Application of an RBF blending interpolation method to problems with shocks

Abstract

Radial basis functions (RBF) have become an area of research in recent years, especially in the use of solving partial differential equations (PDE). Radial basis functions have an impressive capability in interpolating scattered data, even for data with discontinuities. Although, for infinitely smooth radial basis functions such as the multi-quadrics and inverse multi-quadrics, the shape parameter must be chosen properly to obtain accurate approximations while avoiding ill-conditioning of the interpolating matrices. The optimum shape parameter can vary depending on the field, such as in locations of sharp gradients or shocks. Typically, the shape parameter is chosen to maintain a high conditioning number for the interpolation matrix, rendering the RBF smooth [1-10]. However, this strategy fails for a problem with a shock or sharp discontinuity. Instead, in such cases the conditioning number must be kept small. The focus of this work is then to demonstrate the use of RBF interpolation in the approximation of sharp gradients or shocks by use of a RBF blending interpolation approach. This RBF blending interpolation approach is used to maintain the optimum shape parameter depending on the field. The approach is able to sense gradients or shocks in the field and adjust the shape parameter accordingly to keep excellent accuracy. Presented in this work, is an explanation of the RBF blending interpolation methodology and testing of the RBF blending interpolation approach by solving the Burger's equation using the virtual finite difference method.