RBF Based Grid-Free Local Scheme With Spatially Variable Optimal Shape Parameter for Steady Convection-Diffusion Equations

Abstract

In this work, a local algorithm has been proposed to obtain an optimal shape parameter for the infinitely smooth Radial Basis Functions (RBF) when they have been used to solve Convection-Diffusion Equations (CDE) under grid-free environment. The algorithm is based on re-construction of the forcing term in the CDE using the collocation over the centers of the local support domain. The residual errors are calculated using the Rippa's “leave one out cross validation” algorithm originally been developed for interpolation using RBF functions. It has been shown that the cost function and RMS error functions obtained with the developed local scheme are oscillation free unlike the existing global collocation schemes. It has been also observed that, for most of the diffusion dominated problems, the pattern of the (global) Cost function of the proposed algorithm is appeared to be similar to the (R.M.S) error function, however, the same is not been found true for convection dominated problems. Therefore, for the latter case, an (near) optimal variable shape parameter has been obtained by minimizing the local Cost function at each center (node) of the computational domain. The Local RBF (LRBF) scheme with the proposed local optimization algorithm has been tested over several one and two-dimensional linear convection-diffusion problems with strong boundary layer and found to be accurate.