Adaptive radial basis function methods for time dependent partial differential equations

Abstract

Radial basis function (RBF) methods have shown the potential to be a universal grid free method for the numerical solution of partial differential equations. Both global and compactly supported basis functions may be used in the methods to achieve a higher order of accuracy. In this paper, we take advantage of the grid free property of the methods and use an adaptive algorithm to choose the location of the collocation points. The RBF methods produce results similar to the more well known and analyzed spectral methods, but while allowing greater flexibility in the choice of grid point locations. The adaptive RBF methods are most successful when the basis functions are chosen so that the PDE solution can be approximated well with a small number of the basis functions.

An Investigation of Radial Basis Function-Finite Difference (RBF-FD) Method for Numerical Solution of Elliptic Partial Differential Equations

“…We will next present an outline of the RBF along with finite-difference (RBF-FD) formulation for solving PDE Equation (3).…”

“…Unlike the case of interpolation, the system matrix of this case does not guarantee to be nonsingular and is also ill-conditioned for small values of shape parameter [1,6]. Additionally, according to numerical evidence, the global RBF scheme requires high computational cost and memory requirements for large scale problems [3,7]. Therefore, the radial basis function-finite difference (RBF-FD) concept, which is a local RBF scheme, was developed by combining many benefits of the RBF method and traditional finite difference approximations.…”

“…These traditional numerical methods require the data to be arranged in a structured pattern and to be contained in a simply shaped region, and, therefore, these approaches does not suit multidimensional practical problems. The Radial basis function (RBF) method is viewed as an important alternative numerical technique for multidimensional problems because it contains more beneficial properties in high-order accuracy and flexibility on geometry of the computational domain than classical techniques [1][2][3]. The main idea of the RBF method is approximating the solution in terms of linear combination of infinitely differentiable RBF φ, which is the function that depends only on the distance to a center point x and shape parameter ε.…”

A New Method for Solving Steady Flow of a Third-Grade Fluid in a Porous Half Space Based on Radial Basis Functions

“…The RBFs can be compactly and globally supported, are infinitely differentiable, and contain a free parameter c, called the shape parameter [12,18,19]. For more basic details about compactly and globally supported RBFs and convergence rate of them, the interested reader can refer to the recent books and paper by Buhmann [18,20] and Wendland [21].…”

Radial Basis Function Interpolation on Irregular Domain through Conformal Transplantation

“…In that case, nodes can be denser around high activity areas and coarser in other region. Indeed, study about adaptive process for automatically refining/coarsening RBF centers based on the profile of the functions/solutions is becoming an active research area [10,21,22,29].…”