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
F.A. Radu
-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
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About this paper
Journal
Journal of Computational and Applied Mathematics
Publisher Name
Elsevier
Print ISSN
0377-0427
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[1] N. Suciu, Diffusion in random velocity fields with applications to contaminant transport in groundwater, Advances in Water Resources, 69 (2014), pp. 114-133.
CrossRef (DOI)
[2] C. Vamoş, M. Crăciun, Separation of components from a scale mixture of Gaussian white noises, Physical Review E, 81 (2010) no. 5
CrossRef (DOI)
[3] C. Vamoş, M. Crăciun, Automatic trend estimation, Jan 2013, Springer, Dordrecht, 2012, pp. 131, ISBN: 978-94-007-4824-8 (Print), 978-94-007-4825-5 (Online),
CrossRef (DOI)
[4] N. Suciu, F.A. Radu, A. Prechtel, F. Brunner, P. Knabner, A coupled finite element–global random walk approach to advection-dominated transport in porous media with random hydraulic conductivity, Journal of Computational and Applied Mathematics, 246 (2013), pp. 27-37
CrossRef (DOI)
[5] F.A. Radu, N. Suciu, J. Hoffmann, A. Vogel, S. Attinger, Accuracy of numerical simulations of contaminant transport in heterogeneous aquifers: A comparative study, Advances in Water Resources, 34 (2011) no. 1, pp. 47-61.
CrossRef (DOI)
[6] R.W. Bilger, The Structure of Diffusion Flames, 1976, Combust Sci Technol
CrossRef (DOI)
[7] S.B. Pope, The Statistical Theory of Turbulent Flames, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 291 (1979) no. 1384, pp. 529-568
CrossRef (DOI)
[8] D.W. Meyer, P. Jenny, H.A. Tchelepi, A joint velocity-concentration PDF method for tracer flow in heterogeneous porous media, Water Resources Research, 46 (2010) no. 12
CrossRef (DOI)
[9] F.A Jaberi, P. J. Colucci, S. James, P. Givi, S. B. Pope, Filtered Mass Density Function for Large-Eddy Simulation of Turbulent Reacting Flows, J Fluid Mech (1999)
CrossRef (DOI)
[10] Haworth D.C., Progress in probability density function methods for turbulent reacting flows, Prog. Energy Combust. Sci., 36 (2010), 168-259
CrossRef (DOI)
[11] N. Suciu, C. Vamoş, J. Vanderborght, H. Hardelauf, H. Vereecken, Numerical investigations on ergodicity of solute transport in heterogeneous aquifers, Water Resour. Res., 42, W04409, 2006.
CrossRef (DOI)
[12] S. B. Pope, PDF Methods for Turbulent Reacting Flows, Prog Energ Combust, (1985)
CrossRef (DOI)
[13] R. McDermott, S. B. Pope, A particle formulation for treating differential diffusion in filtered density function methods, Journal of Computational Physics, 226 (2007) no 1, pp. 947-993
CrossRef (DOI)
[14] C. Vamoş, N. Suciu, H. Vereecken, Generalized random walk algorithm for the numerical modeling of complex diffusion processes, Journal of Computational Physics, 186 (2003), 527-544
CrossRef (DOI)
[15] Haifeng Wang, Pavel P. Popov, Stephen B. Pope, Weak second-order splitting schemes for Lagrangian Monte Carlo particle methods for the composition PDF/FDF transport equations, Journal of Computational Physics, 229 (2010) issue 5, pp. 1852-1878
CrossRef (DOI)
[16] Venkatramanan Raman, Heinz Pitsch, Rodney O., Fox, Hybrid large-eddy simulation/Lagrangian filtered-density-function approach for simulating turbulent combustion, Combust Flame (2005)
CrossRef (DOI)
[17] S. B. Pope, The probability approach to the modelling of turbulent reacting flows, Combustion and Flame, 27 (1976), pp. 299-312
CrossRef (DOI)
[18] R. O. Fox, Computational Models for Turbulent Reacting Flow, 2003
[19] P.E. Kloeden, E. Platen, Numerical Solutions of Stochastic Differential Equations, Springer, Berlin, 1999.
[20] A.Y. Klimenko, On simulating scalar transport by mixing between Lagrangian particles, Physics of Fluids, 19 (2007) no. 3, p 031702
CrossRef (DOI)
[21] S. B. Pope, Advances in PDF Methods for Turbulent Reactive Flows
CrossRef? yr? J? bk? ISBN