A RBF partition of unity collocation method based on finite difference for initial-boundary value problems
For researchers solving time-dependent PDEs with meshfree methods, this work offers a localized approach that mitigates the high computational cost and ill-conditioning of global RBF methods.
The authors propose a new RBF partition of unity collocation method based on finite differences (RBF-PUM-FD) to solve time-dependent PDEs, achieving reduced ill-conditioning and computational cost while maintaining high accuracy. Numerical experiments on convection-diffusion and pseudo-parabolic equations demonstrate the method's effectiveness.
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.