Numerical solution of Feller's equation

Denys Dutykh (1, 2, 3)

1 LAMA - Laboratoire de Mathématiques
2 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
3 USMB [Université de Savoie] [Université de Chambéry] - Université Savoie Mont Blanc

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.

Mots clés: numerical simulation; Lagrangian scheme; probability density function; Feller equation; parabolic equations

Domaines: Physique Numérique [physics.comp-ph]; Analyse numérique [math.NA]; Analyse numérique [cs.NA]; Equations aux dérivées partielles [math.AP]

Denys Dutykh. Numerical solution of Feller's equation. Mathematics , 2019, 7 (11), pp.1067. ⟨10.3390/math7111067⟩. ⟨hal-02290284⟩