A numerical study of divergence-free kernel approximations

Applied Numerical Mathematics - Tập 96 - Trang 94-107 - 2015
Arthur A. Mitrano1, Rodrigo B. Platte1
1School of Mathematical and Statistical Sciences, Arizona State University, Tempe, USA

Tài liệu tham khảo

Amodei, 1991, A vector spline approximation, J. Approx. Theory, 67, 51, 10.1016/0021-9045(91)90025-6 Dodu, 2002, Vectorial interpolation using radial-basis-like functions, Comput. Math. Appl., 43, 393, 10.1016/S0898-1221(01)00294-2 Dodu, 2004, Irrotational or divergence-free interpolation, Numer. Math., 98, 477, 10.1007/s00211-004-0541-x Driscoll, 2002, Interpolation in the limit of increasingly flat radial basis functions, Comput. Math. Appl., 43, 413, 10.1016/S0898-1221(01)00295-4 Driscoll, 2014 Flyer, 2012, A guide to RBF-generated finite differences for nonlinear transport: shallow water simulations on a sphere, J. Comput. Phys., 231, 4078, 10.1016/j.jcp.2012.01.028 Fornberg, 2011, Stable computations with Gaussian radial basis functions, SIAM J. Sci. Comput., 33, 869, 10.1137/09076756X Fornberg, 2011, Stabilization of RBF-generated finite difference methods for convective PDEs, J. Comput. Phys., 230, 2270, 10.1016/j.jcp.2010.12.014 Fornberg, 2007, A stable algorithm for flat radial basis functions on a sphere, SIAM J. Sci. Comput., 30, 60, 10.1137/060671991 Fornberg, 2004, Stable computation of multiquadric interpolants for all values of the shape parameter, Comput. Math. Appl., 48, 853, 10.1016/j.camwa.2003.08.010 Fornberg, 2004, Some observations regarding interpolants in the limit of flat radial basis functions, Comput. Math. Appl., 47, 37, 10.1016/S0898-1221(04)90004-1 Fornberg, 2007, The Runge phenomenon and spatially variable shape parameters in RBF interpolation, Comput. Math. Appl., 54, 379, 10.1016/j.camwa.2007.01.028 Fuselier, 2008, Sobolev-type approximation rates for divergence-free and curl-free RBF interpolants, Math. Comput., 77, 1407, 10.1090/S0025-5718-07-02096-0 Fuselier, 2009, Error and stability estimates for surface-divergence free RBF interpolants on the sphere, Math. Comput., 78, 2157, 10.1090/S0025-5718-09-02214-5 Fuselier, 2008, Improved stability estimates and a characterization of the native space for matrix-valued RBFs, Adv. Comput. Math., 29, 269, 10.1007/s10444-007-9046-3 Gerbeau, 1997, Spurious velocities in the steady flow of an incompressible fluid subjected to external forces, Int. J. Numer. Methods Fluids, 25, 679, 10.1002/(SICI)1097-0363(19970930)25:6<679::AID-FLD582>3.0.CO;2-Q Handscomb, 1991 Handscomb, 1993, Local recovery of a solenoidal vector field by an extension of the thin-plate spline technique, Numer. Algorithms, 5, 121, 10.1007/BF02212043 Hirsch, 2007 Kosloff, 1993, A modified Chebyshev pseudospectral method with an O(N−1) time step restriction, J. Comput. Phys., 104, 457, 10.1006/jcph.1993.1044 Larsson, 2005, Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions, Comput. Math. Appl., 49, 103, 10.1016/j.camwa.2005.01.010 Lee, 2007, Convergence of increasingly flat radial basis interpolants to polynomial interpolants, SIAM J. Math. Anal., 39, 537, 10.1137/050642113 Lowitzsch, 2005, Error estimates for matrix-valued radial basis function interpolation, J. Approx. Theory, 137, 238, 10.1016/j.jat.2005.09.008 Lowitzsch, 2005, Matrix-valued radial basis functions: stability estimates and applications, Adv. Comput. Math., 23, 299, 10.1007/s10444-004-1786-8 A.A. Mitrano, divfree-src, 2015, URL: http://dx.doi.org/10.5281/zenodo.15647. Narcowich, 1994, Generalized Hermite interpolation via matrix-valued conditionally positive definite functions, Math. Comput., 63, 661, 10.1090/S0025-5718-1994-1254147-6 Narcowich, 2007, Divergence-free RBFs on surfaces, J. Fourier Anal. Appl., 13, 643, 10.1007/s00041-006-6903-2 Platte, 2011, How fast do radial basis function interpolants of analytic functions converge?, IMA J. Appl. Math., 31, 1578 Platte, 2005, Polynomials and potential theory for Gaussian radial basis function interpolation, SIAM J. Numer. Anal., 43, 750, 10.1137/040610143 Platte, 2010, Chebfun: a new kind of numerical computing, 69 Platte, 2011, Impossibility of fast stable approximation of analytic functions from equispaced samples, SIAM Rev., 53, 308, 10.1137/090774707 Schaback, 2005, Multivariate interpolation by polynomials and radial basis functions, Constr. Approx., 21, 293, 10.1007/s00365-004-0585-2 Tolstykh, 2003, On using radial basis functions in a “finite difference mode” with applications to elasticity problems, Comput. Mech., 33, 68, 10.1007/s00466-003-0501-9 Townsend, 2013, An extension of Chebfun to two dimensions, SIAM J. Sci. Comput., 35, C495, 10.1137/130908002 Wendland, 2005, vol. 17 Wright, 2006, Scattered node compact finite difference-type formulas generated from radial basis functions, J. Comput. Phys., 212, 99, 10.1016/j.jcp.2005.05.030