A RBF partition of unity collocation method based on finite difference for initial-boundary value problems

TL;DR

This paper introduces a localized RBF partition of unity collocation method combined with finite difference schemes to efficiently solve time-dependent PDEs, reducing computational costs and improving stability while maintaining high accuracy.

Contribution

It presents a novel RBF-PUM-FD collocation method that decreases ill-conditioning and produces sparse systems for solving unsteady PDEs, enhancing computational efficiency.

Findings

Successfully applied to convection-diffusion equations

Achieves high accuracy with sparse matrix systems

Demonstrates efficiency on benchmark unsteady PDE problems

Abstract

Meshfree radial basis function (RBF) methods are popular tools used to numerically solve partial differential equations (PDEs). They take advantage of being flexible with respect to geometry, easy to implement in higher dimensions, and can also provide high order convergence. Since one of the main disadvantages of global RBF-based methods is generally the computational cost associated with the solution of large linear systems, in this paper we focus on a localizing RBF partition of unity method (RBF-PUM) based on a finite difference (FD) scheme. Specifically, we propose a new RBF-PUM-FD collocation method, which can successfully be applied to solve time-dependent PDEs. This approach allows to significantly decrease ill-conditioning of traditional RBF-based methods. Moreover, the RBF-PUM-FD scheme results in a sparse matrix system, reducing the computational effort but maintaining at the same time a high level of accuracy. Numerical experiments show performances of our collocation scheme on two benchmark problems, involving unsteady convection-diffusion and pseudo-parabolic equations.