Unsymmetric and symmetric meshless schemes for the unsteady convection-diffusion equation

Computer Methods in Applied Mechanics and Engineering

Volumn 195, Issue 19-22, 2006, Pages 2432-2453

  Chinchapatnam, P P ^a   Djidjeli, K ^a   Nair, P B ^a  

^a UNIVERSITY OF SOUTHAMPTON   (United Kingdom)

Author keywords

Convection diffusion; Radial basis functions; Symmetric collocation; Unsymmetric collocation

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; HEAT CONVECTION; NUMERICAL ANALYSIS; THERMAL DIFFUSION;

CONVECTION-DIFFUSION; RADIAL BASIS FUNCTIONS; SYMMETRIC COLLOCATION; UNSYMMETRIC COLLOCATION;

UNSTEADY FLOW;

CONVERGENCE OF NUMERICAL METHODS; HEAT CONVECTION; NUMERICAL ANALYSIS; THERMAL DIFFUSION; UNSTEADY FLOW;

EID: 33644584897     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2005.05.015     Document Type: Article

References (45)

1. H.-G. Roos M. Stynes L. Tobiska Numerical Methods for Singularly Perturbed Differential Equations (Convection-Diffusion and Flow Problems) 1996 Springer-Verlag Berlin
2. K.W. Morton Numerical Solution of Convection-Diffusion Equation 1995 Chapman and Hall London
3. J.H. Ferziger M. Peric Computational Methods for Fluid Dynamics 1999 Springer-Verlag Berlin
4. O.C. Zienkiewicz R.L. Taylor Finite Element Method fifth ed. Fluid Dynamics vol. 3 2000 Elsevier Amsterdam
5. E.J. Kansa Multiquadrics - a scattered data approximation scheme with applications to computation fluid dynamics: I. Surface approximations and partial derivatives estimates Comput. Math. Appl. 19 8/9 1990 127-145
6. E.J. Kansa Multiquadrics - a scattered data approximation scheme with applications to computation fluid dynamics: II. Solution to parabolic, hyperbolic and elliptic partial differential equations Comput. Math. Appl. 19 8/9 1990 147-161
7. B. Fornberg G. Wright E. Larsson Some observations regarding interpolants in the limit of flat radial basis functions Comput. Math. Appl. 1 2004 37-55
8. E.J. Kansa Y.C. Hon Circumventing the ill-conditioning problem with multiquadric radial basis functions Comput. Math. Appl. 39 2000 123-137
9. H. Wendland Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree Adv. Comput. Math. 4 1995 389-396
10. Z. Wu Compactly supported positive definite radial functions Adv. Comput. Math. 4 1995 283-292
11. M.D. Buhmann A new class of radial basis functions with compact support Math. Comput. 70 233 2000 307-318
12. S.M. Wong Y.C. Hon M.A. Golberg Compactly supported radial basis functions for shallow water equations Appl. Math. Comput. 127 2002 79-101
13. R. Schaback Error estimates and condition numbers for radial basis interpolation Adv. Comput. Math. 3 1995 251-264
14. Y.C. Hon R. Schaback On unsymmetric collocation by radial basis functions Appl. Math. Comput. 119 2001 177-186
15. G.E. Fasshauer Solving partial differential equations by collocation with radial basis functions A.L. Méchauté Proceedings of Chamonix 1996 Vanderbilt University Press Nashville, TN 1-8
16. Z. Wu, Solving PDE with radial basis function and the error estimation, in: Z. Chen, Y. Li, C.A. Micchelli, Y. Xu (Eds.), Advances in Computational Mathematics, Lecture Notes on Pure and Applied Mathematics, vol. 202, Guang Zhou,1998
17. Z. Wu Hermite-Birkhoff interpolation of scattered data by radial basis functions J. Approx. Theory 8 2 1992 1-11
18. C. Franke R. Schaback Convergence order estimates of meshless collocation methods using radial basis functions Adv. Comput. Math. 8 4 1998 381-399
19. R.A. Gingold J.J. Moraghan Smooth particle hydrodynamics: Theory and application to non-spherical stars Month. Notices Roy. Astron. Soc. 181 1997 375-389
