteration, Redlich-Kister finite difference, Finite difference, Hyperbolic Telegraph problem

INTRODUCTION

Nowadays, a numerical solution plays a fundamental approach in mathematics and computer science. It involves developing, analysing, and implementing an algorithm to solve mathematical problems, particularly linear partial differential equations (PDEs) [1,2]. Hyperbolic type PDEs is one class of PDEs that uses a wide range of aerospace, industry, and engineering applications, such as the vibration of structure and fundamental equations of atomic physics [3,4]. In recent years, many authors have given much attention to the literature on developing, analysing, and implementing the stable method to get the numerical solution of the telegraph equation [5,6,7]. For instance, the author [5] proposed the Sinc-Galerkin method to solve the telegraph equation. Also,

32 the authors [6,7] applied the numerical solution based on the cubic B-spline method for approximating the telegraph equation.

From the proposed numerical methods listed in the previous paragraph, in this work, a new method based on the Redlich-Kister polynomial function is proposed to approximate the telegraph equation. This function is commonly found in physics and chemistry fields [8,9,10], but there are few studies about this function in numerical analysis. Earlier works of the Redlich-Kister in the numerical analysis were introduced by the study [11]. The authors [11] proposed constructing the first and third-order piecewise Redlich-Kister polynomials with concern to analyze the relation of the Gauss-Seidel iteration over mesh sizes. As a result, they found that third-order Redlich-Kister accuracy is more accurate than the first-order Redlich-Kister models. This study [12] has continued to solve two-point boundary value problems. Also, the finding has pointed out that the accuracy of the Redlich-Kister polynomial solution is more accurate than other control solutions. Inspired by the high accuracy from the findings in previous studies [11,12] and findings of previous works [13-15], which have established other finite-difference discretization schemes. The primary purpose of this paper is to investigate the effectiveness of newly established Redlich-Kister finite difference (RKFD) schemes being used to solve one-dimensional hyperbolic telegraph problems iteratively via SOR iteration. Before emphasizing the discretization process of using RKFD schemes over the proposed problem, let us consider the following one-dimensional hyperbolic telegraph equation as

( ) ( ) ( ) ( )

2 2

2

2 , 2 , , 2 , ( , ),

U U U

x t x t U x t x t f x t

t  t  x

 +  + = +

   (1)

with the initial conditions

( )

,0 1

( )

, U

( )

,0 2

( )

,



0, _n

  

0,

U x g x x g x x x x

t 

=  =  =

 ,

and boundary conditions,

( )

0, 3

( )

,

U t =g t U

( )

1,t =g t4

( )

.

REDLICH-KISTER FINITE DIFFERENCE APPROXIMATION EQUATION

As explained in the first section, the newly established RKFD discretisation schemes will be considered in this investigation. In order to obtain a corresponding RKFD approximation equation, problem (1) needs to be discretised using the RKFD discretisation schemes in the x-direction and standard finite difference discretisation schemes in the time direction. Before the discretisation process starts, the general formula of an RK approximation function is defined as

( ) ( ) ( )

0

,

n

n k k

k

U x t a t T x

=

=





, (2)

wherea_k

( )

t ^,k =0,1, 2,...,n, are to be calculated for the value of unknown parameters considered.

33 Figure 1. Distribution of mesh sizes considered at three-time levels.

To calculate the unknown parameters, a_k

( )

t ^,k =0,1, 2,...,n in equation (2), firstly, we need to consider the distribution of mesh sizes at three-time levels, as depicted in Figure 1. This distribution of mesh sizes helps us to construct the first three RK functions. To define these functions, let us define a second-order RK approximation function is obtained as

( )

^, ₀

( ) ( )

_{0 1}

( ) ( )

_{1 2}

( ) ( )

₂

U x t =a t T x +a t T x +a t T x , (3)

where the first three RK functions are defined as

( ) ( ) ( ) ( )

0 1, 1 , 2 1 .

