A numerical method for the solution of two-dimensional diffusion equations

Buleev, N. I.

doi:10.1016/s0368-3273(15)30051-1

Cited by 17 publications

(31 citation statements)

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“…Implicit factorization iterative procedures were first introduced by Buleev [6], Oliphant [16] and Varga [31] and our formalism for describing OBV methods arose from a tentative synthesis of their and later works, hence the name; our OBV methods include however both explicit and implicit techniques. We also attempt to classify somehow the OBV methods that have been considered in the literature, trying to make the distinction between basic methods and relaxation procedures.…”

Section: Factorization Iterative Methodsmentioning

confidence: 99%

Factorization iterative methods,M-operators andH-operators

Beauwens

1978

Numer. Math.

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Summary. We set up here a general formalism for describing factorization iterative methods of the first order and we use it to review various methods that have been proposed in the literature; next we introduce the notions of M-and H-operators which generalize those of block-M-and block-Hmatrices; finally we discuss the properties of factorization iterative methods in relation with characteristic properties of M-and H-operators.Subject Classifications. AMS(MOS): 65F10; CR: 5.14. Factorization Iterative MethodsIn this section, we define factorization iterative methods in general and OliphantBuleev-Varga or OBV methods in particular. Implicit factorization iterative procedures were first introduced by Buleev [6], Oliphant [16] and Varga [31] and our formalism for describing OBV methods arose from a tentative synthesis of their and later works, hence the name; our OBV methods include however both explicit and implicit techniques. We also attempt to classify somehow the OBV methods that have been considered in the literature, trying to make the distinction between basic methods and relaxation procedures. Definitions and NotationsWe consider here the solution of a linear system Ax=b, xEV, b~V, A~Sf(V) ( 1.1) where V denotes a real or complex n-dimensional vector space and 2'(V) the space of endomorphisms of V, by iterative schemes of the first order, i.

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Section: Factorization Iterative Methodsmentioning

confidence: 99%

Factorization iterative methods,M-operators andH-operators

Beauwens

1978

Numer. Math.

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“…Other notations are defined in section 2 of [1]. This definition is a generalization of the method used by Dupont Kendall and Rachford [4] for the multidimensional diffusion equation and the equivalence between this method and the method first introduced by Buleev [5] has been shown in [61.…”

Section: The Point Buleey Methodsmentioning

confidence: 99%

Point and block factorization iterative methods for arbitrary matrices and the characterization of M-matrices

Beauwens¹

1974

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“…A suitable choice of M can significantly accelerate the convergence of the method. In particular, Buleev [66], Stone [67], and Dupont et al [68] proposed an approximate factorization method for elliptic problems. It is convenient for problem (14) to compute an incomplete…”

Section: Milu Preconditionermentioning

confidence: 99%

Accelerated convergence of the numerical simulation of incompressible flow in general curvilinear co‐ordinates by discretizations on the double‐staggered grids

Shklyar

Arbel

2007

Numerical Methods in Fluids

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SUMMARYThe convergence rate of a methodology for solving incompressible flow in general curvilinear co-ordinates is analyzed. Double-staggered grids (DSGs), each defined by the same boundaries as the physical domain, are used for discretization. Both grids are MAC quadrilateral meshes with scalar variables (pressure, temperature, etc.) arranged at the center and the Cartesian velocity components at the middle of the sides of the mesh cells. The problem was checked against benchmark solutions of natural convection in a squeezed cavity, heat transfer in concentric horizontal cylindrical annuli, and a hot cylinder in a duct.Poisson's pressure-correction equations that arise from the SIMPLE-like procedure are solved by several methods: successive overrelaxation, symmetric overrelaxation, modified incomplete factorization preconditioner, conjugate gradient (CG), and CG with preconditioner. A genetic algorithm was developed to solve problems of numerical optimization of SIMPLE-like calculation time in a space of iteration numbers and relaxation parameters. The application provides a means of making an unbiased comparison between the DSGs method and the widely used interpolation method. Furthermore, the convergence rate was demonstrated by application to the calculation of natural convection heat transfer in concentric horizontal cylindrical annuli. Calculation times when DSGs were used were 2-10 times shorter than those achieved by interpolation. With the DSGs method, calculation time increases slightly with increasing non-orthogonality of the grids, whereas an interpolation method calls for very small iteration parameters that lead to unacceptable calculation times.

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A numerical method for the solution of two-dimensional diffusion equations

Cited by 17 publications

References 0 publications

Factorization iterative methods,M-operators andH-operators

Factorization iterative methods,M-operators andH-operators

Point and block factorization iterative methods for arbitrary matrices and the characterization of M-matrices

Accelerated convergence of the numerical simulation of incompressible flow in general curvilinear co‐ordinates by discretizations on the double‐staggered grids

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