Applying multiquadric quasi-interpolation to solve Burgers' equation

Applied Mathematics and Computation

Volumn 172, Issue 1, 2006, Pages 472-484

  Chen, Ronghua ^a,b   Wu, Zongmin ^a  

^a FUDAN UNIVERSITY   (China)

^b HUNAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (China)

Author keywords

Burgers' equation; Multiquadric quasi interpolation; Radial basis function; Shape parameter

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; INTERPOLATION; MATHEMATICAL MODELS; REYNOLDS NUMBER; VISCOSITY;

BURGERS' EQUATION; MULTIQUADRIC QUASI-INTERPOLATION; RADIAL BASIS FUNCTION; SHAPE PARAMETER;

NONLINEAR PROGRAMMING;

EID: 31144472588     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2005.02.027     Document Type: Article

References (44)

1. R.K. Beatson, and N. Dyn Multiquadric B-splines J. Approx. Theory 87 1996 1 24
2. R.K. Beatson, and M.J.D. Powell Univariate multiquadric approximation: quasi-interpolation to scattered data Constr. Approx. 8 1992 275 288
3. M.D. Buhmann, and C.A. Micchelli Multiquadric interpolation improved Comput. Math. Appl. 24 1992 21 25
4. R.E. Carlson, and T.A. Foley Interpolation of track data with radial basis methods Comput. Math. Appl. 24 12 1992 27 34
5. C.S. Chen, G. Kuhn, J.C. Li, and G. Mishuris Radial basis functions for solving near singular Poisson problems Commun. Numer. Meth. En. 19 2003 333 347
6. R.H. Chen, Z.M. Wu, Solving hyperbolic conservation laws using multiquadric quasi-interpolation, Numer. Method PDE, submitted for publication.
7. I. Christie, and A.R. Mitchell Upwinding of high order Galerkin methods in conduction-convection problems Int. J. Numer. Methods Eng. 12 1978 1764 1771
8. C.K. Chui, J.D. Ward, and K. Jetter Cardinal interpolation with differences of tempered functions Comput. Math. Appl. 24 12 1992 35 48
9. J.D. Cole On a quasi-linear parabolic equation occurring in aerodynamics Quart. Appl. Math. 9 1951 225 236
10. G.E. Fasshauer Solving differential equations with radial basis functions: Multilevel methods and smoothing Adv. Comput. Math. 11 1999 139 159
11. G.E. Fasshauer Newton iteration with multiquadrics for the solution of nonlinear PDEs Comput. Math. Appl. 43 2002 423 438
12. A.I. Fedoseyev, M.J. Friedman, and E.J. Kansa Continuation for nonlinear elliptic partial differential equations discretized by the multiquadric method Int. J. Bifurcat. Chaos 10 2000 481 492
13. B. Fornberg, G. Wright, and E. Larsson Some observations regarding interpolants in the limit of flat radial basis functions Comput. Math. Appl. 47 2004 37 55
14. R. Franke Scattered data interpolation: test of some methods Math. Comput. 38 1982 181 200
15. M.J. Friedman Improved detection of bifurcations in large nonlinear system via the continuation of invariant subspace algorithm Int. J. Bifurcat. Chaos 11 2000 2277 2285
16. M.A. Golberg, and C.S. Chen Improved multiquadric approximation for partial differential equations Eng. Anal. Bound. Elem. 18 1996 9 17
17. Y.C. Hon Multiquadric collocation method with adaptive technique for problem with boundary layer Int. J. Appl. Sci. Comput. 6 1999 173 184
18. Y.C. Hon, M.W. Lu, W.M. Xue, and Y.M. Zhu Multiquadric method for the numerical solution of a biphasic model Appl. Math. Comput. 88 1997 153 175
19. Y.C. Hon, and X.Z. Mao A multiquadric interpolation method for solving initial value problems J. Sci. Comput. 12 1997 51 55
20. Y.C. Hon, and X.Z. Mao An efficient numerical scheme for Burgers' equation Appl. Math. Comput. 95 1998 37 50
21. Y.C. Hon, and Z.M. Wu A quasi-interpolation method for solving ordinary differential equations Int. J. Numer. Methods Eng. 48 2000 1187 1197