T x = T x =x T x =x −x

Then, let the node points,x_c =x⁰+ch c^, =0,1, 2,...,n be defined with their uniform step size as

0, 2 ,^p 1

h=^n⁻ n= p and

0 T

t m⁻

 = as depicted in Figure 1. By considering any group of three node points, determination of three unknown parameters, ^a_k

( )

^{t ,k = −c 1, ,c c+1} can be calculated by solving the following linear system

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

0 1 1 1 2 1 0 1

0 1 2 1

0 1 1 1 2 1 2 1

c c c c

c c c c

c c c c

T x T x T x a t U t

T x T x T x a t U t

T x T x T x a t U t

− − − −

+ + + +

     

    = 

     

      

  , (4)

where ^{U x t}

(

_c^,

)

^=U_c

( )

^t . All these approximation functions (4) will be solved using the matrix approach to obtain the expression of unknown parameters in equation (3). Then, these expressions obtained will be substituted into equation (3) to get the following RK approximation function

( )

^, ₀^{( )} _{c 1}

( )

₁^{( )} _c

( )

₂^{( )} _{c 1}

( )

U x t =N x U ₋ t +N x U t +N x U ₊ t _, (5) where the second-order RKFD shape functions,^{Nk( ),x k =0,1, 2} can be defined, respectively as

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

2 2 2 2 2 2 2 ,

0

2 2 2 2 2

2 ,

1

2 2 2 2 2 2 2 .

2

N x x - xhc - xh+ h c + h c h N x xhc - x - h c + h h N x x - xhc + xh+ h c - h c h

= 

= 



=  (6)

34 Then the first and second derivatives of the second-order RKFD shape functions,^{Nk( ),x k=0,1, 2} can be derived based on equation (6). Also based on equation (5), it can show that the first and second derivatives of the second-order RKFD approximation function can be given as

( ) ( ) ( )

( ) ( ) ( )

2 2

0 1 1 2 1

0 1 1 2 1

( ) ( ) ( ) ,

( ) ( ) ( ) .

Ux c c c c c c c

U c c c c c c

x c

N x U t N x U t N x U t

N x U t N x U t N x U t

 − +

 − +

   

= + +



  

= + + 

 (7)

whereU x t

(

_c^,

)

=U_c

( )

t ^,c=0,1, 2,...,n represent as the approximation solution of function^{U x t}

( )

^{, .}

The obtained expressions in equation (7) show that this process, in proportion to the main objective in this investigation, proposed two newly established RKFD discretization schemes to construct the RKFD approximation equation for solving the proposed problem (1). By substituting equation (7) into the proposed problem (1) and imposing the second-order central difference schemes for discretizing in time (12), we can derive a second-order RKFD approximation equation (5) over the one-dimensional telegraph equation as follows

1, 1 , 1 1, 1 , 1, 0,1, 2, , 1

cUc j cUc j cUc j Rc j j m

 _{− +}  ₊  _{+ + +}

− + − = = −

, (8)

where

( )   ( ) ( )  

( )   ( ) ( )

2 2 2 2

0 1

2 2

2 , 1 , 1 , , 1

( ) , 1 ( ) ,

( ) , 2 1 .

c c c c

c c c j c j c j c j

t N x t t t N x

t N x R t f U t U

   

 _{+ +}  ₋

 

=  = +  +  − 

=   =  + +  −

Again, we can use the RKFD approximation equation (8) to construct a sequence of the RKFD linear systems in matrix form as follows:

1

1 _j , 0,1, 2, , 1

W U j₊ =R ₊ j= m−

. (9)

DERIVATION OF SOR ITERATIVE METHOD

From the previous section, the generated large-scale and sparse linear systems (9) can be identified from the RKFD discretization scheme process completed. According to the studies [16- 18], the best way to solve these linear systems is by using iterative methods since the coefficient matrix of these linear systems has involved many large mesh sizes. Therefore, the SOR iterative method has been considered a linear solver to solve these linear systems (9). The use and efficiency of the SOR method have been studied by [16-18], in which this method is an improvement of one classical point iterative approach, known as the Gauss-Seidel (GS) method. The implementation of the SOR method influenced by the determination the value of weighted parameter, to approximate the proposed problem (1) very well and then the range should be in^{1  2.} However, the SOR method can be reduced to the GS method when the weighted parameter considers ^=1[16]. Before starting to implement the SOR iteration, let the coefficient matrix, ^W of the linear systems (9) at

35 any time level,

(

^j+1

)

be decomposed into the three summations matrices as

(

^{F+ + J L U}

)

_j+1^=R_j+1^{, j=0,1, 2, ,m−1}_{, (10)}

where^J is a diagonal matrix,^F and^L are triangular lower and upper matrices, respectively. By manipulating equation (10), the formulation of the SOR method in point iteration form can be expressed as [14,17,18]

( )

_j^q₁¹

(

¹

) ^{( )}

_j^q₁

( )

¹ _{j 1}

^{( )}

_j^q₁ ^{, 0,1, 2, , 1}

U ₊⁺ = − U ₊ + J +F ^{− }R ₊ −LU ₊ ^ j= m−

  _{, (11)}

where

( )¹

1 q

U j₊⁺

indicates the current value of ^{U x t}

( )

^, _{at the}(q+1)^th iteration.

