A new higher-order RBF-FD scheme with optimal variable shape parameter for partial differential equation

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 (epsilon), and the solution accuracy is strongly dependent on the value of this epsilon. In this study, the MQ-based RBF finite difference (RBF-FD) method is derived in a polynomial form. The optimal value of epsilon 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 epsilon is computed by using finite difference (FD) and combined compact differencing (CCD) schemes. From the analyses, the optimal epsilon 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 epsilon. In general, the solution obtained by using the epsilon 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 epsilon is limited to two, the present RBF-FD with epsilon by CCD scheme is as effective as the one using FD scheme.