Global–local Hermite RBF Collocation

Abstract

A global-local Hermite radial basis function (RBF) collocation method is proposed for time-dependent convection-diffusion equations with Neumann boundary conditions. This method directly incorporates information from the boundary derivatives into the approximation framework and avoids the need for mesh generation. A global formulation and a local version using unit partitioning are investigated. The global method offers high accuracy, while the local formulation improves conditioning and computational efficiency for large sets of nodes. Numerical experiments on the unit square demonstrate the accuracy, stability, and scalability of the proposed approach, as well as the influence of node distribution and shape parameters.