QSMSOR Iterative Method for the Solution of 2 D Homogeneous Helmholtz Equations

In this paper, we consider the numerical solutions of homogeneous Helmholtz equations of the second order. The Quarter-Sweep Modified Successive Over-Relaxation (QSMSOR) iterative method is applied to solve linear systems generated form discretization of the second order homogeneous Helmholtz equations using quarter sweep finite difference (FD) scheme. The formulation and implementation of the method are also discussed. In addition, numerical results by solving several test problems are included and compared with the conventional iterative methods.


Introduction
Many problems in engineering and science involve Helmholtz equation, occur in real time application.On the other hand, the applications of Helmholtz equation are encountered in many fields such as time harmonic acoustic and electromagnetic fields, optical waveguide, acoustic wave scattering, noise reduction in silencer, water wave propagation, radar scattering and lightwave propagation problems (Muthuvalu et al., 2014a; Nabavi et al., 2007;Kassim et al., 2006;Yokota and Sugio, 2002).There is a high important in improving the performance of the methods for solving Helmholtz equation.Hence, the development of fast methods is essential in this research area.
Consider the second order Helmholtz equation which is the elliptic equation with Dirichlet boundary conditions and function f are given.Here, we assume that the domain is the square unit.Assume that the grid spacing is Eq. ( 2) can also can be discretized using the same formula with grid spacing 2h and leads to the following formula Eq. ( 5) is also known as the quarter-sweep FD approximate equation Othman and Abdullah (1998).Another type of approximation derived from the rotated FD approximate equation (Abdullah 1991;Dahlquist and Bjork, 1974) can be constructed by the following transformation Therefore, the scheme of central difference using the rotated FD approximate equation (Dahlquist and Bjorck, 1974) can be expressed as The standard rotated FD approximation ( 4) is also called half-sweep FD approximate equation.The article is organized in the following form.The latter section of this article will discuss the formulations of the Full-Sweep Modified Successive Over-Relaxation (FSMSOR), Half-Sweep Modified Successive Over-Relaxation (HSMSOR) and QSMSOR iterative methods in solving the SLE obtained from discretization of the two-dimensional Helmholtz equations.The computational complexity analysis will be shown in Section 4 to assert the performance of the proposed methods.Then, the numerical results and discussion are given in the final section 2 Point MSOR Methods

1 FSMSOR Method for Helmholtz Equation
To derive the FSMSOR point iterative method, we use full-sweep approach, in which the domains are divided into two types of points (i.e., and ) as shown in Fig. 1.By applying MSOR method (Akhir et al., 2011a;Kincaid and Young, 1972;De Vogelaere, 1958) into Eq.( 2), we will obtain the FSMSOR method for Helmholtz equation as    Step 1: The iterations (using Eq. ( 5)) implemented on the red point first using the relaxation parameter r  .
Step 2: After the red points sweep are completed, the iterations are done on the black points using the relaxation parameter b  .
Step 3: Check the convergence.If converge go to Step 4, otherwise repeat the iteration cycle, (i.e Step 1).

HSMSOR Method for Helmholtz Equation
To derive the HSMSOR iterative method, we use half-sweep approach, in which the domains are divided into three type of points (i.e., , and ) as shown in Fig. 2. By applying MSOR method (Akhir et al., 2011b(Akhir et al., , 2011c) ) into Eq.( 4), we get the HSMSOR method for Helmholtz equation as where 2 1 42 h   .Eq. ( 6) allows us to iterate through half of the points, lying on the 2h -grid.Again, it can be observed that Eq. ( 6) involves points of type and .Therefore the iteration can be carried out autonomously involving only this type of point.The algorithm of HSMSOR method is display in Algorithm 2.2: Algorithm 2.2 Discretize the solution domain into point of three types , and as shown in Figure 2.
Step 1.The iterations (using Eq. 6)) implemented on the red point first using the relaxation parameter r  .
Step 2. After the red points sweep are completed, the iterations are done on the Step 5. Display approximate solutions.

QSMSOR Method for Helmholtz Equation
To derive the QSMSOR iterative method, we use quarter-sweep approach, in which the domains are divided into four type of points (i.e., , , and ) as shown in Fig. 3.By applying MSOR method (Akhir et al., 2012b) into Eq.( 3), we will obtain the QSMSOR method for Helmholtz equation as where 2 2 44 h   .Eq. ( 7) allows us to iterate through quarter of the points, lying on the 2h -grid.Again, it can be observed that Eq. ( 7) involves points of type and .Therefore, the iteration can be carried out autonomously involving only this type of point.The algorithm of QSMSOR method is display in Algorithm 2.3: Algorithm 2. 3 The solution domain must be labeled for the four types of points (i.e., , , and ), as shown in Fig. 3.
Step 1.The iterations (using Eq. ( 7)) implemented on the red point first using the relaxation parameter r  .
Step 2. After the red points sweep are completed, the iterations are done on the black points using the relaxation parameter a. points of type using the full-sweep FD approximate formula (7) on the grid 2h .
points of type using the full-sweep FD approximate formula (3) on the grid h.

