TL;DR
This paper introduces a novel numerical scheme for Feller's diffusion equation, addressing stability issues caused by unbounded diffusion coefficients, using Lagrangian fluid mechanics concepts, and provides an open-source MATLAB implementation.
Contribution
It reformulates Feller's diffusion equation with Lagrangian ideas to develop a stable, efficient numerical scheme and shares an accessible MATLAB code.
Findings
Stable numerical scheme for Feller's diffusion equation
Open-source MATLAB implementation provided
Overcomes divergence issues at the origin
Abstract
This article is devoted to Feller's diffusion equation which arises naturally in probabilities and physics (e.g. wave turbulence theory). If discretized naively, this equation may represent serious numerical difficulties since the diffusion coefficient is practically unbounded and most of its solutions are weakly divergent at the origin. In order to overcome these difficulties we reformulate this equation using some ideas from the Lagrangian fluid mechanics. This allows us to obtain a numerical scheme with a rather generous stability condition. Finally, the algorithm admits an elegant implementation and the corresponding Matlab code is provided with this article under an open source license.
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