Splitting and alternating direction methods

Marchuk, G. I.

doi:10.1016/s1570-8659(05)80035-3

Cited by 367 publications

(317 citation statements)

References 54 publications

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“…As the amplitude of the displacement is increased, the area occupied by the vortices is enlarged, but the qualitative properties of the flow are the same. See figure (19). In all cases presented up to this point, the flow distributions display the axial and cyclic symmetries defined at the beginning of section 4.…”

Section: Case Re = 1000mentioning

confidence: 64%

“…There are many operator-splitting methods that can be employed to solve this type of problems. Here we consider the fractional step scheme of Marchunk-Yaneuko [19] type:…”

Section: Time Discretization By Operator Splittingmentioning

confidence: 99%

“…This very simple scheme is only first order accurate (see [19]), but its low order is compensated by good stability and robustness properties. Actually, this scheme can be made second order accurate by symmetrization (see [20] and [21] for the application of symmetrized splitting schemes to the solution of the Navier-Stokes equations).…”

Section: Time Discretization By Operator Splittingmentioning

confidence: 99%

“…Figures (18) and (19) show the velocity fields and vortices cores for Y = 0.2 and 0.4, for two phases of the oscillation. The vortex distribution and evolution are similar to those of Re=500, only the vortical structures are thinner and confined to the regions near the walls.…”

Section: Case Re = 1000mentioning

confidence: 99%

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Vortex formation in a cavity with oscillating walls

et al. 2009

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The vortex formation in a two-dimensional Cartesian cavity, which their vertical walls move simultaneously with an oscillatory velocity and the horizontal walls are fixed pistons, is studied numerically.The governing equations were solved with a finite element method combined with an operator splitting scheme. We analyzed the behavior of vortical structures occurring inside a cavity with an aspect ratio of height to width of 1.5 for three different displacement amplitudes of the vertical oscillatory walls (amplitude/width= Y = 0.2, 0.4 and 0.8) and Reynolds numbers based on the cavity width of 50, 500 and 1000. Two vortex formation mechanisms are identified: a) the shear, oscillatory motion of the moving boundaries coupled with the fixed walls that provide a translational symmetry-breaking effect and b) the sharp changes in the flow motion when the flow meets the corners of the cavity. The vortices cores were identified using the Jeong-Hussain criterium and it is found that they occupy smaller areas as the Reynolds number increases. All flows studied are cyclic symmetric and for low Y and Re values they are also symmetric with respect to the vertical axis dividing the cavity in two sides. In asymmetric flows, the unbalance between the vortices on each side of the mid-vertical line generates a vortex that occupies the central part of the cavity. The breakdown of the axial symmetry was studied for a fixed value of the oscillation amplitude, Y = 0.8, taking Re as the bifurcation parameter. The results indicates that the symmetry is broken through a supercritical pitchfork bifurcation.

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Section: Case Re = 1000mentioning

confidence: 64%

“…There are many operator-splitting methods that can be employed to solve this type of problems. Here we consider the fractional step scheme of Marchunk-Yaneuko [19] type:…”

Section: Time Discretization By Operator Splittingmentioning

confidence: 99%

Section: Time Discretization By Operator Splittingmentioning

confidence: 99%

Section: Case Re = 1000mentioning

confidence: 99%

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Vortex formation in a cavity with oscillating walls

et al. 2009

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“…There are different methods belonging to this class; see for example Young [27], Conrad and Wallach [8], Marchuk [16], and Benzi and Szyld [2] for the case p = 2 and µ (1, k…”

Section: Sequential Alternating Iterative Methodsmentioning

confidence: 99%

Sequential and parallel synchronous alternating iterative methods

Climent¹,

Perea²,

Tortosa³

et al. 2003

Math. Comp.

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Abstract. The so-called parallel multisplitting nonstationary iterative Model A was introduced by Bru, Elsner, and Neumann [Linear Algebra and its Applications 103:175-192 (1988)] for solving a nonsingular linear system AÜ = using a weak nonnegative multisplitting of the first type. In this paper new results are introduced when A is a monotone matrix using a weak nonnegative multisplitting of the second type and when A is a symmetric positive definite matrix using a P -regular multisplitting. Also, nonstationary alternating iterative methods are studied. Finally, combining Model A and alternating iterative methods, two new models of parallel multisplitting nonstationary iterations are introduced. When matrix A is monotone and the multisplittings are weak nonnegative of the first or of the second type, both models lead to convergent schemes. Also, when matrix A is symmetric positive definite and the multisplittings are P -regular, the schemes are also convergent.

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Use of Eigenvalue Technique in Finite Element Tidal Computations

Patil

Rao

1997

Int. J. Numer. Meth. Fluids

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Splitting and alternating direction methods

Cited by 367 publications

References 54 publications

Vortex formation in a cavity with oscillating walls

Vortex formation in a cavity with oscillating walls

Sequential and parallel synchronous alternating iterative methods

Use of Eigenvalue Technique in Finite Element Tidal Computations

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