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

Meshfree Local Radial Basis Function Collocation Method with Image Nodes