Generalized random walk algorithm for the numerical modeling of complex diffusion processes C alin Vamos ß a , Nicolae Suciu a, * , Harry Vereecken b a Romanian Academy, ‘‘T. Popoviciu’’ Institute of Numerical Analysis, Republicii Str. 37, P.O. Box 68, 3400 Cluj-Napoca 1, Romania b Forschungszentrum J€ ulich GmbH, Institut f€ ur Agrosph€ are (ICG-IV), D-52425, J€ ulich, Germany Received 11 December 2001; received in revised form 10 January 2003; accepted 25 January 2003 Abstract 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. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Diffusion; Random walk; Groundwater; Contaminant transport 1. Introduction It is well known that diffusion processes can be numerically simulated with random walk (RW) algo- rithm. For simple diffusion processes the RW algorithm is equivalent with the finite difference (FD) scheme [1] but, as we shall discuss in the following, this equivalence is not valid for more complex diffusion pro- cesses. The RW algorithm can be used to model the transport of arbitrary physical quantities if parts of the transported quantity are associated with fictitious particles obeying the RW law. To reduce the compu- tational effort and to improve the smoothness of the numerical solution, the gradient of the transported quantity can be associated with the particles. Integrating the gradient transported by each particle over the computation domain the simulated field is obtained with a higher accuracy [8]. The ‘‘gradient random walk’’ algorithm was first developed by Chorin [7] for the simulation of turbulence, the transported quantity being the vorticity. In other applications, mainly for transport in porous media, to reduce memory and to avoid boundary effects, a grid free algorithm called ‘‘particle tracking’’ (PT) is used [22]. The Journal of Computational Physics 186 (2003) 527–544 www.elsevier.com/locate/jcp * Corresponding author. E-mail addresses: cvamos@ictp.acad.ro (C. Vamos ß), n.suciu@fz-juelich.de (N. Suciu), h.vereecken@fz-juelich.de (H. Vereecken). 0021-9991/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0021-9991(03)00073-1