NUMERICAL PROBLEM AND DISCUSSION

In this section, the numerical experiments were performed for solving the problem (1) via the SOR iterative method with the RKFD approximation equation as derived in the previous section. In order to obtain the applicability of two proposed point iterative methods, two model examples have been tested and solved by considering five different mesh sizes. Also, the numerical comparisons are made in terms of the number of iterations (Iter), execution time (Time), and maximum norm (MaxNorm). Then, the use of tolerance error is constant for each example considered.

Example 1

Consider the one-dimensional Telegraph equation (1) with^=1 and^=1 as [19]

( )( )

2 2

2 2 2

2 2 2 2 ^t 2 ^t,

U U U

U t t x x e t e

t t x

  ^{− −}

 +  + = + − + − +

   (12)

with the initial conditions, boundary conditions and the analytical solution of problem (12) are

( )

^,

(

²

)

^{2 t.}

U x t = x−x t e⁻ Example 2

Consider the one-dimensional Telegraph equation (1) with⁼⁴ and ^=2 as [20]

( ) ( )

2 2

2 2 2 ^tsin ,

U U U

U e x

t t x

    ⁻

  

+ + = + − +

   (13)

with the initial conditions, boundary conditions and the analytical solution of problem (13) are

36

( )

^{, tsin}

( )

^.

U x t =e⁻ x

Table 1. Numerical results for Examples 1 and 2

n Method Example 1 Example 2

Iter Time MaxNorm Iter Time MaxNorm

256 GS 122 0.25 7.1168e-04 141 0.18 9.8406e-04

KSOR 44 0.16 7.1097e-04 59 0.09 9.8443e-04

512 GS 425 1.28 7.1460e-04 498 1.71 9.8259e-04

KSOR 84 0.37 7.1125e-04 114 0.22 9.8427e-04

1024 GS 1528 6.70 7.2633e-04 1818 8.72 9.7694e-04 KSOR 159 1.05 7.1184e-04 224 0.75 9.8390e-04 2048 GS 5508 43.86 7.7329e-04 6676 52.36 9.5463e-04 KSOR 303 3.41 7.1304e-04 437 2.75 9.8336e-04 4096 GS 19692 310.61 9.6112e-04 24403 379.93 8.6619e-04 KSOR 579 7.00 7.1548e-04 856 10.48 9.8209e-04

As observed in Table 1, the computational results of the SOR method with the RKFD approximation equation are superior to the bench-marking method in this investigation, the GS method, and approximate the exact solution well. As shown in Table 1, the SOR method enables fewer iterations and speeds up execution time to converge all the model examples considered. For two examples in Table 1, the SOR required 44 iterations with 0.16 seconds while the GS method required 122 iterations with 0.25 seconds to converge problem (12) at 256 mesh sizes. The significant advantage for the SOR method in terms of iteration and time when involving the large mesh sizes. Based on the computational results in Table 1 at ⁿ⁼⁴⁰⁹⁶, the SOR requires 579 iterations with 7.00 seconds compared to the GS method with 196922 iterations and 310.61 seconds.

The computational results from implementing the proposed SOR method for solving problems (12) and (13) have the same pattern of their findings. In line with the findings [18], the SOR method has improved the GS method according to iteration and time. In terms of accuracy, both methods have a good agreement with their exact solution.

CONCLUSION

In this investigation, the new approach using the SOR method with two newly established RKFD discretization schemes has been used to solve the one-dimensional Telegraph equation. The RKFD solution of problem (1) has started from the discretization process to derive the corresponding RKFD approximation equation based on the proposed problem. Then this approximation equation leads to a sequence of linear systems, which is solved numerically by using the iterative method. To demonstrate the applicability of the proposed SOR method based on newly established RKFD discretization schemes, numerical results showed and revealed that the SOR method is superior in terms of less number of iterations and execution time compared to the GS method.

