On convergence of integrated radial basis function approximations on specific stencils with numerical applications

One strategy to enhance the performance of the localized meshless radial basis function scheme in finite difference mode (RBF-FD) is to incorporate a polynomial term into the RBF-FD framework, known as the integrated radial basis function (IRBF) approach. In this work, we propose a comprehensive framework for obtaining the approximation weights for the IRBF method. Then, we derive analytical expressions for the weighting coefficients in the one-dimensional scenario for specific stencils, enabling us to explore the theoretical high-order convergence properties at the nodal points utilizing the Gaussian RBF. Furthermore, these weights are extended to be used for solving high-dimensional partial differential problems, such as the Poisson equation and the four-dimensional time-dependent Black-Scholes equation.