On optimization of the RBF shape parameter in a grid-free local scheme for convection dominated problems over non-uniform centers

Global optimization techniques exist in the literature for finding the optimal shape parameter of the infinitely smooth radial basis functions (RBF) if they are used to solve partial differential equations. However these global collocation methods, applied directly, suffer from severe ill-conditioning when the number of centers is large. To circumvent this, we have used a local optimization algorithm, in the optimization of the RBF shape parameter which is then used to develop a grid-free local (LRBF) scheme for solving convection–diffusion equations. The developed algorithm is based on the re-construction of the forcing term of the governing partial differential equation over the centers in a local support domain. The variable (optimal) shape parameter in this process is obtained by minimizing the local Cost function at each center (node) of the computational domain. It has been observed that for convection dominated problems, the local optimization scheme over uniform centers has produced oscillatory solutions, therefore, in this work the local optimization algorithm has been experimented over Chebyshev and non-uniform distribution of the centers. The numerical experiments presented in this work have shown that the LRBF scheme with the local optimization produced accurate and stable solutions over the non-uniform points even for convection dominant convection–diffusion equations.