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Local RBFs Based Collocation Methods for Unsteady Navier-Stokes Equations

Published online by Cambridge University Press:  29 May 2015

Xueying Zhang,
Xin An and
C. S. Chen
Xueying Zhang*
Affiliation:
College of Science, Hohai University, Nanjing, Jiangsu 210098, China
Xin An
Affiliation:
College of Science, Hohai University, Nanjing, Jiangsu 210098, China
C. S. Chen
Affiliation:
Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS 39406, U.S.ADepartment of Engineering Mechanics, Hohai University, Nanjing, Jiangsu 210098, China
*
*Corresponding author. Email: zhangxy.math@gmail.com (X. Y. Zhang)

Abstract

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The local RBFs based collocation methods (LRBFCM) is presented to solve two-dimensional incompressible Navier-Stokes equations. In avoiding the ill-conditioned problem, the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low-order linear systems within local supporting domain. Then, we reformulate local matrix in the global and sparse matrix. The obtained large sparse linear systems can be directly solved instead of using more complicated iterative method. The numerical experiments have shown that the developed LRBFCM is suitable for solving the incompressible Navier-Stokes equations with high accuracy and efficiency.

Keywords

65M1035L03Navier-Stokes equationslocal collocation methodparticular solutionsradial basis functions (RBFs)large sparse linear systems

Information

Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 4 , August 2015, pp. 430 - 440
Copyright
Copyright © Global-Science Press 2015 

References

[1]Hardy, R. L., Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res., 76 (1971), pp. 1905–1915.Google Scholar
[2]Kansa, E. J., A scattered scattered data approximation scheme with applications to computational fluiddynamics, I. Surface approximations and partial derivative estimates, Comput. Math. Appl., 19 (1990), pp. 127–145.Google Scholar
[3]Kansa, E. J., Multiquadrics, a scattered data approximation scheme with applications to computational fuid dynamics, II. Solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Appl., 19 (1990), pp. 147–161.Google Scholar
[4]Franke, R., Scatterd data interpolation: tests of some methods, Math. Comput., 38 (1982), pp. 181–199.Google Scholar
[5]Islam, S. U., Haqb, S. and Ali, A., A meshfree method for the numerical solution of the RLW equation, J. Comput. Appl. Math., 223 (2009), pp. 997–1012.Google Scholar
[6]Khattak, A. J., Tirmizi, S. I. A. and Islam, S. U., Application of meshfree collocation method to a class of nonlinear partial differential equations, Eng. Anal. Bound. Elem., 33 (2009), pp. 661–667.CrossRefGoogle Scholar
[7]Mai-Duy, N., Ho-Minh, D. and Tran-Cong, T., A Galerkin approach incorporating integrated radial basis function networks for the solution of 2D biharmonic equations, int. J. Comput. Math., 86 (2009), pp. 1746–1759.Google Scholar
[8]Parand, K., Abbasbandy, S., Kazem, S. and Rezaei, A., Comparison between two common collocation approaches based on radial basis functions for the case of heat transfer equations arising in porous medium, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), pp. 1396–1407.CrossRefGoogle Scholar
[9]Shu, C., Ding, H. and Yeo, K. S., Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 192 (2003), pp. 941–954.Google Scholar
[10]Shu, C., Yao, Q. and Yeo, K. S., Block-marching in time with DQ discretization: an efficient method for time-dependent problems, Comput. Methods Appl. Mech. Eng., 191 (2002), pp. 4587–4597.CrossRefGoogle Scholar
[11]Kosec, G. and Sarler, B., Solution of a low Prandtl number natural convection benchmark by a local meshless method, int. J. Numer. Methods Heat Fluid Flow, 23 (2012), pp. 189–204.Google Scholar
[12]Chen, C. S., Fan, C. M. and Wen, P. H., The method of particular solutions for solving elliptic problems with variable coefficients, int. J. Comput. Methods, 8 (2011), pp. 545–559.CrossRefGoogle Scholar
[13]Larsson, E. and Fornberg, B., A numerical study of radial basis functions based solution methods for elliptic pdes, Comput. Math. Appl., 46 (2003), pp. 891–902.Google Scholar
[14]Wen, P. H. and Chen, C. S., The method of particular solutions for solving scalar wave equations, int. J. Numer. Methods Biomed. Eng., 26 (2010), pp. 1878–1889.Google Scholar
[15]Islam, S. U., Vertnik, R. and Sarler, B., Local radial basis function collocation method along with explicit time stepping for hyperbolic partial differential equations, Appl. Numer. Math., 67 (2013), pp. 136–151.Google Scholar
[16]Yao, Guangming, Chen, C. S. and Kolibal, Joseph, A localized approach for the method of approximate particular solutions, Comput. Math. Appl., 61 (2011), pp. 2376–2387.Google Scholar
[17]Li, J. C. and Hon, Y. C., Domain decomposition for radialbasis meshless methods, Numer. Meth. Part. Differ. Eq., 20(3) (2004), pp. 450–462.CrossRefGoogle Scholar
[18]Chen, C. S., Golberg, M. A., Ganesh, M. and Cheng, A. H.-D., Multilevel compact radial functions basedcomputational schemes for some elliptic problems, Comput. Math. Appl., 43 (2002), pp. 359–378.Google Scholar
[19]Libre, N. A., Emdadi, A. and Kansa, E. J.ET AL., A stabilized rbf collocation scheme for neumann type boundary value problems, Comput. Model. Eng. Sci., 24 (2008), pp. 61–80.Google Scholar
[20]Lee, C. K., Liu, X. and Fan, S. C., Local muliquadric approximation for solving boundary value problems, Comput. Mech., 30 (2003), pp. 395–409.Google Scholar
[21]Sarler, B. and Vertnik, R., Meshfree explicit local radial basis function collocation method for diffusion problems, Comput. Math. Appl., 21 (2006), pp. 1269–1282.CrossRefGoogle Scholar