Numerical Results
In this section, we exemplify two numerical examples to illustrate the effectiveness of the methods prescribed in previous section.The algorithms were tested on the following model problems: Problem 1 (El-Sayed and Kaya, 2004) 5 0  was chosen to within 0.01  that gave the minimum number of iterations.The computer language used programming is C++, and the program performed on a personal PC Intel(R) Core (TM) i7 CPU 860@3.00Ghz,6.00GB RAM.The operation system used was Window 7 with the installation Borland C++ compiler version 5.5.The numerical results of the experiment for different value of mesh size are given in Tables 1 and 2, respectively

Computational Complexity Analysis of MSOR Methods for Helmholtz Equation
The computational effort measured by number of computer operations needed to obtain a solution by the three methods discussed for solving problem (1) can be assessed.Undertake the solution domain is large with 2 m number of internal mesh points with 1 mn .In their iterative manner, the FSMSOR and HSMSOR methods require   internal mesh points.Note that our valuation on this computational complexity is based on the arithmetic operations performed per iteration and execution time for the additions/subtraction (ADD/SUB) and multiplications/divisions (MUL/DIV) operations.Therefore the number of operations of operations required (excluding red and black equations, convergence test and direct solution) for FSMSOR, HSMSOR and QSMSOR methods as described in Section 3 are correspondingly given as follows in

Discussions of Results
In Section 5, three types of pointwise MSOR methods are applied into a Helmholtz equation model to check the execution times and number of iterations.From the numerical result, QSMSOR methods is the fastest method among the other two MSOR methods (HSMSOR and FSMSOR) if we compare with either number of iterations or execution time.This can also be verified if we compared the computational complexity of all three MSOR methods where QSMSOR method has the least computational complexity.
It can be pragmatic that the accuracies of the QSMSOR methods remain as good as the HSMSOR and FSMSOR methods but they oblige lesser number of iterations and computing timing to attain the result.For example, the number of iterations of QSMSOR is merely about 22-31% and 22-39% as well as 46-52% and 43-70% compared to HSMSOR and FSMSOR methods in Problems 1 and 2, respectively.Again, the execution times of QSMSOR are much faster just about 15-21% and 30-32% along with 64-73% and 63-84% compared to HSMSOR and FSMSOR methods in Problems 1 and 2 respectively.
Experimental results also show promising results that make them as alternative to conventional FD scheme.From the number of iterations and timing obtained, it can be seen that among three MSOR iterative methods presented, the QSMSOR method requires the least time for all n compared with the other two MSOR iterative methods.This is due to the fact that among the three methods, the QSMSOR method requires least number of numbers of iterations and computational operations.This is reflected by total arithmetic operations required by the method given in Table 3.
Moreover, the accuracy are significantly good, since all the method used the descriptive stencil   2 Oh .Overall, the numerical results show that the QSMSOR method is superior than HSMSOR and FSMSOR methods.This is mainly because of computational complexity of the QSMSOR method which is approximately 50% and 75% less than HSMSOR and FSMSOR methods respectively.For future works, the capability of octo-sweep iteration (Akhir et al., 2012d) should be scrutinized for solving homogeneous (El-Sayed and Kaya, 2004) and nonhomogeneous Helmholtz equations (Akhir et al., 2012c).Also, advance studies for innumerable point block iterative methods can be also scrutinized (Akhir et al., 2012e).

1 : 2 . 1
(5) allows us to iterate through all of the points, lying on the h -grid.It can be observed that Eq. (5) involves points of type and .Therefore the iteration can be carried out independently involving only this types of point.The algorithm of FSMSOR method is display in Algorithm 2.Algorithm Discretize the solution domain into point of two types and as shown in Fig.1.

Step 3 .
black points using the relaxation parameter b  .Check the convergence.If converge go to Step 4, otherwise repeat the iteration cycle, (i.e Step 1).Step 4. Evaluate the solutions at the remaining points type using the fullsweep FD approximate formula (3) on the grid h (Akhir et al., 2012a, b; 2011b, c).

Step 3 .
Check the convergence.If converge go to Step 4, otherwise repeat the iteration cycle, (i.e Step 1).Step 4. Evaluate the solutions at the remaining points according to the following sequence.(Aruchuan et al.,2014; Akhir et al., 2012b; Othman and Abdullah, 1998).

Table 3 Table 1
Number of iterations, execution time and maximum absolute error for the proposed iterative methods in solving Problem 1.

Table 2
Number of iterations, execution time and maximum absolute error for the proposed iterative methods in solving Problem 2.

Table 3
The total computing costs for the three MSOR methods.