Abstract
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.
Highlights
Many problems in engineering and science involve Helmholtz equation, occur in real time application
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)
Eq (5) is known as the quarter-sweep finite difference (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 i, j 1 i 1, j 1 i 1, j i 1, j 1 x, y 2h
Summary
Many problems in engineering and science involve Helmholtz equation, occur in real time application. Eq (2) can can be discretized using the same formula with grid spacing 2h and leads to the following formula. Eq (5) is 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 i, j 1 i 1, j 1 i 1, j i 1, j 1 x, y 2h. 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 numerical results and discussion are given in the final section
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