Gaussian RBF-FD weights and its corresponding local truncation errors

Abstract

In this work we derive analytical expressions for the weights of Gaussian RBF-FD and Gaussian RBF-HFD formulas for some differential operators. These weights are used to derive analytical expressions for the leading order approximations to the local truncation error in powers of the inter-node distance h and the shape parameter є. We show that for each differential operator, there is a range of values of the shape parameter for which RBF-FD formulas and RBF-HFD formulas are significantly more accurate than the corresponding standard FD formulas. In fact, very often there is an optimal value of the shape parameter є+ for which the local error is zero to leading order. This value can be easily computed from the analytical expressions for the leading order approximations to the local error. Contrary to what is generally believed, this value is, to leading order, independent of the internodal distance and only dependent on the value of the function and its derivatives at the node.