Node adaptive domain decomposition method by radial basis functions

González–Casanova, Pedro; Muñoz-Gómez, José Antonio; Rodríguez-Gómez, Gustavo

doi:10.1002/num.20410

Cited by 16 publications

(4 citation statements)

References 22 publications

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“…A dynamic adaptive scheme was proposed by Wu for time‐dependent PDEs. Moreover, Gonzalez‐Casanova et al have formulated a node adaptive domain decomposition method by RBFs. An adaptive algorithm with local thin plate spline(TPS) RBFs interpolation was developed by Behrens and Iske , and Behrens et al successfully applied, in a semi‐Lagrangian context, to linear evolutionary PDEs.…”

Section: Introductionmentioning

confidence: 99%

A shift‐adaptive meshfree method for solving a class of initial‐boundary value problems with moving boundaries in one‐dimensional domain

Esmaeilbeigi

Garmanjani

2016

Numerical Methods Partial

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A new shift‐adaptive meshfree method for solving a class of time‐dependent partial differential equations (PDEs) in a bounded domain (one‐dimensional domain) with moving boundaries and nonhomogeneous boundary conditions is introduced. The radial basis function (RBF) collocation method is combined with the finite difference scheme, because, unlike with Kansa's method, nonlinear PDEs can be converted to a system of linear equations. The grid‐free property of the RBF method is exploited, and a new adaptive algorithm is used to choose the location of the collocation points in the first time step only. In fact, instead of applying the adaptive algorithm on the entire domain of the problem (like with other existing adaptive algorithms), the new adaptive algorithm can be applied only on time steps. Furthermore, because of the radial property of the RBFs, the new adaptive strategy is applied only on the first time step; in the other time steps, the adaptive nodes (obtained in the first time step) are shifted. Thus, only one small system of linear equations must be solved (by LU decomposition method) rather than a large linear or nonlinear system of equations as in Kansa's method (adaptive strategy applied to entire domain), or a large number of small linear systems of equations in the adaptive strategy on each time step. This saves a lot in time and memory usage. Also, Stability analysis is obtained for our scheme, using Von Neumann stability analysis method. Results show that the new method is capable of reducing the number of nodes in the grid without compromising the accuracy of the solution, and the adaptive grading scheme is effective in localizing oscillations due to sharp gradients or discontinuities in the solution. The efficiency and effectiveness of the proposed procedure is examined by adaptively solving two difficult benchmark problems, including a regularized long‐wave equation and a Korteweg‐de Vries problem. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1622–1646, 2016

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Section: Introductionmentioning

confidence: 99%

A shift‐adaptive meshfree method for solving a class of initial‐boundary value problems with moving boundaries in one‐dimensional domain

Esmaeilbeigi

Garmanjani

2016

Numerical Methods Partial

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“…In their adaptive algorithm, nodes can be added and removed based on residuals evaluated at a finer points set [9]. In [10], a RBF dynamical domain decomposition algorithm which simultaneously performs nodes adaptive strategy is introduced. A review of center choosing algorithms is given in [11].…”

Section: Introductionmentioning

confidence: 99%

Application of variational mesh generation approach for selecting centers of radial basis functions collocation method

Barfeie

Soheili

Ameri

2013

Engineering Analysis with Boundary Elements

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“…It is worth to mention that, an important alternative to the ill-conditioned problem of RBFs collocation methods, is that they can be solved by using domain decomposition methods, see, [2], [1], [12]. In this work, however, we are interested in comparing the results of the local methods with those obtained by solving the global asymmetric collocation (AC) method by using direct solvers with quad precision which is a current alternative that has been used by Kansa [3] and Sarra, [8] among others.…”

Section: Introductionmentioning

confidence: 99%

Radial basis function methods for optimal control of the convection–diffusion equation: A numerical study

Casanova

Gout

Zavaleta

2019

Engineering Analysis with Boundary Elements

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Node adaptive domain decomposition method by radial basis functions

Cited by 16 publications

References 22 publications

A shift‐adaptive meshfree method for solving a class of initial‐boundary value problems with moving boundaries in one‐dimensional domain

A shift‐adaptive meshfree method for solving a class of initial‐boundary value problems with moving boundaries in one‐dimensional domain

Application of variational mesh generation approach for selecting centers of radial basis functions collocation method

Radial basis function methods for optimal control of the convection–diffusion equation: A numerical study

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