Generalized random walk algorithm for the numerical modeling of complex diffusion processes


A generalized form of the random walk algorithm to simulate diffusion processes is introduced.

Unlike the usual approach, at a given time all the particles from a grid node are simultaneously scattered using the Bernoulli repartition. This procedure saves memory and computing time and no restrictions are imposed for the maximum number of particles to be used in simulations.

We prove that for simple diffusion the method generalizes the finite difference scheme and gives the same precision for large enough number of particles.

As an example, simulations of diffusion in random velocity field are performed and the main features of the stochastic mathematical model are numerically tested.


Călin Vamoş
Tiberiu Popoviciu Institute of Numerical Analysis (Romanian Academy)

Nicolae Suciu
Tiberiu Popoviciu Institute of Numerical Analysis (Romanian Academy)

Harry Vereecken


Diffusion; random walk; groundwater; contaminant transport

Cite this paper as

C. Vamoş, N. Suciu, H. Vereecken (2003), Generalized random walk algorithm for the numerical modeling of complex diffusion processes, J. Comp. Phys., 186(2), 527-544, doi: 10.1016/S0021-9991(03)00073-1


see the expanding block below.



About this paper


J. Comp. Phys.

Publisher Name
Print ISSN

Not available yet.

Online ISSN

Not available yet.

Google Scholar Profile

[1] W.F. Ames, Numerical Methods for Partial Differential Equations, second ed., Academic Press, New York, 1977.

[2] S. Attinger, M. Dentz, H. Kinzelbach, W. Kinzelbach, Temporal behavior of a solute cloud in a chemical heterogeneous porous medium, J. Fluid Mech. 386 (1999) 77.
CrossRef (DOI)

[3] M. Avellaneda, A.J. Majda, Superdiffusion in nearly stratified flows, J. Stat. Phys. 69 (3/4) (1992) 689.
CrossRef (DOI)

[4] A. Bellin, P. Salandin, A. Rinaldo, Simulation of dispersion in heterogeneous porous formations: statistics, first-order theories, convergence of computations, Water Resour. Res. 28 (9) (1992) 2211.
CrossRef (DOI)

[5] C.W. Gardiner, Handbook of Stochastic Methods, Springer, New York, 1983. C. Vamos et al. / Journal of Computational Physics 186 (2003) 527–544 543

[6] S.K. Godunov, V.S. Ryabenkii, Difference Schemes: An Introduction to the Underlying Theory, North-Holland, Amsterdam, 1987.

[7] A.J. Chorin, Vortex sheet approximation of boundary layers, J. Comput. Phys. 27 (1978) 428.
CrossRef (DOI)

[8] A.F. Ghoniem, F.S. Sherman, Grid-free simulation of diffusion using random walk methods, J. Comput. Phys. 61 (1985) 1.
CrossRef (DOI)

[9] W. Horsthemke, R. Lefever, Noise-induced Transitions. Theory and Applications in Physics, Chemistry and Biology, Springer, Berlin, 1984.

[10] C. Kapoor, L.W. Gelhar, Transport in three-dimensionallity heterogeneous aquifers 1. Dynamics of concentration fluctuations, Water Resour. Res. 30 (6) (1994) 1775.
CrossRef (DOI)

[11] C. Kapoor, L.W. Gelhar, Transport in three-dimensionallity heterogeneous aquifers 2. Prediction and observations of concentration fluctuations, Water Resour. Res. 30 (6) (1994) 1789.
CrossRef (DOI)

[12] W. Kinzelbach, Numerische Methoden zur Modellierung des Transports von Schadstoffen im Grundwasser, Oldenbourg Verlag, Munchen, 1992. €

[13] G.L. Moltyaner, M.H. Klukas, C.A. Willis, R.W.D. Killey, Numerical simulations of twin lake natural-gradient tracer tests: A comparison of methods, Water Resour. Res. 29 (10) (1992) 3443.
CrossRef (DOI)

[14] O. Neuendorf, Numerische 3D-Simulation des Stofftransport in einem heterogenen Aquifer. Ph.D. Thesis, Jul-3421 Forschungs- € zentrum Julich, 1997. €

[15] A. Papoulis, Probability, Random Variables and Stochastic Processes, McGraw-Hill, New York, 1991.

[16] K. Roth, K. Hammel, Transport of conservative chemical through an unsaturated two-dimensional Miller-similar medium with steady state flow, Water Resour. Res. 32 (6) (1996) 1653.
CrossRef (DOI)

[17] P. Salandin, V. Fiorotto, Solute transport in highly heterogeneous aquifers, Water Resour. Res. 34 (5) (1998) 949.
CrossRef (DOI)

[18] H. Schwarze, U. Jaekel, H. Vereecken, Estimation of macrodispersivity by different approximation methods for flow and transport in randomly heterogeneous media, Transp. Porous Media 43 (2001) 265.

[19] Ne.-Z. Sun, Mathematical Modeling in Groundwater Pollution, Springer, New York, 1996.
CrossRef (DOI)

[20] A.F.B. Tompson, L.W. Gelhar, Numerical simulation of solute transport in three-dimensional, randomly heterogeneous porous media, Water Resour. Res. 26 (10) (1990) 2541.
CrossRef (DOI)

[21] A.F.B. Tompson, R.B. Knapp, Reactive Geochemical Transport Problems in Nuclear Waste Analyses, Preprint UCRL-00552, Lawrence Livermore National Laboratory, Livermore, 1989.

[22] A.F.B. Tompson, R.D. Falgout, S.G. Smith, W.J. Bosl, S.F. Asby, Analysis of subsurface contaminant migration and remediation using high performance computing, Adv. Water Resour. 22 (3) (1998) 203.
CrossRef (DOI)

[23] C. Vamos, N. Suciu, H. Vereecken, O. Nitzsche, H. Hardelauf, Global random walk simulations of diffusion, in: W. Kramer, J.W.V. Gudenberg (Eds.), Scientific Computing, Validated Numerics, Interval Methods, 343, Kluwer Academic Publishers, Dordrecht, 2001.
CrossRef (DOI)

[24] R. Zhang, K. Huang, M.T. van Genuchten, An efficient Eulerian–Lagrangian method for solving solute transport problems in steady and transient flow fields, Water Resour. Res. 29 (12) (1993) 4131.
CrossREf (DOI)


Related Posts