A collocation method based on localized radial basis functions with reproducibility for nonlocal diffusion models

In this paper, a kind of localized radial basis function-based collocation method with reproducibility has been designed for nonlocal diffusion models. The basic idea of the method is to localize the RBF shape function by a corrected kernel with compact support, and meanwhile make the interpolation function to meet the reproducing conditions by modifying the coefficient contained in such kernel. Three types of nonlocal diffusion problems including constant and singular kernels are solved by our method in numerical experiments, which indicates that our method shows almost the same convergent behavior compared with RBF collocation methods, but it is much better conditioning and more time-efficient. It also overcomes the shortcoming that the RK-enhanced RBF method is not convergent for nonlocal diffusion models.