An RBF Interpolation Blending Scheme for Effective Shock-Capturing

Abstract

In recent years significant focus has been given to the study of Radial basis functions (RBF), especially in their use on solving partial differential equations (PDE). RBF have an impressive capability of interpolating scattered data, even when this data presents localized discontinuities. However, for infinitely smooth RBF such as the Multiquadrics, inverse Multiquadrics, and Gaussian, 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 significantly depending on the field, particularly in locations of steep gradients, shocks, or discontinuities. Typically, the shape parameter is chosen to be high value to render flatter RBF therefore yielding a high condition number for the ensuing interpolation matrix. However, this optimization strategy fails for problems that present steep gradients, shocks or discontinuities. Instead, in such cases, the optimal interpolation occurs when the shape parameter is chosen to be low in order to render steeper RBF therefore yielding low condition number for the interpolation matrix. The focus of this work is to demonstrate the use of RBF interpolation to capture the behaviour of steep gradients and shocks by implementing a blending scheme that combines high and low shape parameters. A formulation of the RBF blending interpolation scheme along with testing and validation through its implementation in the solution of the Burger's linear advection equation and compressible Euler equations using a Localized RBF Collocation Meshless Method (LRC-MM) is presented in this paper.