Owing to its high accuracy, the radial basis function (RBF) is gaining popularity in function interpolation and for solving partial differential equations (PDEs). The implementation of RBF methods is independent of the locations of the points and the dimensionality of the problems. However, the stability and accuracy of RBF methods depend significantly on the shape parameter, which is mainly affected by the basis function and the node distribution. If the shape parameter has a small value, then the RBF becomes accurate but unstable. Several approaches have been proposed in the literature to overcome the instability issue. Changing or expanding the radial basis function is one of the most commonly used approaches because it addresses the stability problem directly. However, the main issue with most of those approaches is that they require the optimization of additional parameters, such as the truncation order of the expansion, to obtain the desired accuracy. In this work, the Hermite polynomial is used to expand the RBF with respect to the shape parameter to determine a stable basis, even when the shape parameter approaches zero, and the approach does not require the optimization of any parameters. Furthermore, the Hermite polynomial properties enable the RBF to be evaluated stably even when the shape parameter equals zero. The proposed approach was benchmarked to test its reliability, and the obtained results indicate that the accuracy is independent of or weakly dependent on the shape parameter. However, the convergence depends on the order of the truncation of the expansion. Additionally, it is observed that the new approach improves accuracy and yields the accurate interpolation, derivative approximation, and PDE solution.Mathematics 2019, 7, 979 2 of 18
Changing the InterpolantRadial basis function methods are used to obtain an accurate grid-free approximation of a function with a strong gradient [1]. Kansa [2] used an RBF to solve PDEs by direct collocation. Fasshauer [3] changed the interpolant by using Hermite interpolation to improve the condition number of the matrix of the differential operator of PDE. This matrix is symmetric, which reduces the amount of computation, particularly for large matrices. Compared with Kansa's approach, Fasshauer's approach requires a higher derivative of the basis function to evaluate the same differential operator, hence, Kansa's approach is simpler. The convergence of Fasshauer's approach was verified and compared with the theoretical one [4]. The compactly supported RBF was used with Fasshauer's method to enhance the accuracy near the boundary [5]. Kansa's and Fasshauer's approaches have been compared in the literature to solve the steady-state convection-diffusion equation at high Peclet number values [6]. The comparison results reveal that both approaches are more convenient to implement than conventional methods, such as the finite difference, finite element, and finite volume. It was also observed that, for steady-state problems, Fasshauer's approach was marginally ...