RBF-FD formulas and convergence properties

The local RBF is becoming increasingly popular as an alternative to the global version that suffers from ill-conditioning. In this paper, we study analytically the convergence behavior of the local RBF method as a function of the number of nodes employed in the scheme, the nodal distance, and the shape parameter. We derive exact formulas for the first and second derivatives in one dimension, and for the Laplacian in two dimensions. Using these formulas we compute Taylor expansions for the error. From this analysis, we find that there is an optimal value of the shape parameter for which the error is minimum. This optimal parameter is independent of the nodal distance. Our theoretical results are corroborated by numerical experiments.

[1]  Shmuel Rippa,et al.  An algorithm for selecting a good value for the parameter c in radial basis function interpolation , 1999, Adv. Comput. Math..

[2]  Benjamin Seibold,et al.  Minimal positive stencils in meshfree finite difference methods for the Poisson equation , 2008, 0802.2674.

[3]  P. Lancaster,et al.  Surfaces generated by moving least squares methods , 1981 .

[4]  Somchart Chantasiriwan,et al.  Solutions to harmonic and biharmonic problems with discontinuous boundary conditions by collocation methods using multiquadrics as basis functions , 2007 .

[5]  Robert Vertnik,et al.  Meshfree explicit local radial basis function collocation method for diffusion problems , 2006, Comput. Math. Appl..

[6]  Bengt Fornberg,et al.  Scattered node compact finite difference-type formulas generated from radial basis functions , 2006, J. Comput. Phys..

[7]  S. Chantasiriwan.  Investigation of the use of radial basis functions in local collocation method for solving diffusion problems , 2004 .

[8]  R. Franke.  Scattered data interpolation: tests of some methods , 1982 .

[9]  B. Fornberg,et al.  Radial basis function interpolation: numerical and analytical developments , 2003 .

[10]  B. Fornberg,et al.  Some observations regarding interpolants in the limit of flat radial basis functions , 2003 .

[11]  Manuel Kindelan,et al.  Use of singularity capturing functions in the solution of problems with discontinuous boundary conditions , 2009 .

[12]  S. C. Fan,et al.  Local multiquadric approximation for solving boundary value problems , 2003 .

[13]  T. Driscoll,et al.  Interpolation in the limit of increasingly flat radial basis functions , 2002 .

[14]  Guirong Liu,et al.  A point interpolation meshless method based on radial basis functions , 2002 .

[15]  E. Kansa.  Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates , 1990 .

[16]  R. L. Hardy.  Multiquadric equations of topography and other irregular surfaces , 1971 .

[17]  H. Ding,et al.  Error estimates of local multiquadric‐based differential quadrature (LMQDQ) method through numerical experiments , 2005 .

[18]  A. Cheng,et al.  Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method , 2007 .

[19]  Robert Schaback,et al.  Error estimates and condition numbers for radial basis function interpolation , 1995, Adv. Comput. Math..

[20]  B. Fornberg,et al.  Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions , 2003 .

[21]  B. Fornberg.  CALCULATION OF WEIGHTS IN FINITE DIFFERENCE FORMULAS∗ , 1998 .

[22]  Thomas C. Cecil,et al.  Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions , 2004 .

[23]  Bengt Fornberg,et al.  Classroom Note: Calculation of Weights in Finite Difference Formulas , 1998, SIAM Rev..

[25]  C. Shu,et al.  Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations , 2003 .

[26]  Gregory E. Fasshauer,et al.  Meshfree Approximation Methods with Matlab , 2007, Interdisciplinary Mathematical Sciences.

[27]  T. Liszka,et al.  The finite difference method at arbitrary irregular grids and its application in applied mechanics , 1980 .

[28]  R. E. Carlson,et al.  The parameter R2 in multiquadric interpolation , 1991 .

[29]  E. Kansa.  MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS-- II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 1990 .