An improved radial basis-pseudospectral method with hybrid Gaussian-cubic kernels

TL;DR

This paper introduces a hybrid radial basis-pseudospectral method combining Gaussian and cubic kernels, which improves stability and accuracy in numerical solutions of PDEs while mitigating conditioning issues.

Contribution

It proposes a novel hybrid RBF approach with parameter optimization that enhances the stability and accuracy of pseudospectral methods for PDEs.

Findings

Reduces ill-conditioning in RBF-PS methods

Maintains stability and accuracy with small shape parameters

Outperforms pure Gaussian and cubic kernels

Abstract

While pseudospectral (PS) methods can feature very high accuracy, they tend to be severely limited in terms of geometric flexibility. Application of global radial basis functions overcomes this, however at the expense of problematic conditioning (1) in their most accurate flat basis function regime, and (2) when problem sizes are scaled up to become of practical interest. The present study considers a strategy to improve on these two issues by means of using hybrid radial basis functions that combine cubic splines with Gaussian kernels. The parameters, controlling Gaussian and cubic kernels in the hybrid RBF, are selected using global particle swarm optimization. The proposed approach has been tested with radial basis-pseudospectral method for numerical approximation of Poisson, Helmholtz, and Transport equation. It was observed that the proposed approach significantly reduces the ill-conditioning problem in the RBF-PS method, at the same time, it preserves the stability and accuracy for very small shape parameters. The eigenvalue spectra of the coefficient matrices in the improved algorithm were found to be stable even at large degrees of freedom, which mimic those obtained in pseudospectral approach. Also, numerical experiments suggest that the hybrid kernel performs significantly better than both pure Gaussian and pure cubic kernels.