Application of the conjugate gradient method to the COMMIX-1B three dimensional momentum equation Public Deposited

http://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/r781wj24k

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  • The conjugate gradient method is an efficient means of solving large sparse symmetric positive definite systems of linear equations which arise from finite difference approximations to self-adjoint elliptic partial differential equations. In obtaining a solution, the conjugate gradient method successively minimizes a certain norm of the error in different orthogonal directions, causing an exact solution to be obtained in less than N steps for an NxN system of equations. Because the conjugate gradient method is not widely known, it is seldom used in engineering applications in comparison to the successive over-relaxation (S.O.R.) method. Although comparisons between the conjugate gradient and S.O.R. methods have been made, these comparisons usually focus on the solution of a single system of equations often arising from one or two dimensional problems. For this reason the purpose of this research was to compare the performance of these methods in the context of the C0MMIX-1B three dimensional thermal hydraulics code where these methods are required to solve many different systems of equations in a given problem to the same level of convergence. To accomplish its purpose, this thesis has three main objectives. The first is to give the reader sufficient background to understand the conjugate gradient method used in COMMIX-IB. The second is to show how the conjugate gradient method fits into the overall solution strategy of COMMIX-IB. The last is to compare the running times of the conjugate gradient and S.O.R. methods for general problems run with COMMIX-IB, and to discuss several factors affecting this comparison. It is concluded that under many circumstances, the conjugate gradient method is more efficient than S.O.R.
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