Hybrid GFD-RBF Method for Convection-Diffusion Problems
Priyal Garg, T. V. S. Sekhar
TL;DR
The paper tackles solving non-homogeneous PDEs with both lower and higher derivatives on domains where mesh generation is difficult. It introduces a meshless GFD-RBF-FD hybrid that assigns first-order convective terms $\partial U/\partial x$ and $\partial U/\partial y$ to GFD and second- or higher-order diffusive terms $\partial^2 U/\partial x^2$, $\partial^2 U/\partial y^2$, etc., to RBF-FD, enabling direct high-order PDE discretization without order-reduction. The approach is validated on linear, nonlinear, and coupled convection-diffusion problems with both uniform and Chebyshev node layouts, showing second-order accuracy with five nodes per star $(N_G,N_R)=(5,5)$ and meaningful CPU-time savings compared with GFD or standard RBF-FD, and competitive performance against CD2. This meshless framework is robust, scalable to higher dimensions, and particularly advantageous when convective terms are highly nonlinear, reducing mesh-generation overhead and expanding applicability to complex geometries.
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.