Radial Basis Function Collocation Method

Abstract

The radial basis function (RBF) collocation method, and particularly the multiquadric collocation method, has some amazing properties for solving partial differential equations. First, it is a “meshless” method, which is a much desired characteristic in modern day numerical methods. Advantages include the easy preparation of input data and the flexibility in solving moving boundary problems. The basis functions are of “global” nature, and the approximate solution is infinitely smooth, that is, it can be differentiated infinite number of times everywhere. It is based on the point collocation of a simple series expression, hence the mathematical algorithm is simple, and easy to program. Radial basis function is based on the Euclidean norm r _i =∥x−x _i ,∥ which means that the numerical algorithm can be easily extended to solve n-space problems. For a class of ill-posed boundary value problems, traditional methods require that it be converted into well-posed problems and be solved iteratively by optimization. RBF collocation method can solve these problems without iteration. The most amazing characteristic of multiquadric collocation is its amazing accuracy. Unlike the popular numerical methods, such as the Finite Element Method (FEM) and the Finite Difference Method (FDM), which use piece-wise, low-degree polynomials for interpolation, collocation methods use global and highly smooth functions. As a consequence, multiquadric collocation achieves exponential error convergence of $O\left( {e^{ac} \lambda ^{ch^{ - 1} } } \right)$, as compared to the power law O(h ^k) of traditional numerical methods. Furthermore, by varying the constant c in the multiquadric, (r ²+c)^1/2 , increasing accuracy of solution can be achieved without refining the mesh. For a 2-D Poisson equation, it is demonstrated in a numerical example that a uniform accuracy of 10^-15 can be accomplished using merely a 20x20 mesh. This accuracy is impossible to reach with the O(h ^k) numerical scheme