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

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. * Corresponding Author

A local hybrid kernel meshless method for numerical solutions of two‐dimensional fractional cable equation in neuronal dynamics

“…Hybrid Gaussian‐cubic radial basis function (HGC‐RBF) was initially introduced in [55] and then in [56, 57] they were used successfully for solving PDEs. As stated in [55–57], combination of the cubic kernel in the Gaussian kernel decreases the condition number and increases accuracy, remarkably. Moreover, such a hybridization of the kernels makes the used algorithm well‐posed and stabilizes interpolation.…”

“…The HGC‐RBF was first developed in [55] and has been shown to be stabilized for scattered data interpolation problems. Later in works [56, 57] it is shown that the HGC‐RBF is a plausible approach for numerical treatment of PDEs. The HGC‐RBF is defined as: where ε is shape parameter corresponds to Gaussian RBF and α , β are coefficients which govern contribution of the Gaussian and cubic kernels in the hybrid kernel.…”

Hybrid radial basis function methods of lines for the numerical solution of viscous Burgers’ equation

“…These HRBF methods do not suffer on the limitations of RBF such as ill-conditioning due to an increase in problem size (number of degrees of freedom). In fact, it has been shown that such HRBF family improves RBF pseudospectral (RBF-PS) method results and provides a reasonable compromise on the accuracy (Mishra et al 2017(Mishra et al , 2019. In another earlier study, a hybrid RBF method was also developed by taking linear combination of multiquadric (MQ) and thin plate spline (TPS) RBFs (Ahmed 2006).…”

“…This later HRBF method has been found to be a well-behaved alternative to Kansa method in a recent study (Hussain and Haq 2020b). In addition, both HRBF families [GS+S3 (Mishra et al 2019) and MQ+TPS (Ahmed 2006)] correspond to a change in function space where good stability properties from the cubic (S3)/TPS RBF as well as high accuracy from the Gaussian (GS)/MQ RBF is utilized. It has been also concluded that the computational cost of HRBF approach is the same as standard RBF method which implies that HRBF method leads to a fast and stable algorithm (Mishra et al 2019).…”

“…In addition, both HRBF families [GS+S3 (Mishra et al 2019) and MQ+TPS (Ahmed 2006)] correspond to a change in function space where good stability properties from the cubic (S3)/TPS RBF as well as high accuracy from the Gaussian (GS)/MQ RBF is utilized. It has been also concluded that the computational cost of HRBF approach is the same as standard RBF method which implies that HRBF method leads to a fast and stable algorithm (Mishra et al 2019). In this paper, we present a family of HRBF methods by combining infinite smooth RBFs defined by shape parameter (c) with piecewise smooth RBFs, where the latter is independent of shape parameters (see Table 1).…”