Solution to a Convection-Diffusion Problem Using a New Variable Inverse-Multiquadric Parameter in a Collocation Meshfree Scheme
DOI:
https://doi.org/10.21152/1750-9548.11.4.359Abstract
The lack of reliable judgment on the choice of shape parameter is acknowledged as a drawback in the methodology of collocation meshfree by radial basis function (RBF). Some attempts have been proposed and tested only with specific classes of PDEs. Moreover, while most of this focusses on multiquadric (MQ) RBF, there has not been work done on invers-multiquadric (IMQ) RBF despite its’ increasing popularity. As a consequence, our main tasks in this work are, firstly, to numerically investigate the quality of each adaptive/variable RBF-shape parameter approaches presented in literature by applying them to the same type of problem, convection-diffusion class. Secondly, we proposed a new form of shape-parameter scheme to be used with the inverse-multiquadric (IMQ) type of RBF. The Kansa meshless method is implemented and it is interestingly found that the proposed form produces good results quality in terms of both matrix condition number, and the accuracy.
References
Gu, Y.T., Liu, G.R., Meshless techniques for convection dominated problems, Computational Mechanics, 2006, 38(2), 171-182. https://doi.org/10.1007/s00466-005-0736-8
Zhang, S.H., Wang, W.Q., A stencil of the finite-difference method for the 2D convection diffusion equation and its new iterative scheme, International Journal of Computer Mathematics, 2010, 87(11), 2588-2600. https://doi.org/10.1080/00207160802691637
Sun, H. W., Li, L.Z., A CCD–ADI method for unsteady convection–diffusion equations, Computer Physics Communications, 2014, 185(3), 790-797. https://doi.org/10.1016/j.cpc.2013.11.009
Chanthawara, K., Kaennakham, S. and Toutip, W., The numerical study and comparison of radial basis functions in applications of the dual reciprocity boundary element method to convection-diffusion problems: Proceeding of Progress in Applied Mathematics in Sciences and Engineering, AIP Publishing LLC, Melville, NY, 2016, 1705(1). https://doi.org/10.1063/1.4940277
Kansa, E.J., Multiquadrics{a scattered data approximation scheme with applications to computational fluid dynamics-I, Computers & Mathematics with Applications, 1990, 19(8-9), 127-145. https://doi.org/10.1016/0898-1221(90)90270-t
Gu, Y.T., Liu, G.R., An Introduction to Meshfree Methods and Their Programming, Springer, The Netherlands, 2005.
Buhmann, M.D., Multivariate cardinal interpolation with radial-basis functions, Constructive Approximation, 1990, 6(3), 225-255. https://doi.org/10.1007/bf01890410
Haung, C.S., Lee, C.F., Cheng, A.H.D., Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method, Engineering Analysis with Boundary Elements, 2007, 31(7), 614-623. https://doi.org/10.1016/j.enganabound.2006.11.011
Gherlone, M., Lurlaro, L., Sciuva, M.D., A novel algorithm for shape parameter selection in radial basis functions collocation method, Composite Structures, 2012, 94(2), 453-461. https://doi.org/10.1016/j.compstruct.2011.08.001
Rippa, S. An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Advances in Computational Mathematics, 1999, 11(2), 193-210.
Hardy, R.L., Multiquadric equations of topography and other irregular surfaces, Journal of Geophysical Research Atmospheres, 1971, 76(8), 1905-1915. https://doi.org/10.1029/jb076i008p01905
Franke,C, Schaback, R.A., Convergence order estimates of meshless collocation methods using radial basis functions, Advances in Computational Mathematics, 1998, 8(4), 381-399.
Zhang, K., Song, K.Z., Lu, M.W., Liu, X. Meshless methods based on collocation with radial basis functions , Computational Mechinics, 2000, 26(4), 333-343. https://doi.org/10.1007/s004660000181
Wang, J.G., Liu, G.R., On the optimal shape parameters of radial basis functions used for 2-D meshless methods, Computer Methods in Applied Mechanics and Engineering, 2002, 191(23-24), 2611-2630. https://doi.org/10.1016/s0045-7825(01)00419-4
Lee, C.K., Liu, X., Fan, S.C., Local multiquadric approximation for solving boundary value problems, Computational Mechanics, 2003, 30(5), 396-409. https://doi.org/10.1007/s00466-003-0416-5
Sarra, S.A., Sturgill, D. A random variable shape parameter strategy for radial basis function approximation methods, Engineering Analysis with Boundary Elements, 2009, 33(11), 1239-1245. https://doi.org/10.1016/j.enganabound.2009.07.003
Kansa, E.J., Carlson, R.E., Improved accuracy of multiquadric interpolation using variable shape parameters, Computers & Mathematics with Applications, 1992, 24(12), 99-120. https://doi.org/10.1016/0898-1221(92)90174-g
Sarra, S.A., Adaptive radial basis function methods for time dependent partial differential equations, Applied Numerical Mathematics, 2005, 54(1), 79-94. https://doi.org/10.1016/j.apnum.2004.07.004
Gu, Y.T., Liu, G.R., Meshless techniques for convection dominated problems, Computational Mechanics, 2006, 38(2), 171-182. https://doi.org/10.1007/s00466-005-0736-8
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