Hybrid GFD-RBF Method for Convection-Diffusion Problems

Abstract: In this paper, we present a meshless hybrid method combining the Generalized Finite Difference (GFD) and Finite Difference based Radial Basis Function (RBF-FD) approaches to solve non-homogeneous partial differential equations (PDEs) involving both lower and higher order derivatives. The proposed method eliminates the need for mesh generation by leveraging the strengths of both GFD and RBF-FD techniques. The GFD method is robust and stable, effectively handling ill-conditioned systems, while the RBF-FD method excels in extending to higher-order derivatives and higher-dimensional problems. Despite their individual advantages, each method has its limitations. To address these, we developed a hybrid GFD-RBF approach that combines their strengths. Specifically, the GFD method is employed to approximate lower order terms (convective terms), and the RBF method is used for higher order terms (diffusive terms). The performance of the proposed hybrid method is tested on both linear and nonlinear PDEs, considering uniform and non-uniform distributions of nodes within the domain. This approach demonstrates the versatility and effectiveness of the hybrid GFD-RBF method in solving second and higher order convection-diffusion problems.