# A Fokker-Planck approach for probability distributions of species concentrations transported in heterogeneous media

## Abstract

We identify sufficient conditions under which evolution equations for probability density functions (PDF) of random concentrations are equivalent to Fokker–Planck equations. The novelty of our approach is that it allows consistent PDF approximations by densities of computational particles governed by Itô processes in concentration–position spaces. Accurate numerical solutions are obtained with a global random walk (GRW) algorithm, stable, free of numerical diffusion, and insensitive to the increase of the total number of computational particles. The system of Itô equations is specified by drift and diffusion coefficients describing the PDF transport in the physical space, provided by up-scaling procedures, as well as by drift and mixing coefficients describing the PDF transport in concentration spaces. Mixing models can be obtained similarly to classical PDF approaches or, alternatively, from measured or simulated concentration time series. We compare their performance for a GRW-PDF numerical solution to a problem of contaminant transport in heterogeneous groundwater systems.

## Authors

N. Suciu
-Mathematics Department, Friedrich-Alexander University of Erlangen-Nuremberg, Cauerstraße 11, 91058 Erlangen, Germany
-Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

-Department of Mathematics, University of Bergen, Allegaten 41, 5008 Bergen, Norway

S. Attinger
-Faculty for Chemistry and Earth sciences, University of Jena, Burgweg 11, 07749 Jena, Germany
-Department Computational Hydrosystems, UFZ Centre for Environmental Research, Permoserstraße 15, 04318 Leipzig, Germany

L. Schüler
-Faculty for Chemistry and Earth sciences, University of Jena, Burgweg 11, 07749 Jena, Germany
-Department Computational Hydrosystems, UFZ Centre for Environmental Research, Permoserstraße 15, 04318 Leipzig, Germany

P. Knabner
Mathematics Department, Friedrich-Alexander University of Erlangen-Nuremberg, Cauerstraße 11, 91058 Erlangen, Germany

## Keywords

PDF methods; Mixing; Random walk; Porous media.

##### Cite this paper as:

N. Suciu, F.A. Radu, S. Attinger, L. Schüler, P. Knabner, A Fokker-Planck approach for probability distributions of species concentrations transported in heterogeneous media, J. Comput. Appl. Math., 289 (2015), 241-252,
doi:10.1016/j.cam.2015.01.030

## PDF

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##### Journal

Journal of Computational and Applied Mathematics

Elsevier

0377-0427

##### Online ISSN

Not available yet.