TEKNIK PEMECUTAN KAEDAH LELARAN TITIK DAN BERKUMPULAN UNTUK MENYELESAIKAN MASALAH POISSON
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Date
2005-09
Authors
CHONG, LEE SlAW
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Abstract
Numerical methods are often used to solve partial differential equations (POE) arising
from fluid mechanics and thermodynamic problems due to the difficulty in obtaining its
complicated analytical solution. The finite difference method is among the most
commonly used numerical method. Such method will lead to a large and sparse system
of linear equations that can be solved by iterative methods. The common problem
faced by researchers is the large amount of time needed to solve such large systems.
Hence, this thesis will investigate several accelerated techniques which can be used to
overcome this problem, specifically in solving the Poisson equation.
In solving the Poisson equation, two common approaches often taken are the
point iterative scheme and the group iterative scheme. The point iterative schemes
which are commonly used are iterative methods based on the standard five point and
rotated five point (Dahlquist and Bjorck (1974)) discretizations such as the standard
five point method and the rotated five point method. Among the group iterative scheme
which have been proven to work efficiently are the Explicit Group method/ EG (Yousif
and Evans (1986)) and Explicit Decoupled Group method/ EDG (Abdullah (1990)).
These schemes will lead to large system of linear equations where discretization
involving large grid size will consume large amount of time to solve. Therefore
accelerating methods such as SOR method(Successive Over Relaxation) and AOR
method(Accelerated Over Relaxation) are applied to improve its rate of convergence.
Recently researchers have taken an alternative approach towards this aim by
implementing preconditioning methods. Preconditioning method is any form of
modifying the original linear system so the iterative process will converge faster without
altering its exact solution. The main objective of this thesis is to investigate how the
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preconditioning method proposed by Martins, et.al (2000) affects the performance of
several point and group iterative schemes through SOR and AOR iterations.
Elliptic POE such as the Poisson problem is used as the model problem
studied. Numerical experiments will be implemented on each developed preconditioned
scheme so that comparison with the original scheme may be carried out. All results
obtained will be justified by the eigen value and computational complexity analysis so
that the effect of the preconditioning method can be seen clearly. At the end of this
thesis, the development cost of preconditioned systems will be discussed and several
suggestions are given for future research.
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TEKNIK PEMECUTAN KAEDAH LELARAN TITIK DAN BERKUMPULAN