LAUSR

Abstract Radial basis functions (RBFs) with multiquadric (MQ) kernel have been commonly used to solve partial differential equation (PDE). The MQ kernel contains a user-defined shape parameter (ε), and the solution accuracy is strongly dependent on the value of this ε. In this study, the MQ-based RBF finite difference (RBF-FD) method is derived in a polynomial form. The optimal value of ε is computed such that the leading error term of the RBF-FD scheme is eliminated to improve the solution accuracy and to accelerate the rate of convergence. The optimal ε is computed by using finite difference (FD) and combined compact differencing (CCD) schemes. From the analyses, the optimal ε is found to vary throughout the domain. Therefore, by using the localized shape parameter, the computed PDE solution accuracy is higher as compared to the RBF-FD scheme which employs a constant value of ε. In general, the solution obtained by using the ε computed from CCD scheme is more accurate, but at a higher computational cost. Nevertheless, the cost-effectiveness study shows that when the number of iterative prediction of ε is limited to two, the present RBF-FD with ε by CCD scheme is as effective as the one using FD scheme.