22. R.L. Hardy Theory and applications of the multiquadric-biharmonic method, 20 years of discovery 1968-1988 Comput. Math. Appl. 19 1990 163 208
23. E.J. Kansa Multiquadrics-a scattered data approximation scheme with applications to computational fluid dynamics I Comput. Math. Appl. 19 1990 127 145
24. E.J. Kansa Multiquadrics-a scattered data approximation scheme with applications to computational fluid dynamics II Comput. Math. Appl. 19 1990 147 161
25. E.J. Kansa, and R.E. Carlson Improved accuracy of multiquadric interpolation using variable shape parameters Comput. Math. Appl. 24 12 1992 99 120
26. E.J. Kansa, and Y.C. Hon Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to ellipitic partial differential equations Comput. Math. Appl. 39 2000 123 137
27. J.C. Li, and C.S. Chen Some observations on unsymmetric radial basis function collocation methods for convection-diffusion problems Int. J. Numer. Methods Eng. 57 2003 1085 1094
28. J.C. Li, A.H.D. Cheng, and C.S. Chen A comparison of efficiency and error convergence of multiquadric collocation method and finite element method Eng. Anal. Bound. Elem. 27 2003 251 257
29. W.R. Madych, and S.A. Nelson Multivariate interpolation and conditionally positive definite functions I J. Approx. Theory Appl. 4 1988 77 89
30. W.R. Madych Miscellaneous error bounds for multiquadric and related interpolators Comput. Math. Appl. 24 12 1992 121 138
31. C.A. Micchelli Interpolations of scattered data: distance matrices and conditionally positive definite functions Constr. Approx. 2 1986 11 22
32. F.J. Narcowich, and J.D. Ward Norm estimates for the inverses of a general class of scattered-data radial-function interpolation matrices J. Approx. Theory 69 1992 84 109
33. F.J. Narcowich, R. Schaback, and J.D. Ward Multilevel interpolation and approximation Appl. Comput. Harmonic Anal. 7 1999 243 261
34. F.J. Narcowich, J.D. Ward, and H. Wendland Refined error estimates for radial basis function interpolation Constr. Approx. 19 2003 541 564
35. R. Schaback, and Z.M. Wu Operators on radial basis functions J. Comput. Appl. Math. 73 1996 257 270
36. M. Sharan, E.J. Kansa, and S. Gupta Application of the multiquadric method for numerical solution of elliptic partial differential equations Appl. Math. Comput. 84 1997 275 302
37. H. Wendland Piecewise polynomial positive definite and compactly supported radial basis functions of minimal degree Adv. Comput. Math. 4 1995 389 396
38. A.S.M. Wong, Y.C. Hon, T.S. Li, S.L. Chung, and E.J. Kansa Multizone decomposition for simulation of time-dependent problem using the multiquadric scheme Comput. Math. Appl. 37 1999 23 43
39. Z.M. Wu Compactly supported radial functions and the Strang-Fix condition Appl. Math. Comput. 84 1997 115 124
40. Z.M. Wu Solving differential equation with radial basis function Advances in Computational Mathematics Lecture note in pure and applied mathematics vol. 202 1999 Dekker 537 544
41. Z.M. Wu Dynamically knots setting in meshless method for solving time dependent propagations equation Comput. Methods Appl. Mech. Eng. 193 2004 1221 1229
42. Z.M. Wu, and R. Schaback Local error estimates for radial basis function interpolation of scattered data IMA J. Numer. Anal. 13 1993 13 27
43. Z.M. Wu, and R. Schaback Shape preserving properties and convergence of univariate multiquadric quasi-interpolation ACTA Math. Appl. Sinica 10 1994 441 446
44. W.X. Zhang, Z.M. Wu, Some shape-preserving quasi-interpolations to non-uniformly distributed data constructed by MQ-B-splines, preprint Fudan University, 2003.