# A stabilized radial basis-finite difference (RBF-FD) method with hybrid kernels

## TL;DR

This paper introduces a stabilized hybrid kernel RBF-FD method that enhances system matrix conditioning, enabling efficient and stable meshless solutions to PDEs like the acoustic wave equation in complex domains.

## Contribution

The paper proposes a hybrid Gaussian-cubic kernel for RBF-FD, improving stability and computational efficiency over existing methods.

## Key findings

- Eigenvalue spectra remain stable regardless of irregularity and stencil size.
- The hybrid kernel reduces computational cost by enabling direct solvers.
- Effective absorption boundary conditions suppress spurious reflections.

## Abstract

Recent developments have made it possible to overcome grid-based limitations of finite difference (FD) methods by adopting the kernel-based meshless framework using radial basis functions (RBFs). Such an approach provides a meshless implementation and is referred to as the radial basis-generated finite difference (RBF-FD) method. In this paper, we propose a stabilized RBF-FD approach with a hybrid kernel, generated through a hybridization of the Gaussian and cubic RBF. This hybrid kernel was found to improve the condition of the system matrix, consequently, the linear system can be solved with direct solvers which leads to a significant reduction in the computational cost as compared to standard RBF-FD methods coupled with present stable algorithms. Unlike other RBF-FD approaches, the eigenvalue spectra of differentiation matrices were found to be stable irrespective of irregularity, and the size of the stencils. As an application, we solve the frequency-domain acoustic wave equation in a 2D half-space. In order to suppress spurious reflections from truncated computational boundaries, absorbing boundary conditions have been effectively implemented.