Fast quarter sweep using modified successive over-relaxation iterative methods for solving two dimensional helmholtz equation

New over-relaxation methods for the solution of two-dimensional Helmholtz partial deferential equations (PDEs) are described. In the Helmholtz equation, when solving the resulting PDEs using a finite difference (FD) scheme, the computations involve large sparse systems of linear equations (SLEs). These require considerable computation time. Hence, to overcome this problem, the development of faster iterative techniques is desirable. Point iterative methods, which are based on full-, half-, and quarter-sweep discretization, are commonly used to solve the Helmholtz equation. Due to the large scale of the resulting SLE, many studies have attempted to speed up the convergence rate of the solution. Hence, Young (1971) has already elaborated and discussed the concepts behind various iterative methods. In addition, block (or group) iterative methods, whereby the mesh points are grouped into blocks, have been shown to reduce the number of iterations and execution time, because the solution at the mesh points can be updated in groups instead of pointwise. Among these group iterative methods, Explicit Group (EG), Explicit Decoupled Group (EDG), and Modified Explicit Group (MEG) methods have been expansively researched, and have been shown to converge faster than their pointwise counterparts. Apart from this approach, in order to improve the rate of convergence of these techniques, conjoint accelerated methods, such as Successive Over-Relaxation (SOR), may be applied to reduce the number of iterations. Whereas the above methods have already been implemented with SOR, the quarter-sweep pointwise and MEG methods have never been implemented with Modified Successive Over-Relaxation (MSOR). This thesis explains the construction and formulation of a quarter-sweep method combined with MSOR, namely QSMSOR. In addition, a computational complexity analysis is presented, and the method is compared with half-sweep MSOR (HSMSOR) and full-sweep MSOR (FSMSOR). Next, the derivation of a combined MEG and MSOR method for solving the two-dimensional Helmholtz equation iteratively is discussed in detail, and a computational complexity analysis of the proposed method is conducted. The numerical results illustrate the improvement of the MEGMSOR method over the combined EDGMSOR and EGMSOR methods in terms of number of iterations, execution timing and maximum absolute error. This is shown to be true for both nonhomogeneous and homogeneous problems in second-order schemes. In conclusion, the newly developed method is a viable alternative for solving the Helmholtz equation iteratively.

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