Meshfree Local Radial Basis Function Collocation Method with Image Nodes

TL;DR

This paper introduces a meshfree local RBF collocation method with image nodes for efficiently solving 2D heat diffusion problems, improving accuracy at boundaries by incorporating image nodes similar to electrostatics.

Contribution

The paper proposes a novel boundary treatment using image nodes in local RBF collocation, enhancing boundary condition handling in meshfree methods for heat diffusion.

Findings

Significant error reduction in boundary value problems

Increased numerical stability with sufficient nodes and careful parameters

Method is computationally efficient for local collocation

Abstract

We numerically solve two-dimensional heat diffusion problems by using a simple variant of the meshfree local radial-basis function (RBF) collocation method. The main idea is to include an additional set of sample nodes outside the problem domain, similarly to the method of images in electrostatics, to perform collocation on the domain boundaries. We can thereby take into account the temperature profile as well as its gradients specified by boundary conditions at the same time, which holds true even for a node where two or more boundaries meet with different boundary conditions. We argue that the image method is computationally efficient when combined with the local RBF collocation method, whereas the addition of image nodes becomes very costly in case of the global collocation. We apply our modified method to a benchmark test of a boundary value problem, and find that this simple modification reduces the maximum error from the analytic solution significantly. The reduction is small for an initial value problem with simpler boundary conditions. We observe increased numerical instability, which has to be compensated for by a sufficient number of sample nodes and/or more careful parameter choices for time integration.