On the structure of function spaces in optimal recovery of point functionals for ENO-schemes by radial basis functions

Abstract

Radial basis functions are used in the recovery step of finite volume methods for the numerical solution of conservation laws. Being conditionally positive definite such functions generate optimal recovery splines in the sense of Micchelli and Rivlin in associated native spaces. We analyse the solvability to the recovery problem of point functionals from cell average values with radial basis functions. Furthermore, we characterise the corresponding native function spaces and provide error estimates of the recovery scheme. Finally, we explicitly list the native spaces to a selection of radial basis functions, thin plate splines included, before we provide some numerical examples of our method. Classification (1991): 65M60 41A15 41A29 41A63; 65N30 76M25 Mathematics Subject

Numerical methods for high dimensional Hamilton–Jacobi equations using radial basis functions

“…where / is a RBF, M is the number of cells in the reconstruction stencil, and the second sum is over polynomials fp j g which form a basis of the kernel of the seminorm ½Á; Á of the native space in which / lives [18]. In general a spline, U, in a semi-Hilbert space, V , interpolating data,…”

“…The familiarity of polynomials and the ability to simply construct their derivatives using divided differences made the ENO methods for conservation laws and Hamilton-Jacobi equations very popular [16,22,26]. In multiple dimension on nonuniform grids there has been some progress using polynomials [4,31], and RBFs [18]. Here we present an incremental stencil selection method which exploits the LU factorization of the RBF coefficient matrix.…”

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

“…More specifically, in [2] and in related work [1,23], the mesh-free feature of Radial Basis Functions (RBFs) is successfully combined with ENO/WENO ideas for scalar conservation laws in general geometries using unstructured grids. In the past decades, interpolation techniques with RBFs have been actively studied (see [4]) to approximate functions of more than one variable.…”

Adaptive meshless centres and RBF stencils for Poisson equation

“…The local shape functions are not necessarily polynomials, they can be generated by moving least squares for example, but typically they are nevertheless guaranteed to reproduce polynomials of certain degree [24,34,20,32,35]. Local numerical differentiation with RBF rather than polynomials has been explored for solving certain time dependent PDEs in [18,6].…”