The efficient use of vector computers with emphasis to computational fluid dynamics

Schönauer, Willi; Gentzsch, Wolfgang

doi:10.1016/0167-8191(85)90020-1

Cited by 5 publications

(13 citation statements)

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“…6 shows that when the number of variables (proportional to the number of grid points) is increased, the vector speed up increases. When the dimensions of the arrays containing the variables of the problem increase, the calculus can take a better advantage of the vector hardware and in particular of the vector registers and of the pipeline mathematical unit 25 .…”

Section: Vectorizationmentioning

confidence: 99%

Parallel solution of three-dimensional Marangoni flow in liquid bridges

Lappa

Savino

1999

Int. J. Numer. Meth. Fluids

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This paper describes the implementation and performances of a parallel solver for the direct numerical simulation of the three-dimensional and time-dependent Navier Stokes equations on distributed-memory, massively parallel computers. The feasibility of this approach to study Marangoni flow instability in half zone liquid bridges is examined. The results indicate that the incompressible, non linear Navier-Stokes problem, governing the Marangoni flows behaviour, can effectively be parallelized on a distributedmemory parallel machine by remapping the distributed data structure. The numerical code is based on a three-dimensional Simplified Marker and Cell primitive variable method applied to a staggered finitedifference grid. Using this method, the problem is split in two problems, one parabolic and the other elliptic. A parallel algorithm, explicit in time, is utilized to solve the parabolic equations. A parallel multisplitting kernel is introduced for the solution of the pseudo-pressure elliptic equation, representing the most time-consuming part of the algorithm. A grid-partition strategy is used in the parallel implementations of both the parabolic equations and the multisplitting elliptic kernel. A Message Passing Interface (MPI) is coded for the boundary conditions; this protocol is portable to different systems supporting this interface for interprocessor communications. Numerical experiments illustrate good numerical properties and parallel efficiency. In particular, good scalability on a large number of processors can be achieved as long as the granularity of the parallel application is not too small. However, increasing the number of processors, the Speed-Up is ever smaller than the ideal linear Speed Up. The communication timings indicate that complex practical calculations, such as the solutions of the Navier-Stokes equations for the numerical simulation of the instability of Marangoni flows, can be expected to run on a massively parallel machine with good efficiency.

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Section: Vectorizationmentioning

confidence: 99%

Parallel solution of three-dimensional Marangoni flow in liquid bridges

Lappa

Savino

1999

Int. J. Numer. Meth. Fluids

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“…The minimal residual smoothing technique was introduced by Schönauer [31] and analyzed extensively by Weiss [32]. In this approach, an auxiliary sequence {v k } (with the associated residual sequence {s k }) is generated from {u h k } by a recurrence relations…”

Section: Acceleration By Minimal Residual Smoothingmentioning

confidence: 99%

Accelerated multigrid high accuracy solution of the convection-diffusion equation with high Reynolds number

Zhang

1997

Numer. Methods Partial Differential Eq.

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A fourth-order compact finite difference scheme is employed with the multigrid algorithm to obtain highly accurate numerical solution of the convection-diffusion equation with very high Reynolds number and variable coefficients. The multigrid solution process is accelerated by a minimal residual smoothing (MRS) technique. Numerical experiments are employed to show that the proposed multigrid solver is stable and yields accurate solution for high Reynolds number problems. We also show that the MRS acceleration procedure is efficient and the acceleration cost is negligible.

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“…We first present a simple residual smoothing scheme, which is a naive extension of the classical smoothing technique in [16,21] to the case of multiple right-hand sides (cf. [22]).…”

Section: Simple Smoothing Schemementioning

confidence: 99%

“…At this point, we reconsider the standard recursion formulas used in the global methods. When solving (16), the recursions (2) are replaced with…”

Section: Subsequently the Linear Transformation By ã For An Arbitrarymentioning

confidence: 99%

“…Residual smoothing is a well-known technique for generating a smoothed residual sequence from the primary residual sequence. However, as noted in [7], conventional smoothing schemes [16,21,24] do not aid in improving the attainable accuracy. This is due to the fact that the smoothed sequences are computed from the primary (non-smoothed) sequences, and they do not affect the primary iterations; thus, rounding errors that are accumulated in the primary sequences propagate to the smoothed sequences one-sidedly.…”

Section: Introductionmentioning

confidence: 99%

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Cross-interactive residual smoothing for global and block Lanczos-type solvers for linear systems with multiple right-hand sides

Aihara¹,

Imakura²,

Morikuni³

2021

Preprint

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Global and block Krylov subspace methods are efficient iterative solvers for large sparse linear systems with multiple right-hand sides. However, global or block Lanczos-type solvers often exhibit large oscillations in the residual norms and may have a large residual gap relating to the loss of attainable accuracy of the approximations. Conventional residual smoothing schemes suppress the oscillations but do not aid in improving the attainable accuracy, whereas a novel residual smoothing scheme enables the attainable accuracy for single right-hand side Lanczos-type solvers to be improved. The underlying concept of this scheme is that the primary and smoothed sequences of the approximations and residuals influence one another, thereby avoiding the severe propagation of rounding errors. In the present study, we extend this cross-interactive residual smoothing to the case of solving linear systems with multiple right-hand sides. The resulting smoothed methods can reduce the residual gap with few additional costs compared to their original counterparts. We demonstrate the effectiveness of the proposed approach through rounding error analysis and numerical experiments.

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The efficient use of vector computers with emphasis to computational fluid dynamics

Cited by 5 publications

References 0 publications

Parallel solution of three-dimensional Marangoni flow in liquid bridges

Parallel solution of three-dimensional Marangoni flow in liquid bridges

Accelerated multigrid high accuracy solution of the convection-diffusion equation with high Reynolds number

Cross-interactive residual smoothing for global and block Lanczos-type solvers for linear systems with multiple right-hand sides

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