Proceedings of ISP RAS

OpenFOAM iterative methods efficiency analysis for linear systems solving.

I.K. Marchevsky, V.V. Puzikova.


Significant part of the computational work in numerical simulations of technical systems, physical phenomena and technological processes is solving of linear algebraic equations systems arising from the discretization of the corresponding differential or integrodifferential equations. There are several classes of iterative methods for linear algebraic equations systems solving, which differ by the approach to the construction of the next iterative approximation. These classes are methods based on splitting, variational-type methods and projection-type methods. The aim of this study is the approach development for computational efficiency analysis of iterative methods for linear algebraic equations systems solving, which are difference analogues of continuum mechanics equations, and the approaches for method a’priori choice for linear algebraic equation solving with high computational efficiency. To choose the optimal numerical method for linear systems solving, in addition to the rate of convergence such characteristics of a linear system and numerical method, as condition number, smoothing factor and cost-coefficient should be considered. The smoothing factor and cost-coefficient can be computed through the amplification factors of the modes. The performance of a smoothing method is measured by its smoothing factor, but the cost of a numerical method is measured through its costcoefficient which shows the difference between amplitudes vanishing speeds of smooth modes and rough modes. The method for modes amplification factors computing using the discrete Fourier transform is proposed. The cost-coefficient usage allows to choose the optimal parameters of the multigrid preconditioner. Some test problems are considered and the efficiency of BiCGStab (BiConjugate Gradient Stabilized) method with the Incomplete LU and multigrid preconditioners is investigated for linear systems solving which follow from discrete forms of Helmholtz and Poisson equations. These linear algebraic equations systems arise in numerical simulation of incompressible viscous flow in a square cavity by using the LS-STAG cut-cell immersed boundary method with level-set function.


Sparse linear systems; preconditioning; smoothers; modes amplification factors; multigrid methods.


Proceedings of the Institute for System Programming, vol. 24, 2013, pp. 71-86.

ISSN 2220-6426 (Online), ISSN 2079-8156 (Print).

DOI: 10.15514/ISPRAS-2013-24-4

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