The selection of shape parameter and fictitious radius for RBF collocation method using the modified Franke formula and effective condition number

The shape parameter $c$ and the fictitious radius $R$ are important parameters that affect the performance of the polynomial-augmented RBF collocation method (RBFCM) with fictitious centers. It is known that the RBFCM can reduce the accuracy dependency on shape parameter by using extra polynomial constraints. Moreover, we find that calculation inaccuracies exhibit a strong association with the effective condition numbers for various fictitious radii. While there have been some methods for selecting the shape parameter, an approach that allows for the simultaneous selection of $c$ and $R$ has not yet been widely researched. In this paper, we propose a systematic method to choose parameters for the polynomial-augmented RBFCM with fictitious centers. The method utilizes the effective condition number to find an appropriate fictitious radius $R$ and the modified Franke formula to select a good shape parameter $c$. Five examples of second and fourth order PDEs in 2D and 3D are presented to demonstrate the effectiveness of the proposed method.