Rotational transport on a sphere: Local node refinement with radial basis functions

This paper develops an algorithm for radial basis function (RBF) local node refinement and implements it for vortex roll-up and transport on a sphere. A heuristic based on an electrostatic repulsion type principle is used to re-distribute the nodes, clustering in areas where higher resolution is needed. It is then important to have a scheme that varies the shape of the RBFs over the domain so as to counteract the effects of Runge phenomena where the nodes are sparse. The roll-up of two diametrically opposed moving vortices are studied. The performance differences between near-uniform and refined nodes are addressed in terms of convergence, time stability, and computational cost. RBF results are put into context by comparison with published results for methods such as finite volume and discontinuous Galerkin.

[1]  I. J. Schoenberg.  Metric spaces and completely monotone functions , 1938 .

[2]  Robert Schaback,et al.  Interpolation by basis functions of different scales and shapes , 2004 .

[3]  Simon Hubbert,et al.  Lp-error estimates for radial basis function interpolation on the sphere , 2004, J. Approx. Theory.

[4]  C. Jablonowski,et al.  Moving Vortices on the Sphere: A Test Case for Horizontal Advection Problems , 2008 .

[5]  J. Wertz,et al.  The role of the multiquadric shape parameters in solving elliptic partial differential equations , 2006, Comput. Math. Appl..

[6]  Bengt Fornberg,et al.  A Stable Algorithm for Flat Radial Basis Functions on a Sphere , 2007, SIAM J. Sci. Comput..

[7]  R. E. Carlson,et al.  Improved accuracy of multiquadric interpolation using variable shape parameters , 1992 .

[8]  Ian H. Sloan,et al.  How good can polynomial interpolation on the sphere be? , 2001, Adv. Comput. Math..

[9]  Bengt Fornberg,et al.  Stable Computation of Multiquadric Interpolants for All Values of the Shape Parameter , 2004 .

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

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

[12]  I. J. Schoenberg.  Positive definite functions on spheres , 1942 .

[13]  Gregory E. Fasshauer,et al.  On choosing “optimal” shape parameters for RBF approximation , 2007, Numerical Algorithms.

[14]  A. Kuijlaars.  Ward Cheney and Will Light, A Course in Approximation Theory , 2001 .

[15]  Bengt Fornberg,et al.  The Runge phenomenon and spatially variable shape parameters in RBF interpolation , 2007, Comput. Math. Appl..

[16]  Bengt Fornberg,et al.  On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere , 2008, J. Comput. Phys..

[17]  Kurt Jetter,et al.  Error estimates for scattered data interpolation on spheres , 1999, Math. Comput..

[18]  Natasha Flyer,et al.  Transport schemes on a sphere using radial basis functions , 2007, J. Comput. Phys..

[19]  Shian-Jiann Lin,et al.  Finite-volume transport on various cubed-sphere grids , 2007, J. Comput. Phys..

[20]  Natasha Flyer,et al.  A radial basis function method for the shallow water equations on a sphere , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[21]  M. J. D. Powell,et al.  Radial basis function methods for interpolation to functions of many variables , 2001, HERCMA.

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

[23]  Stephen J. Thomas,et al.  A Discontinuous Galerkin Transport Scheme on the Cubed Sphere , 2005 .

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

[25]  Ward Cheney,et al.  A course in approximation theory , 1999 .

[26]  Martin D. Buhmann,et al.  Radial Basis Functions: Theory and Implementations: Preface , 2003 .

[27]  Vladimir Cherkassky,et al.  Learning from Data: Concepts, Theory, and Methods , 1998 .

[28]  Scott A. Sarra,et al.  A random variable shape parameter strategy for radial basis function approximation methods , 2009 .

[29]  Bengt Fornberg,et al.  Locality properties of radial basis function expansion coefficients for equispaced interpolation , 2007 .

[30]  Armin Iske,et al.  Multiresolution Methods in Scattered Data Modelling , 2004, Lecture Notes in Computational Science and Engineering.

[31]  R. Wyatt,et al.  Radial basis function interpolation in the quantum trajectory method: optimization of the multi-quadric shape parameter , 2003 .

[32]  Tobin A. Driscoll,et al.  Adaptive residual subsampling methods for radial basis function interpolation and collocation problems , 2007, Comput. Math. Appl..

[33]  W. Madych,et al.  Bounds on multivariate polynomials and exponential error estimates for multiquadratic interpolation , 1992 .