The method of incomplete factorization for solving two-dimensional and three-dimensional equations of the diffusion type

Buleev, N. I.

doi:10.1016/0041-5553(70)90027-3

Cited by 4 publications

(4 citation statements)

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“…The resulting algebraic equations systems for momentum equations were solved applying the line by line method with the implicit scheme of altering directions. The resulting algebraic equations system for energy equation was solved applying the explicit Buleev method for three-dimensional seven-point equations [27]. If values of mass balance for each control volume as well as the residual values of the different equations are sufficiently low overall convergence is obtained (typically 10´6).…”

Section: Numerical Methods and Validationmentioning

confidence: 99%

Analysis of Entropy Generation in Natural Convection of Nanofluid inside a Square Cavity Having Hot Solid Block: Tiwari and Das’ Model

Sheremet

Öztop

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et al. 2015

Entropy

106

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A computational work has been performed in this study to investigate the effects of solid isothermal partition insertion in a nanofluid filled cavity that is cooled via corner isothermal cooler. Mathematical model formulated in dimensionless primitive variables has been solved by finite volume method. The study is performed for different geometrical ratio of solid inserted block and corner isothermal cooler, Rayleigh number and solid volume fraction parameter of nanoparticles. It is observed that an insertion of nanoparticles leads to enhancement of heat transfer and attenuation of convective flow inside the cavity.

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Section: Numerical Methods and Validationmentioning

confidence: 99%

Analysis of Entropy Generation in Natural Convection of Nanofluid inside a Square Cavity Having Hot Solid Block: Tiwari and Das’ Model

Sheremet

Öztop

Pop

et al. 2015

Entropy

106

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“…Articles [2,3,5,8,9,10,11,12,14] are published in Russian. Scarce articles of the author [8,9,10,11,12,13] can be found at the web-address https://yadi.sk/d/plXWGEUGJRIkWQ.…”

Section: Comment To Referencesmentioning

confidence: 99%

“…The Incomplete Factorization iterative method (IF) for solving 2D and 3D elliptic finite-difference (FD) equations was suggested by Buleev [2]. In the method, schemes of factorization can be of Explicit (IFE) or Implicit (IFI) type, see [6].…”

Section: Introductionmentioning

confidence: 99%

On Incomplete Factorization Implicit Technique for 2d Elliptic Fd Equations

Sabinin

2020

Mathematical Modelling and Analysis

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A new variant of Incomplete Factorization Implicit (IFI) iterative technique for 2D elliptic finite-difference (FD) equations is suggested which is differed by applying the matrix tridiagonal algorithm. Its iteration parameter is shown be linked with the one for Alternating Direction Implicit method. An effective set of values for the parameter is suggested. A procedure for enhancing the set of iteration parameters for IFI is proposed. The technique is applied to a 5-point FD scheme, and to a 9-point FD scheme. It is suggested applying the solver for 5-point scheme to solving boundary-value problems for the 9-point scheme, too. The results of numerical experiment with Dirichlet and Neumann boundary-value problems for Poisson equation in a rectangle, and in a quasi-circle are presented. Mixed boundary-value problems in square are considered, too. The effectiveness of IFI is high, and weakly depends on the type of boundary conditions.

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“…In [12], N. I. Buleev suggested an iterative process where recurrence formula (0. In this case, z itj is assumed to be computed from the sequence z ij = <*ij z i-u + <Pij· On eliminating z itj from these equalities, we obtain the expression >ij -<*u w i-ι j -AJ"IJ-ι -)Wi + ι j ( 1 26 ) 1) which if only compared with original equation (0.1) [the 'compensating' term oCijO?,·.…”

Section: Componentwise Representation Of Implicit Algorithmsmentioning

confidence: 99%

Incomplete factorization methods

Ильин¹

1988

Russian Journal of Numerical Analysis and Mathematical Modelling

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The paper analyses explicit and implicit algorithms for incomplete factorization. The structure of preconditioning matrices and properties of their entries are considered as well as the conditions for the correctness of methods and for the convergence of iterative processes. The convergence rate estimates for iterations are given, and the results of numerical experiments are discussed.The incomplete factorization iterative methods have a thirty-year long history formally started in 1958 by publishing in [1] the Buleev method (later considered and developed in [2][3][4][5][6] and also in other papers) of solving a linear system of five-point grid equationsapproximating the boundary value problem for the second order elliptic equation. In (0.1), the boundary conditions are assumed to be taken into account at the values of coefficients a ij9 b ij9 c ij9 d ij9 e itj and of right-hand sides f itj . All coefficients are assumed to be non-negative, and the coefficient matrix is assumed to possess the properties of diagonal predominance, i.e. e itj ^ a itj + b itj + c itj + d ij9 and at least in one case the inequality is strict. It means that, since the coefficient matrix A is indecomposable, this matrix is monotonic (to be more precise, A belongs to the class of M-matrices; see [4]).Under the natural row-by-row ordering of nodes, the matrix of the system can be presented in the formwhere D = {Dj} is a block diagonal matrix, each block being a tridiagonal matrix with non-zero entries -a i>j9 e itj9 -c itj9 while L-{Lj} and U = {£/,·} are, correspondingly, lower and upper triangular matrices with non-zero entries b itj and d itj . If we exploit the fact that the grid nodes are ordered by diagonals, A is also a block tridiagonal matrix but now in equality (0.2), D is a diagonal matrix, while L and t/, in the general case, have in the rows two non-zero entries, each. The incomplete factorization methods were given rise to and developed on the basis of the approximation of A by a matrix K which can be presented in the form of a product of easily invertible matrices. The iterative process can be reduced to the formwhere u n = {w? tj ·}, /={//.,/}, and τ η are parameters determined from the Chebyshev acceleration, the conjugate gradient method and others. The diversity of choice of preconditioning matrices has resulted in a large number of publications containing different algorithms of incomplete factorization. These publications can be divided into two groups. The first group deals with the

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The method of incomplete factorization for solving two-dimensional and three-dimensional equations of the diffusion type

Cited by 4 publications

References 0 publications

Analysis of Entropy Generation in Natural Convection of Nanofluid inside a Square Cavity Having Hot Solid Block: Tiwari and Das’ Model

Analysis of Entropy Generation in Natural Convection of Nanofluid inside a Square Cavity Having Hot Solid Block: Tiwari and Das’ Model

On Incomplete Factorization Implicit Technique for 2d Elliptic Fd Equations

Incomplete factorization methods

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