ACKNOWLEDGEMENT

The authors would like to express sincere gratitude to Universiti Malaysia Sabah for funding this research under the UMSGreat research grant for postgraduate student: GUG0494-1/2020.

REFERENCES

37 [1] Bernatz, R. Fourier series and numerical methods for partial differential equations. (John Wiley and Sons, 2010).

[2] Ray, S, S. Numerical analysis with algorithms and programming. (Taylor & Francis Group, LLC, USA, 2016).

[3] Saadatmandi, A. and Dehghan, M. Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method. Numer. Methods Partial Differerential Equation 26 (2010), 239–252.

[4] Dehghan, M. and Shokri, A. A numerical method for solving the hyperbolic telegraph equation. Numerical Methods Partial Differential Equation 24 (2008), 1080–1093.

[5] Hesameddini, E. and Asadolahifard, E. The sinc-collocation method for solving the telegraph equation. Journal of Computer Engineering and Informatics (1)1, (2013), 13–17.

[6] Mittal, R.C. and Bhatia, R. Numerical Solution of second order hyperbolic telegraph equation via new cubic trigonometric Bsplines approach. Applied Mathematics and Computation 220 (2013), 496–506.

[7] Rashidinia, J., Jamalzadeh, S. and Esfahani, F. Numerical solution of one-dimensional telegraph equation using cubic B-spline collocation method. Journal of Interpolation and Approximation in Scientific Computing 2014 (2014).

[8] Babu, S., Trabelsi, R., Srinivasa Krishna, T., Ouerfelli, N. and Toumi, A. Reduced Redlich-Kister functions and interaction studies of Dehpa+ Petrofin binary mixtures at 298.15 K. Physics and Chemistry of Liquids 57(4) (2019), 536-546.

[9] Gayathri, A., Venugopal, T. and Venkatramanan, K. Redlich-Kister coefficients on the analysis of physico- chemical characteristics of functional polymers. Materials Today: Proceedings 17 (2019), 2083-2087.

[10] Komninos, N.P. and Rogdakis, E.D. Geometric investigation of the three-coefficient Redlich-Kister expansion global phase diagram for binary mixtures. Fluid Phase Equilibria (2020), 112728.

[11] Hasan, M. K., Sulaiman, J., Ahmad, S., Othman, M. and Abdul Karim, S. A. Approximation of iteration number for Gauss-Seidel using Redlich-Kister polynomial. American Journal of Applied Sciences. 7 (2010), 956-962.

[12] Suardi, M. N., and Sulaiman, J. Solution of One-Dimensional Boundary Value Problem by Using Redlich-Kister Polynomial. Computational Science and Technology: 7^th ICCST 2020, Pattaya, Thailand, 29–30 August 2020, 724, 487-500.

[13] Elbarbary, E. M. and El-Kady, M. Chebyshev finite difference approximation for the boundary value problems.

Applied Mathematics and Computation. 139(2-3) (2003), 513-523.

[14] Pandey, P. K. Rational finite difference approximation of high order accuracy for nonlinear two point boundary value problems. Sains Malaysiana, 43(7) (2014), 1105-1108.

[15] Pandey, P. K. Solving twopoint boundary value problems for ordinary differen-tial equations using exponential finite difference method. Boletim da Sociedade Pa-ranaense de Matemática, 34(1) (2016), 45-52.

[16] Young, D.M. Iterative Solution Of Large Linear Systems. (London: Academic Press, 1971).

[17] Youssef, I. On the successive overrelaxation method. Journal of Mathematics and Statistics 8(2) (2012), 176-184.

[18] Ali, N.A.M., Rahman R. Sulaiman, J. and Ghazali, K. SOR iterative method with wave variable transformation for solving advection-diffusion equations. In AIP Conference Proceedings 2013(1) (2018), 020036.

[19] Dugassa, M., Gemechis, F. and Tesfaye, A. Numerical Solution of Second Order One Dimensional Linear Hyperbolic Telegraph Equation. Ethiopian Journal of Education and Sciences 14(1) (2018), 39-52.

[20] Kurt Bahşı, A. and Salih Y. A new algorithm for the numerical solution of telegraph equations by using Fibonacci polynomials. Mathematical and Computational Applications 21(2) (2016), 15.

38 01-012

STUDY OF ENHANCING THE MODIFIED BAPTISTA TYPE