Stochastic heat diffusion modelling with random walks on the non-uniformly gridded circle

A technique to approximate heat diffusion on Riemannian manifolds is presented. We provide a numerical way to approximate the solution to the heat equation by using the idea of random walks of particles, governed by a continuous-time Markov chain, where the transition rates of the Markov chain are characterized by the distances between nodes on a given grid with non-equally placed nodes. The emphasis lies on the fact that nodes do not need to be distributed equidistant from each other, since such a regular grid is not effective on many manifolds, where some parts of the manifold require less nodes than others due to curvature. In this paper we show how to characterize the Markov chain for a given grid in order to build a framework for the numerical approximation of the solution to the heat equation on Riemannian manifolds. This framework approximates the Laplace-Beltrami operator which is used on such manifolds. Furthermore, we discuss advantages of this technique and provide examples and simulations of our results.