Four Point High Order Compact Iterative Schemes For The Solution Of The Helmholtz Equation
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Date
2015-08
Authors
Teng, Wai Ping
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Abstract
Improved techniques derived from the standard and rotated finite difference operators have been developed over the last few years in solving linear systems that arise from the discretization of various partial differential equations (PDEs) [14]. Furthermore, a higher order system can be generated from discretization of the finite difference scheme by using the fourth order compact scheme generated from the second order central difference. By using compact finite differences, new standard and rotated point schemes with fourth order accuracy for the two-dimensional (2D) Helmholtz equation are formulated in this thesis. The fourth order point schemes in both standard and rotated grids can be further applied to formulate a fourth order system to be used as group iterative method in their respective grid. On the other hand, the multiscale multigrid method combined with Richardson’s extrapolation is first introduced by Zhang [18] to solve the 2D Poisson equation. By combining all the fourth order schemes, multiscale multigrid method and Richardson’s extrapolation in the solution of the 2D Helmholtz equation, the order of accuracy of the approximation can be improved up to sixth order, and with larger mesh size, the convergence rate of these iterative methods is faster as well. Numerical experiments are conducted on all the schemes combined with multiscale multigrid method and Richardson’s extrapolation, and the results are compared with existing point and group methods solved by using the multigrid method. The results show the improvements in the convergence rate and the efficiency of the newly formulated iterative schemes/systems.
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Mathematics