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Article · NEWS OF THE NATIONAL ACADEMY OF SCIENCES OF THE REPUBLIC OF KAZAKHSTAN · April 15, 2021

ANALYSIS OF NUMERICAL SOLUTION OF POISSON EQUATION BY ILU (0)-CG METHOD

А.А. Абдурахимова, Nurislam Kassymbek, Orken J. Mamyrbayev

Orken Mamyrbayev
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Abstract

The problem of generalization of the method is the main question that arises when studying the quality of iterative methods. The efficiency of solving systems using iterative methods directly depends on the assumptions about the system of equations to be solved. Prerequisites are used to provide a more efficient solution. Many types of prerequisites are currently known, for example, prerequisites based on the approximation of the system matrix: ILU, IQR, and ILQ; Prerequisites based on the approximation of the inverse matrix: a polynomial, rarely filled approximation of the inverse matrix (for example, AINV), an approximation in the factorized form of the inverse matrix (for example, FSAI, SPAI, etc.). This article analyzes the CG and CG methods with the preconditioner ILU (0) by the example of solving the two-dimensional Poisson equation. The CG method is usually used to solve any system of linear equations. ILU (0) was selected as a prerequisite for the article. The incomplete LU decomposition (ILU (0)) is an efficient precursor and is easily implemented. This suggests a system that can be solved to speed up the accumulation of CG and other iterative methods, that is, to reduce the number of iterations. The ILU (0) preconditioner is very easy to detect using the LU decomposition. Since the linear matrix was rarely filled, the CSR format was used to store the matrix in memory. ILU (0) + CG, i.e. the algorithm with a precondition, was assembled 5-8 times faster than the CG algorithm. Data on the number of iterations of convergence of the method without a preconditioner and with the ILU(0) preconditioner were obtained and analyzed.

Matrix Theory and AlgorithmsIterative Methods for Nonlinear EquationsAdvanced Numerical Methods in Computational MathematicsPoisson equationincomplete LU preconditionerconjugate gradient method

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