An accurate and stable RBF method for solving partial differential equations

Abstract

Most standard radial basis function (RBF)methods strongly depend on a shape parameter in accuracy and stability. However, the selection of optimal shape parameter in the RBF methods has long been a challenging task. In this paper, a new global RBF, known as coupled radial basis function (CRBF), is presented based on coupling the infinitely smooth RBFs with the conical spline. It is found that the Kansa's method with the CRBF inherits the strengths of the aforementioned two types of RBFs, and achieves an accurate and stable numerical solution almost independent of the shape parameter values. Therefore, the proposed methodology completely overcome the troublesome issue of the optimal shape parameter that is a formidable obstacle to global RBF schemes. Two test examples verify the accuracy and robustness of the proposed CRBF method.