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Open AccessFeature PaperArticle

Numerical Simulation of Feller’s Diffusion Equation

by Denys Dutykh 1,2
1
Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France
2
LAMA, UMR 5127 CNRS, Université Savoie Mont Blanc, Campus Scientifique, F-73376 Le Bourget-du-Lac CEDEX, France
Mathematics 2019, 7(11), 1067; https://doi.org/10.3390/math7111067
Received: 12 September 2019 / Revised: 2 November 2019 / Accepted: 4 November 2019 / Published: 6 November 2019
(This article belongs to the Special Issue The Application of Mathematics to Physics and Nonlinear Science)
This article is devoted to Feller’s diffusion equation, which arises naturally in probability 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.
Keywords: Feller equation; parabolic equations; Lagrangian scheme; Fokker–Planck equation; probability distribution Feller equation; parabolic equations; Lagrangian scheme; Fokker–Planck equation; probability distribution
MDPI and ACS Style

Dutykh, D. Numerical Simulation of Feller’s Diffusion Equation. Mathematics 2019, 7, 1067.

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