On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere

Journal of Computational Physics

Volumn 227, Issue 5, 2008, Pages 2758-2780

  Fornberg, Bengt ^a   Piret, Cécile ^a  

^a UNIVERSITY OF COLORADO   (United States)

Author keywords

Radial basis functions; RBF; RBF QR method; Shape parameter; Sphere; Spherical harmonics; Wendland functions

Indexed keywords

FOURIER ANALYSIS; HARMONIC ANALYSIS; HARMONIC FUNCTIONS; HEAT CONDUCTION; IMAGE SEGMENTATION; RADIAL BASIS FUNCTION NETWORKS;

BASE FUNCTION; QR METHOD; RADIAL BASE FUNCTION; RADIAL BASE FUNCTION-QR METHOD; RADIAL BASIS; SHAPE PARAMETERS; SPHERICAL HARMONICS; WENDLAND FUNCTION;

SPHERES;

EID: 38549128750     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2007.11.016     Document Type: Article

References (46)

1. Boyd J.P. Chebyshev and Fourier Spectral Methods. second ed. (2001), Dover
2. Browning G.L., Hack J.J., and Swarztrauber P.N. A comparison of three different numerical methods for solving differential equations on the sphere. Mon. Wea. Rev. 117 (1989) 1058-1075
3. Buhmann M. Radial Basis Functions (2003), Cambridge University Press, Cambridge
4. Buhmann M., and Dyn N. Spectral convergence of multiquadric interpolation. Proc. Edinburgh Math. Soc. 36 (1993) 319-333
5. Driscoll T.A., and Fornberg B. Interpolation in the limit of increasingly flat radial basis functions. Comp. Math. Appl. 43 (2002) 413-422
6. Fasshauer G.E. Meshfree Approximation Methods with Matlab (2007), World Scientific Publishing, Singapore
7. Flyer N., and Wright G. Transport schemes on a sphere using radial basis functions. J. Comput. Phys. 226 (2007) 1059-1084
8. N. Flyer, G. Wright, Solving the nonlinear shallow water wave equations using radial basis functions (submitted for publication).
9. Fornberg B. On a Fourier method for the integration of hyperbolic equations. SIAM J. Numer. Anal. 12 4 (1975) 509-528
10. Fornberg B. A Practical Guide to Pseudospectral Methods (1996), Cambridge University Press, Cambridge
11. Fornberg B., and Flyer N. Accuracy of radial basis function interpolation and derivative approximations on 1-D infinite grids. Adv. Comput. Math. 23 (2005) 5-20
12. Fornberg B., Larsson E., and Wright G. A new class of oscillatory radial basis functions. Comput. Math. Appl. 51 (2006) 1209-1222
13. Fornberg B., and Merrill D. Comparison of finite difference and pseudospectral methods for convective flow over a sphere. Geophys. Res. Lett. 24 (1997) 3245-3248
14. Fornberg B., and Piret C. A stable algorithm for flat radial basis functions on a sphere. SIAM J. Sci. Comp. 30 (2007) 60-80
15. Fornberg B., and Wright G. Stable computation of multiquadric interpolants for all values of the shape parameter. Comp. Math. Appl. 48 (2004) 853-867
16. Fornberg B., Wright G., and Larsson E. Some observations regarding interpolants in the limit of flat radial basis functions. Comp. Math. Appl. 47 (2004) 37-55
17. Fornberg B., and Zuev J. The Runge phenomenon and spatially variable shape parameters in RBF interpolation. Comp. Math. Appl. 54 (2007) 379-398
18. Giraldo F.X., and Rosmond T.E. A scalable spectral element Eulerian atmospheric model (SEE-AM) for NWP: Dynamical core tests. Mon. Wea. Rev. 132 (2004) 133-153
19. J.J. Hack, R. Jakob, Description of a global shallow water model based on the spectral transform method, NCAR Technical Note TN 343 STR, 1992.
20. Hubbert S., and Baxter B. Radial basis functions for the sphere. Progress in Multivariate Approximation. International Series of Numerical Mathematics vol. 137 (2001), Birkhauser 33-47
21. Jakob-Chien R., Hack J.J., and Williamson D.L. Spectral transform solutions to the shallow water test set. J. Comp. Phys. 119 (1995) 164-187
