and Computational Methods

doi: 10.18698/2309-3684-2014-4-3752

When sampling partial differential equations one has to solve a system of linear algebraic equations. To select the optimal in the sense of the computational efficiency of iterative method for solving such equations, in addition to the rate of convergence we should take into account such characteristics of the system and method, as the condition number, the smoothing factor, the indicator "costs on." The last two characteristics are calculated by the coefficients of harmonics amplification that give evidence of the smoothing properties of the iterative method and its "costs on", i. e. how worse the method suppresses frequency components of the error as compared with the highfrequency ones. The suggested method of determining harmonic gain factors is based on of the discrete Fourier transform. As an example, an analysis of the effectiveness of the BiCGStab method with ILU and multigrid preconditioning when solving difference analogues of the Helmholtz and Poisson equations is described.

[1] Zarubin V.S., Kuvyrkin G.N. Matematicheskoe modelirovanie i chislennye metody – Mathematical Modeling and Computational Methods, 2014, no. 1, pp. 5–17.

[2] PETSc – Portable, Extensible Toolkit for Scientific Computation. Available at: http://www.mcs.anl.gov/petsc.

[3] Scalable linear solvers: HYPRE. Available at: http://computation.llnl.gov/casc/linear_solvers/sls_hypre.html.

[4] Intel Math Kernel Library 11.0. Available at: http://software.intel.com/enus/intel-mkl.

[5] Wesseling P. An introduction to multigrid methods. Chichester, John Willey & Sons Ltd., 1991, 284 p.

[6] Ol'shansky M.A. Lektsii i uprazhneniya po mnogosetochnym metodam [Lectures and Exercises on Multigrid Methods]. Moscow, Lomonosov MSU Publ., 2003, 163 p.

[7] Fedorenko R.P. Introduction to Computational Physics. Moscow, Moscow Institute of Physics and Technology Publ., 1994, 528 p.

[8] Galanin M.P., Savenkov E.B. Methods of Numerical Analysis of Mathematical Models. Moscow, BMSTU Publ., 2010, 590 p.

[9] Saad Y. Iterative Methods for Sparse Linear Systems. New-York, PWS Publ., 1996, 547 p.

[10] Sergienko А.B. Digital Signal Processing. St. Petersburg, Piter Publ., 2002, 608 p.

[11] Van der Vorst H.A. SIAM J. Sci. Stat. Comp, 1992, no. 2, pp. 631–644.

[12] Il'in V.P. Sibirskiy Zhurnal Industrial'noy Matematiki, Siberian Journal of Industrial Mathematics, 2008, vol. 9, no. 4, pp. 47–60.

[13] Puzikova V.V. Vestnik MGTU im. N.E. Baumana. Seriya: Estestvennye nauki, Herald of the Bauman Moscow State Technical University, Series: Natural Science, 2011, special issue “Applied Mathematics”, pp. 124–133.

[14] Van Kan J., Vuik C., Wesseling P. Numer. Linear Algebra Appl, 2000, no. 7, pp. 429–447.

[15] OpenFOAM – The Open Source Computational Fluid Dynamics (CFD) Toolbox. Available at: http://www.openfoam.com.

Marchevsky I., Puzikova V. Performance analysis of iterative methods of combined linear algebraic equations solution. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 37-52

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