20. J.J. Moraghan Why particle methods work SIAM J. Sci. Statist. Comput. 3 1982 422-433
21. J.J. Moraghan An introduction to SPH Comput. Phys. Comm. 48 1988 89-96
22. B. Nayroles G. Touzat P. Villon Generalizing the FEM: Diffuse approximation and diffuse elements Comput. Mech. 10 1992 307-318
23. T. Belytschko Y. Lu L. Gu Element free Galerkin methods Internat. J. Numer. Methods Engrg. 37 1994 229-256
24. Y. Lu T. Belytschko L. Gu A new implementation of the element free Galerkin method Comput. Methods Appl. Mech. Engrg. 113 1994 397-414
25. W.K. Liu S. Jun T. Belytschko Reproducing kernel particle methods Internat. J. Numer. Methods Fluids 20 1995 1081-1106
26. I. Boztosun A. Charafi RBF-based meshless schemes for advection-diffusion problems Adv. Bound. Elem. Tech. II 2001 573-581
27. H. Power V. Barraco A comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations Comput. Math. Appl. 43 2002 551-583
28. J. Li C.S. Chen Some observations on unsymmetric radial basis function collocation methods for convection-diffusion problems Internat. J Numer. Methods Engrg. 57 8 2003 1085-1094
29. M. Zerroukat K. Djidjeli A. Charafi Explicit and implicit meshless methods for linear advection-diffusion type partial differential equations Internat. J. Numer. Methods Engrg. 48 2000 19-35
30. I. Boztosun A. Charafi M. Zerroukat K. Djidjeli Thin-plate spline radial basis function scheme for advection-diffusion problems Elect. J. Bound. Elem. BETEQ 2 2002 267-282
31. I. Boztosun A. Charafi An analysis of the linear advection-diffusion equation using mesh-free and mesh-dependent methods Eng. Anal. Bound. Elem. 26 2002 889-895
32. G.E. Fasshauer, Meshfree methods in: M. Rieth, W. Schommers (Eds.), Handbook of Theoretical and Computational Nanotechnology, American Scientific publishers, to appear.
33. C.A. Micchelli Interpolation of scattered data: Distance matrices and conditionally positive definite functions Constr. Approx. 2 1986 11-22
34. R. Franke Scattered data interpolation: Tests of some methods Math. Comput. 38 1982 181-200
35. R.L. Hardy Theory and application of the multiquadric-biharmonic method Comput. Math. Appl. 19 1990 163-208
36. 2 in multiquadric interpolation Comput. Math. Appl. 21 1991 29-42
37. S. Rippa An algorithm for selecting a good value for the parameter c in radial basis function interpolation Adv. Comput. Math. 11 1999 193-210
38. E.J. Kansa R.E. Carlson Improved accuracy of multiquadric interpolation using variable shape parameters Comput. Math. Appl. 24 12 1992 99-120
39. R.L. Hardy Multiquadric equations of topography and other irregular surfaces J. Geophys. Res. 176 1971 1905-1915
40. A.H.D. Cheng M.A. Golberg E.J. Kansa G. Zammito Exponential convergence and H-c multiquadric collocation method for partial differential equations Numer. Methods Partial Diff. Eq. 19 2003 571-594
41. R.P. Brent Algorithms for Minimization Without Derivatives 1973 Prentice-Hall Englewood Cliffs, NJ
42. A.S.M. Wong Y.C. Hon T.S. Li S.L. Cheng E.J. Kansa Multizone decomposition for simulation of time-dependent problems using the multiquadric scheme Comput. Math. Appl. 37 1999 23-43
43. M.J.D. Powell The theory of radial basis function approximation in 1990 W. Light Advances in Numerical Analysis, II: Wavelets, Subdivision Algorithms and Radial Basis Functions 1992 Clarendon Press Oxford 105-210
44. R. Schaback Improved error bounds for scattered data interpolation by radial basis functions Math. Comput. 68 1999 201-206
45. I.M. Sobol On the systematic search in a hypercube SIAM J. Numer. Anal. 16 15 1979 790-793