Numerical Simulation of Feller’s Diffusion Equation

This article is devoted to <span style="font-variant: small-caps;">Feller</span>’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...

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Main Author:	Denys Dutykh
Format:	Article
Language:	English
Published:	MDPI AG 2019-11-01
Series:	Mathematics
Subjects:	feller equation parabolic equations lagrangian scheme fokker–planck equation probability distribution
Online Access:	https://www.mdpi.com/2227-7390/7/11/1067

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spelling	doaj-09c81470ba8d44419081fbfc85977b892020-11-25T01:56:33ZengMDPI AGMathematics2227-73902019-11-01711106710.3390/math7111067math7111067Numerical Simulation of Feller’s Diffusion EquationDenys Dutykh0Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, FranceThis article is devoted to <span style="font-variant: small-caps;">Feller</span>’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 <span style="font-variant: small-caps;">Lagrangian</span> 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 <span style="font-variant: small-caps;">Matlab</span> code is provided with this article under an open source license.https://www.mdpi.com/2227-7390/7/11/1067feller equationparabolic equationslagrangian schemefokker–planck equationprobability distribution
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Denys Dutykh
spellingShingle	Denys Dutykh Numerical Simulation of Feller’s Diffusion Equation Mathematics feller equation parabolic equations lagrangian scheme fokker–planck equation probability distribution
author_facet	Denys Dutykh
author_sort	Denys Dutykh
title	Numerical Simulation of Feller’s Diffusion Equation
title_short	Numerical Simulation of Feller’s Diffusion Equation
title_full	Numerical Simulation of Feller’s Diffusion Equation
title_fullStr	Numerical Simulation of Feller’s Diffusion Equation
title_full_unstemmed	Numerical Simulation of Feller’s Diffusion Equation
title_sort	numerical simulation of feller’s diffusion equation
publisher	MDPI AG
series	Mathematics
issn	2227-7390
publishDate	2019-11-01
description	This article is devoted to <span style="font-variant: small-caps;">Feller</span>’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 <span style="font-variant: small-caps;">Lagrangian</span> 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 <span style="font-variant: small-caps;">Matlab</span> code is provided with this article under an open source license.
topic	feller equation parabolic equations lagrangian scheme fokker–planck equation probability distribution
url	https://www.mdpi.com/2227-7390/7/11/1067
work_keys_str_mv	AT denysdutykh numericalsimulationoffellersdiffusionequation
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Numerical Simulation of Feller’s Diffusion Equation

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