A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations

Lu, Te‐Ling; Neittaanmäki, Pekka; Tai, Xue–Cheng

doi:10.1051/m2an/1992260606731

Cited by 87 publications

(63 citation statements)

References 12 publications

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“…We include here a brief description of the parallel method originally proposed in [19] and later used and analyzed in [8] and [11] to solve the previous PoissonDirichlet problems.…”

Section: The Solution To the Poisson-dirichlet Problems With Sdi Methodsmentioning

confidence: 99%

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Time and space parallelization of the navier-stokes equations

Núñez¹,

Canalejo²,

Soto³

et al. 2005

Mat. apl. comput.

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Abstract. In this paper, we will be mainly concerned with a parallel algorithm (in time and space) which is used to solve the incompressible Navier-Stokes problem. This relies on two main ideas: (a) a splitting of the main differential operator which permits to consider independently the most important difficulties (nonlinearity and incompressibility) and (b) the approximation of the resulting stationary problems by a family of second-order one-dimensional linear systems. The same strategy can be applied to two-dimensional and three-dimensional problems and involves the same level of difficulty. It can be also useful for the solution of other more complicate systems like Boussinesq or turbulence models. The behavior of the method is illustrated with some numerical experiments.Mathematical subject classification: 65M06, 35A35, 68W10.

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“…We include here a brief description of the parallel method originally proposed in [19] and later used and analyzed in [8] and [11] to solve the previous PoissonDirichlet problems.…”

Section: The Solution To the Poisson-dirichlet Problems With Sdi Methodsmentioning

confidence: 99%

“…It will be seen that this method leads to difficulties essentially of the same kind in the 2D and 3D settings. The seminal ideas for this approach can be found in [19].…”

Section: Introductionmentioning

confidence: 99%

Time and space parallelization of the navier-stokes equations

Núñez¹,

Canalejo²,

Soto³

et al. 2005

Mat. apl. comput.

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“…However, the system matrix in (11) is not axis symmetric, a property that may be important in some cases. If such a property is required, one could use the additive operator splitting (AOS) [13] which was actually invented for parallel implementation of splitting methods…”

Section: Operator Splitting Schemesmentioning

confidence: 99%

“…Historically, additive operator splitting (AOS) schemes were first developed for (nonlinear elliptic/parabolic) monotone equations and Navier-Stokes equations [12,13]. In image processing applications, the AOS scheme was found to be an efficient way for approximating the Perona-Malik filter [29], especially if symmetry in scale-space is required.…”

Section: Introductionmentioning

confidence: 99%

“…In image processing applications, the AOS scheme was found to be an efficient way for approximating the Perona-Malik filter [29], especially if symmetry in scale-space is required. The AOS scheme is first order in time, semi-implicit, and unconditionally stable with respect to its time-step [13,29]. In the early 1950's (see [19]) the alternating-direction method (ADI) was introduced, and in [31] the LOD (locally one-dimensional) splitting method was proposed.…”

Section: Introductionmentioning

confidence: 99%

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On Semi-implicit Splitting Schemes for the Beltrami Color Image Filtering

2011

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The Beltrami flow is an efficient nonlinear filter, that was shown to be effective for color image processing. The corresponding anisotropic diffusion operator strongly couples the spectral components. Usually, this flow is implemented by explicit schemes, that are stable only for very small time steps and therefore require many iterations. In this paper we introduce a semi-implicit Crank-Nicolson scheme based on locally one-dimensional (LOD)/additive operator splitting (AOS) for implementing the anisotropic Beltrami operator. The mixed spatial derivatives are treated explicitly, while the non-mixed derivatives are approximated in an implicit manner. In case of constant coefficients, the LOD splitting scheme is proven to be unconditionally stable. Numerical experiments indicate that the proposed scheme is also stable in more general settings. Stability, accuracy, and efficiency of the splitting schemes are tested in applications such as the Beltrami-based scale-space, Beltrami denoising and Beltrami deblurring. In order to further accelerate the convergence of the numerical scheme, the reduced rank extrapolation (RRE) vector extrapolation technique is employed.

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Nonoverlapping domain decomposition and additive splitting up methods for multidimensional parabolic problem

Daoud¹

2005

Numerical Methods Partial

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The explicit implicit domain decomposition methods are noniterative types of methods for nonoverlapping domain decomposition but due to the use of the explicit step for the interface prediction, the methods suffer from inaccuracy of the usual explicit scheme. In this article a specific type of first-and second-order splitting up method, of additive type, for the dependent variables is initially considered to solve the twoor three-dimensional parabolic problem over nonoverlapping subdomains. We have also considered the parallel explicit splitting up algorithm to define (predict) the interface boundary conditions with respect to each spatial variable and for each nonoverlapping subdomains. The parallel second-order splitting up algorithm is then considered to solve the subproblems defined over each subdomain; the correction step will then be considered for the predicted interface nodal points using the most recent solution values over the subdomains. Finally several model problems will be considered to test the efficiency of the presented algorithm.

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A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations

Cited by 87 publications

References 12 publications

Time and space parallelization of the navier-stokes equations

Time and space parallelization of the navier-stokes equations

On Semi-implicit Splitting Schemes for the Beltrami Color Image Filtering

Nonoverlapping domain decomposition and additive splitting up methods for multidimensional parabolic problem

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