The Schwarz alternating sinc domain decomposition method

Morlet, Anne C.; Lybeck, Nancy J.; Bowers, Kenneth L.

doi:10.1016/s0168-9274(97)00068-8

Cited by 14 publications

(5 citation statements)

References 11 publications

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“…Error in the Sinc approximation for the test functions given inFig. 1using increasing values of N. Errors are identical for both cases a) and b2.4E-3 8.7E-5 1.3E-5 2.7E-8 5.6E-11 6.1E-16 1.6E-16 1.5E-16 0.10 9.5E-3 5.7E-4 1.1E-5 1.4E-7 5.4E-11 4.4E-16 2.1E-16 1.5E-16 0.15 7.2E-3 7.3E-4 3.0E-5 1.7E-7 1.6E-11 8.3E-17 3.3E-16 1.7E-16 0.20 4.8E-4 1.0E-3 2.7E-6 1.1E-7 1.5E-11 1.6E-15 2.8E-16 6.1E-16 0.25 7.5E-3 3.7E-4 3.3E-5 1.2E-7 3.8E-11 1.1E-15 1.7E-16 6.1E-16 0.30 1.1E-2 5.2E-4 2.1E-5 1.6E-7 9.6E-11 1.9E-15 6.1E-16 1.7E-16 0.35 1.2E-2 1.1E-3 1.3E-5 1.4E-7 6.9E-11 1.8E-15 3.9E-16 4.4E-16 0.40 9.4E-3 1.1E-3 3.5E-5 1.1E-7 4.8E-11 1.4E-15 2.2E-16 2.2E-16 0.45 5.1E-3 6.9E-4 2.8E-5 1.9E-7 1.0E-10 5.6E-16 1.4.6E-3 6.5E-4 2.6E-5 1.9E-7 9.8E-11 7.8E-16 1.1E-16 0 0.60 7.8E-3 9.9E-4 3.2E-5 1.0E-7 4.6E-11 7.8E-16 8.9E-16 1.1E-16 0.65 8.9E-3 8.9E-4 1.1E-5 1.3E-7 6.4E-11 1.3E-15 1.1E-16 5.6E-16 0.70 7.6E-3 3.9E-4 1.7E-5 1.4E-7 8.7E-11 8.9E-16 5.6E-16 8.9E-16 0.75 4.4E-3 2.6E-4 2.6E-5 9.7E-8 3.3E-11 2.2E-15 1.1E-15 1.1E-16 0.80 2.3E-4 6.4E-4 1.9E-6 8.7E-8 1.3E-11 2.0E-15 7.8E-16 1.1E-15 0.85 2.8E-3 3.9E-4 2.0E-5 1.2E-7 1.3E-11 0 3.3E-16 1.1E-16 0.90 2.6E-3 2.5E-4 6.0E-6 9.8E-8 4.1E-11 3.3E-16 3.9E-16 7.2E-16 0.95 2.5E-4 2.6E-5 5.7E-6 1.6E-8 3.9E-11 4.2E-16 7.2E-16 5.…”

mentioning

confidence: 86%

“…For a second-order, two-point boundary-value problem with multiple domains, Morlet, Lybeck and Bowers 16,17 have combined the Sinc-collocation domain decomposition method with the Schwarz alternating technique to overcome the problem's singularities. In their work, the exponential convergence property was proven for a problem with subdomains.…”

Section: Introductionmentioning

confidence: 99%

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Navier-Stokes Solution of the Driven-Cavity Problem Using Sinc-Collocation Elements

Narasimhan

Majdalani

2002

32nd AIAA Fluid Dynamics Conference and Exhibit

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Different numerical approaches have been proposed in the past to solve the Navier-Stokes equations. Conventional methods have often relied on finite-difference, finite-element, and boundary-element techniques. Multi-grid methods have been recently introduced because they help promote a faster convergence rate of the error residual. A difficulty plaguing numerical methods today is the inability to treat singularities at or near boundaries. Such difficulties become even more pronounced when coupled with the need to handle semiinfinite and infinite domains. Sinc-based numerical algorithms have the advantage of handling singularities, boundary layers, and semi-infinite domains very effectively. In addition, they typically require fewer nodal points while providing an exponential convergence rate in solving linear differential equations. This study involves a first step in applying the Sinc-based algorithm to solve a nonlinear set of partial differential equations. The example we consider arises in the context of a driven-cavity flow in two space dimensions. As such, the steady and incompressible Navier-Stokes equations are solved by means of two-dimensional Sinc collocation in conjunction with the primitive variable method and a pressure correction algorithm based on artificial compressibility. Simulations are also carried out using forward differences, central differences, and a commercial code. Results are compared with one another and with the Sinc-collocation approximation. It is found that the error in the Sinc-collocation approximation outperforms other solutions, especially near the singular corners of the cavity.

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mentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Navier-Stokes Solution of the Driven-Cavity Problem Using Sinc-Collocation Elements

Narasimhan

Majdalani

2002

32nd AIAA Fluid Dynamics Conference and Exhibit

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“…The functions are given in Figure 3 for real values of t. In (14), w = γ(r) and γ −1 (w) = r = e w . We may define γ −1 on the real line as…”

Section: Sinc-approximation Formula For Hyperbolicmentioning

confidence: 99%

Sinc-Galerkin method for solving hyperbolic partial differential equations

Seçer

2018

Int. J. Optim. Control, Theor. Appl. (IJOCTA)

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In this work, we consider the hyperbolic equations to determine the approximate solutions via Sinc-Galerkin Method (SGM). Without any numerical integration, the partial differential equation transformed to an algebraic equation system. For the numerical calculations, Maple is used. Several numerical examples are investigated and the results determined from the method are compared with the exact solutions. The results are illustrated both in the table and graphically.

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“…For the overlapping method, which uses matching conditions (3.3), the Schwarz alternating method may be used to iteratively solve the system of equations. Details are given in [12] for the two-point boundary-value problem. For the patching method, which uses matching condition (3.4), an appropriate iterative procedure is outlined in [ 141.…”

Section: )mentioning

confidence: 99%

Domain decomposition in conjunction with sinc methods for Poisson's equation

Lybeck

Bowers

1996

Numer. Methods Partial Differential Eq.

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Efforts to develop sinc domain decomposition methods for second-order two-point boundary-value problems have been successful, thus warranting further development of these methods. A logical first step is to thoroughly investigate the extension of these methods to Poisson's equation posed on a rectangle. The Sinc-Galerkin and sinc-collocation methods are, for appropriate weight choices, identical for Poisson's equation, and thus only the Sinc-Galerkin system is discussed here. Both the Sincaalerkin patching method and the Sincaalerkin overlapping method are presented in the simple case of decomposition into two subdomains. Numerical results are presented for each of these methods, showing the convergence. As an indication of the capabilities of these domain d e c o m psition techniques, they are applied to Poisson's equation on an L-shaped domain. Restrictions due to the method by which the discrete system is developed require that this problem be solved using nonoverlapping subdomains. Thus only the Sinc-Galerkin patching method is presented. Numerical results are presented that show the convergence of the approximate solutions, even in the presence of boundary singularities. 0 1996 John Wiley & Sons, Inc.

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The Schwarz alternating sinc domain decomposition method

Cited by 14 publications

References 11 publications

Navier-Stokes Solution of the Driven-Cavity Problem Using Sinc-Collocation Elements

Navier-Stokes Solution of the Driven-Cavity Problem Using Sinc-Collocation Elements

Sinc-Galerkin method for solving hyperbolic partial differential equations

Domain decomposition in conjunction with sinc methods for Poisson's equation

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