22. Kansa E.J. Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics. I. Surface approximations and partial derivative estimates. Comput. Math. Appl. 19 8/9 (1990) 127-145
23. Kansa E.J. Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics. II. Solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Appl. 19 8/9 (1990) 147-161
24. Kreiss H.-O., and Oliger J. Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24 (1972) 199-215
25. Larsson E., and Fornberg B. A numerical study of radial basis function based solution methods for elliptic PDEs. Comp. Math. Appl. 46 (2003) 891-902
26. Larsson E., and Fornberg B. Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions. Comp. Math. Appl. 49 (2005) 103-130
27. Madych W.R., and Nelson S.A. Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation. J. Approx. Theory 70 (1992) 94-114
28. Merilees P.E. The pseudospectral approximation applied to the shallow water equation on a sphere. Atmosphere 11 (1973) 13-20
29. Merilees P.E. Numerical experiments with the pseudospectral method in spherical coordinates. Atmosphere 12 (1974) 77-96
30. Nair R.D., Thomas S.J., and Loft R.D. A discontinuous Galerkin transport scheme on the cubed sphere. Mon. Wea. Rev. 133 (2005) 814-828
31. Platte R., and Driscoll T.A. Eigenvalue stability of radial basis function discretizations for time-dependent problems. Comput. Math. Appl. 51 (2006) 1251-1268
32. Powell M.J.D. The theory of radial basis function approximation in 1990. In: Light W. (Ed). Advances in Numerical Analysis. Wavelets, Subdivision Algorithms and Radial Functions vol. II (1990), Oxford University Press, Oxford 105-210
33. Schaback R. Multivariate interpolation by polynomials and radial basis functions. Constr. Approx. 21 (2005) 293-317
34. Schaback R., and Wendland H. Characterization and construction of radial basis functions. In: Dyn N., et al. (Ed). Multivariate Approximation and Applications (2001), Cambridge University Press 1-24
35. Spotz W.F., Taylor M.A., and Swarztrauber P.N. Fast shallow water equation solvers in latitude-longitude coordinates. J. Comput. Phys. 145 (1998) 432-444
36. A. St-Cyr, C. Jablanowski, J.M. Dennis, H.M. Tufo, S.J. Thomas, A comparison of two shallow water models with non-conforming adaptive grids, Mon. Wea. Rev. (in press).
37. Swarztrauber P.N. Spectral transform methods for solving the shallow-water equations on the sphere. Mon. Wea. Rev. 124 (1996) 730-744
38. Taylor M., Tribbia J., and Iskandarani M. The spectral element method for the shallow water equations on the sphere. J. Comput. Phys. 130 (1997) 92-108
39. Temperton C. On scalar and vector transform methods for global spectral models. Mon. Wea. Rev. 119 (1991) 1303-1307
40. Thomas S.J., and Loft R.D. The NCAR spectral element climate dynamical core: semi-implicit, Eulerian formulation. J. Sci. Comp. 25 (2005) 307-322
41. Trefethen L.N. Spectral Methods in Matlab (2000), SIAM, Philadelphia
42. Wendland H. Scattered Data Approximation (2005), Cambridge University Press, Cambridge
43. Williamson D.L., Drake J.B., Hack J.J., Jakob R., and Swarztrauber P.N. A standard test set for numerical approximations to the shallow water equations in spherical geometry. J. Comput. Phys. 102 (1992) 211-224
44. R.S. Womersley, I. Sloan, Interpolation and cubature on the sphere, http://web.maths.unsw.edu.au/rsw/Sphere/.
45. Womersley R.S., and Sloan I. How good can polynomial interpolation on the sphere be?. Adv. Comput. Math. 14 (2001) 195-226
46. Yoon J. Spectral approximation orders of radial basis function interpolation on the Sobolev space. SIAM J. Math. Anal. 23 (2001) 946-958