A RBF partition of unity collocation method based on finite difference for initial–boundary value 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.

Coherent synthesis of ultrashort pulse fiber laser based on balanced difference method

“…Coherent synthesis technology of fiber laser or amplifier array is a research hotspot in the field of optoelectronic technology. It provides an effective way to obtain high power and high beam quality laser output, and will be an important research direction in the field of high-energy laser in the future [1]. Because of the small coupling area and the limited pump power, the output power of ordinary fiber laser is very low, which is usually in milliwatt level.…”

Error indicators and refinement strategies for solving Poisson problems through a RBF partition of unity collocation scheme

“…Recently, this method combined with RBFs has gained popularity in many fields of applied mathematics, scientific computing and engineering, since it was used to effectively solve large scale approximation and PDE problems (see e.g. [7,18,22,16,24,26,27,28]). Nevertheless, as far as we know, in the current literature the adaptivity problem via a RBF-PU scheme has not properly been addressed yet.…”

A stable meshfree PDE solver for source-type flows in porous media

“…In this realm, many methods, such as Radial Basis Function (RBF) collocation via Partition of Unity (PU) method, multiscale methods and RBF-Finite Difference (FD) local approaches, have already been developed, see, e.g. [21,22,23,24]. Here we investigate on the performances, in terms of stability and accuracy, of some collocation schemes on non-uniform distributions and we deal with the stability issues due to the ill-conditioning of the collocation matrices.…”