Scattered node compact finite difference-type formulas generated from radial basis functions

In standard equispaced finite difference (FD) formulas, symmetries can make the order of accuracy relatively high compared to the number of nodes in the FD stencil. With scattered nodes, such symmetries are no longer available. The generalization of compact FD formulas that we propose for scattered nodes and radial basis functions (RBFs) achieves the goal of still keeping the number of stencil nodes small without a similar reduction in accuracy. We analyze the accuracy of these new compact RBF-FD formulas by applying them to some model problems, and study the effects of the shape parameter that arises in, for example, the multiquadric radial function.

[1]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

[2]  R. Schaback.  Multivariate Interpolation by Polynomials and Radial Basis Functions , 2005 .

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

[4]  Ming-Chih Lai,et al.  Fast direct solvers for Poisson equation on 2D polar and spherical geometries , 2002 .

[5]  Leevan Ling,et al.  A least-squares preconditioner for radial basis functions collocation methods , 2005, Adv. Comput. Math..

[6]  Rémi Abgrall,et al.  On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation , 1994 .

[7]  J. Demmel.  Numerical linear algebra , 1993 .

[8]  Benny Y. C. Hon,et al.  An efficient numerical scheme for Burgers' equation , 1998, Appl. Math. Comput..

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

[10]  C. Micchelli.  Interpolation of scattered data: Distance matrices and conditionally positive definite functions , 1986 .

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

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

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

[14]  Robin J. Y McLeod,et al.  Geometry and Interpolation of Curves and Surfaces , 1998 .

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

[16]  A. U.S.,et al.  Stable Computation of Multiquadric Interpolants for All Values of the Shape Parameter , 2003 .

[17]  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 .

[18]  R. L. Hardy.  Theory and applications of the multiquadric-biharmonic method : 20 years of discovery 1968-1988 , 1990 .

[19]  A. I. Tolstykh,et al.  High-accuracy discretization methods for solid mechanics , 2003 .

[20]  Gregory E. Fasshauer,et al.  Solving differential equations with radial basis functions: multilevel methods and smoothing , 1999, Adv. Comput. Math..

[21]  Willi Schönauer,et al.  How we solve PDEs: 473 , 2001 .

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

[23]  L. Collatz.  The numerical treatment of differential equations , 1961 .

[24]  Willi Schönauer,et al.  How We solve PDEs , 2001 .

[25]  Elisabeth Larsson,et al.  A new class of oscillatory radial basis functions , 2006, Comput. Math. Appl..

[26]  Carsten Franke,et al.  Solving partial differential equations by collocation using radial basis functions , 1998, Appl. Math. Comput..

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

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

[29]  Gregory E. Fasshauer,et al.  Hermite interpolation with radial basis functions on spheres , 1999, Adv. Comput. Math..

[30]  Bengt Fornberg,et al.  Accuracy of radial basis function interpolation and derivative approximations on 1-D infinite grids , 2005, Adv. Comput. Math..

[31]  B. Fornberg,et al.  A numerical study of some radial basis function based solution methods for elliptic PDEs , 2003 .

[32]  Ramesh Chand Mittal,et al.  High-Order Finite-Differences Schemes to Solve Poisson's Equation in Polar Coordinates , 1991 .

[33]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[34]  E. Kansa,et al.  Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary , 2002 .

[35]  W. R. Madych,et al.  Miscellaneous error bounds for multiquadric and related interpolators , 1992 .

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

[37]  E. Kansa,et al.  Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations , 2000 .

[38]  G. R. Liu,et al.  1013 Mesh Free Methods : Moving beyond the Finite Element Method , 2003 .

[39]  Xingping Sun,et al.  Scattered Hermite interpolation using radial basis functions , 1994 .

[40]  吴宗敏.  HERMITE—BIRKHOFF INTERPOLATION OF SCATTERED DATA BY RADIAL BASIS FUNCTIONS , 1992 .

[41]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[42]  Jungho Yoon,et al.  Spectral Approximation Orders of Radial Basis Function Interpolation on the Sobolev Space , 2001, SIAM J. Math. Anal..