Kaedah-kaedah lelaran berprasyarat dalam penyelesaian persamaan pembezaan separa
| dc.contributor.author | Teek Ling, Sam | |
| dc.date.accessioned | 2014-11-17T08:27:57Z | |
| dc.date.available | 2014-11-17T08:27:57Z | |
| dc.date.issued | 2007 | |
| dc.description | Master | en_US |
| dc.description.abstract | Numerical methods are techniques for finding approximate solutions to partial differential equations (PDEs) which arise from fluid dynamics and thermodynamics problems. Since the analytical solutions of the problems in these areas are difficult to obtain, the finite difference method is one of the existing methods which is capable of generating numerical solutions for the PDEs. Such methods will generally lead to a large and sparse linear system. Well-known iterative methods in solving this type of system are Jacobi, Gauss-Seidel (GS) and Successive Over Relaxation (SOR). However, the effectiveness of these methods is highly dependent on the diagonally dominant property of the resulting system coefficient matrix. Thus, researchers begin to shift to projection methods which do not require any knowledge of the diagonally dominant properties of the system coefficient matrix. Krylov subspace method is a projection method and it also known to be parameter-free iterative methods. Usually, large size systems which solved by the Krylov subspace methods are applied with a preconditioned matrix to further accelerate the convergence. A linear system of equations applied with a preconditioned matrix is modified to a different system which has the same solution as the original system but has more favourable spectral properties. In this thesis, we consider four preconditioned Krylov subspace methods, such as preconditioned Conjugate Gradient (CG), preconditioned Bi-Conjugate Gradient Stabilized (Bi-CGSTAB), preconditioned Generalized Minimal Residual (GMRES) and preconditioned Transpose-Free Quasi-Minimal Residual (TFQMR) which are applied to solve the linear system obtained from the point and group iterative schemes derived from the standard five-point and rotated five-point discretizations. Particularly, the group iterative methods that are under study are the Explicit Group (EG) method and Explicit Decoupled Group (EDG) method. The main issue in such methods is in choosing the preconditioners which are suitable to be used for the resulting system. The main objective of this thesis is to study the performance of Krylov subspace methods preconditioned by an incomplete LU (ILU) factorization on point iterative methods and a modified 2×2 block version of a diagonal-incomplete LU (D-ILU) factorization on group iterative methods which are derived from either the standard five-point formula or the rotated five-point formula. We will investigate whether these preconditioners are capable of improving the convergence rates of the original methods. In this thesis, a two-dimensional Poisson and convection-diffusion problems have been used as test problems to study the efficiency of the iterative methods. We perform numerical experiments on preconditioned Krylov subspace methods and the original methods, to compare the results between them. Finally, the computational complexity and the condition number are analysed to verify our numerical results. | en_US |
| dc.identifier.uri | http://hdl.handle.net/123456789/539 | |
| dc.language.iso | en | en_US |
| dc.subject | Mathematical science | en_US |
| dc.subject | Partial differential equation | en_US |
| dc.title | Kaedah-kaedah lelaran berprasyarat dalam penyelesaian persamaan pembezaan separa | en_US |
| dc.type | Thesis | en_US |
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