Application of Four-Point EGSOR Iteration with Nonlocal Arithmetic Mean Discretization Scheme for Solving Burger’s Equation

The main objective for this study is to examine the efficiency of block iterative method namely Four-Point Explicit Group Successive Over Relaxation (4EGSOR) iterative method. The nonlinear Burger’s equation is then solved through the application of nonlocal arithmetic mean discretization (AMD) scheme to form a linear system. Next, to scrutinize the efficiency of 4EGSOR with Gauss-Seidel (GS) and Successive Over Relaxation (SOR) iterative methods, the numerical experiments for four proposed problems are being considered. By referring to the numerical results obtained, we concluded that 4EGSOR is more superior than GS and SOR iterative methods in aspects of number of iterations and execution time.


Introduction
In recent yeras, the Burger's equations have been known as one of the most important nonlinear equation. This equation can be found in applied mathematics, physics and engineering. There are many researchers proposed various methods in solving the problems of Burger's equations such as Cole-Hopf Transformation [1], Standard Explicit Method [2], Weak Galerkin Finite Element Method [3] and Adomian Decomposition Method [4].
In this paper, we focus on the application of nonlocal arithmetic mean discretization scheme in getting the linear system of Burger's equation. To solve the linear system since it is sparse and large scale, the family of iterative methods such as point and block iterative methods is used as a linear solver. To scrutinize the efficiency of 4EGSOR iterative method with nonlocal (AMD) scheme, there are four examples of Burger's problems that can be solved by using the block iterative methods.
In order to get the approximation solution, let us consider the general form of nonlinear Burger's equation: is the nonlinear term.
There are many sections in this paper. For the first section, we elaborate the introduction of Burger's equation. In section 2, we state the formulation of nonlocal AMD scheme to form a linear system. Next, the formulation of the proposed iterative methods is shown to solve the linear system. Then, in section 4, we introduced four examples of problem 1 and we also present the numerical results. Finally, in the last section we made the conclusion of this paper.

Formulation of Nonlocal AMD Scheme
To start this section, problem (1) can be rewritten as To discretize problem (2), we need to consider the following general formulations of nonlocal AMD scheme at any time level 1 j which given as follows [5] By considering equations (3) and (4) Actually, the expression shows the nonlinear term of problem (2). Then, we need to use the nonlocal AMD scheme to form a system of linear equation. By substituting equation (4) into equation (6), we get the new expression as Both equations (5) and (7) can be represented in general form as By considering any group of two neighboring node points,

Formulation of EGSOR Method
In the previous section, by referring to the linear system [9], it is obviously show that the feature of its coefficient matrix is sparse and large scale. According to the previous studies, Evans [6] has proposed the Explicit Group iterative method to solve the sparse linear system. Hence, this paper attempts to scrutinize the effectiveness of the 4EGSOR method. Figure 1 shows the implementation of 4EGSOR iterative method and as can be seen in this figure, the last three node points are known as an ungroup case [7]. The general formulation of the EG iterative method can be shown as , By determining the inversion matrix of equation (10), the formulation of the 4-point Explicit Group Gauss-Seidel (4EGGS) iterative method can be shown as By adding the weighted parameter, into equation (11), the general form of 4-Point EGSOR iterative method can be shown as If yes, proceed to next step. Or else repeat step (iii). v. Stop.

Numerical Examples
For the numerical comparison to testify the efficiency of 4EGSOR iterative method, there are four examples that are proposed. There are three aspects will be measured such as number of iterations, execution time (second) and maximum absolute error. The numerical results obtained are presented as shown in Table 1.
The exact solution of problem (13) is specified by

Example 2 [9]
Let the initial value equation be define as The exact solution of problem (15) can be stated as shown 2 , . 1 2 x v x t t (16)

Example 3 [10]
Let the initial value equation be define as sin , 0 2 , for 0. cos The exact solution of problem (17) is shown as
x   Table 2 show the reduction percentage of SOR and 4EGSOR iterative methods with GS iterative method. By referring to Table 1, the number of iterations of SOR iterative method has decreased by approximately and respectively compared to GS iterative method. Hence, it required less execution time than GS iterative method by and .
Meanwhile for the 4EGSOR iterative method, the number of iterations has decreased by approximately and respectively. Thus, the 4EGSOR iterative method also required less execution time than GS iterative method by and . By referring to the reduction percentage in Table 2, it can be concluded that the 4EGSOR iterative method is more superior as compared with GS and SOR iterative methods.

Conclusion
In this study, the nonlocal AMD scheme is successfully applied to solve the nonlinear Burger's equation. The nonlinear equation are then transform to linear system. By referring to the numerical results obtained, it show that 4EGSOR iterative method is efficient than GS and SOR iterative methods in aspects of number of iterations and execution time. For the future research, other block iterative methods such as EDGAOR method can be considered in solving Burger's equation.