Consistency issues in PDF methods

Open access


Concentrations of chemical species transported in random environments need to be statistically characterized by probability density functions (PDF). Solutions to evolution equations for the one-point one-time PDF are usually based on systems of computational particles described by Itô equations. We establish consistency conditions relating the concentration statistics to that of the Itô process and the solution of its associated Fokker-Planck equation to that of the PDF equation. In this frame, we use a recently proposed numerical method which approximates PDFs by particle densities obtained with a global random walk (GRW) algorithm. The GRW-PDF approach is illustrated for a problem of contaminant transport in groundwater.


  • [1] S.B. Pope, PDF methods for turbulent reactive ows, Prog. Energy Combust. Sci. 11(2) (1985) 119{192.

  • [2] R.O. Fox, Computational Models for Turbulent Reacting Flows, Cambridge University Press, New York, 2003.

  • [3] N. Suciu. Diffusion in random velocity fields with applications to contaminant transport in groundwater, Adv. Water Resour. 69 (2014) 114-133.

  • [4] S.B. Pope, Turbulent Flows, Cambridge University Press, Cambridge, 2000.

  • [5] D.C. Haworth, Progress in probability density function methods for turbulent reacting ows, Prog. Energy Combust. Sci., 36 (2010) 168-259.

  • [6] A.Y. Klimenko, R.W. Bilger, Conditional moment closure for turbulent combustion, Progr. Energ. Combust. Sci. 25 (1999) 595{687

  • [7] N. Suciu, F.A. Radu, S. Attinger, L. Schüler, Knabner, A Fokker-Planck approach for probability distributions of species concentrations transported in heterogeneous media, J. Comput. Appl. Math. (2014), in press (doi:10.1016/

  • [8] D.W. Meyer, P. Jenny, H.A. Tchelepi, A joint velocity-concentration PDF method for tracer ow in heterogeneous porous media, Water Resour. Res., 46 (2010) W12522.

  • [9] R. W. Bilger, The Structure of Diffusion Flames, Combust. Sci. Tech. 13 (1976) 155-170.

  • [10] S. Attinger, M. Dentz, H. Kinzelbach, W. Kinzelbach, Temporal behavior of a solute cloud in a chemically heterogeneous porous medium, J. Fluid Mech. 386 (1999) 77-104.

  • [11] N. Suciu, C. Vamos, J. Vanderborght, H. Hardelauf, H. Vereecken, Numerical investigations on ergodicity of solute transport in heterogeneous aquifers, Water Resour. Res. 42 (2006) W04409.

  • [12] C. Vamoș, N. Suciu, H. Vereecken, Generalized random walk algorithm for the numerical modeling of complex diffusion processes, J. Comp. Phys., 186 (2003) 527-544.

  • [13] F. A. Radu, N. Suciu, J. Hoffmann, A. Vogel, O. Kolditz, C-H. Park, S. Attinger, Accuracy of numerical simulations of contaminant transport in heterogeneous aquifers: a comparative study, Adv. Water Resour. 34 (2011) 47-61.

  • [14] 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, J. Comput. Appl. Math. 246 (2013) 27-37.

  • [15] C. Vamoș, M. Crăciun, Separation of components from a scale mixture of Gaussian white noises, Phys. Rev. E 81 (2010) 051125.

  • [16] C. Vamoș, M. Crăciun, Automatic Trend Estimation, Springer, Dortrecht, (2012).

Analele Universitatii "Ovidius" Constanta - Seria Matematica

The Journal of "Ovidius" University of Constanta

Journal Information

IMPACT FACTOR 2016: 0.422

CiteScore 2016: 0.56

SCImago Journal Rank (SJR) 2016: 0.346
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researchers in all fields of pure and applied mathematics


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