Title: A New Technique for Neumann Boundary Conditions in the Radial Basis Function Finite Difference Method

Program: Mathematics MS

Committee Chair: Grady Wright

Committee: Grady Wright, Donna Calhoun, Michal Kopera

Abstract: Partial differential equations (PDEs) play a central role in modeling a wide range of phenomena in science and engineering. Due to the difficulty in obtaining analytical solutions, especially on complex geometries, it is necessary to use numerical methods for determining approximate solutions. Radial Basis Function Finite Differences (RBF-FD) are a relatively recent mesh-free approach that offers high accuracy and flexibility for handling complex domains. RBF-FD methods have been extended to many types of PDEs, enabling applications in fields such as biology, chemistry, geophysics, and computer graphics. These methods can be particularly effective for complex geometries as they use scattered points and do not require meshes or body-fitted parameterizations of the domain, thereby simplifying the discretization of the domain. One issue with this approach, however, has been the lack of a general, flexible method for implementing non-Dirichlet boundary conditions. This thesis addresses this issue by introducing a new RBF-FD method for Neumann boundary conditions through Hermite-Birkhoff interpolation. We compare the performance of this method with the existing ``ghost'' or ``fictitious'' point technique for several problems, finding that it gives solutions with at least the same accuracy without having to resort to introducing unknowns outside of the domain.