Short communication An iterative Langevin solution for contaminant dispersion simulation using the Gram–Charlier PDF Jonas C. Carvalho a, * ,E ´ zio R. Nichimura a , Marco Tu´llio M.B. de Vilhena b , Davidson M. Moreira a , Gerva´sio A. Degrazia c a Universidade Luterana do Brasil, Engenharia Ambiental, PPGEAM, Canoas, RS, Brazil b Universidade Federal do Rio Grande do Sul, Instituto de Matema ´tica, Porto Alegre, RS, Brazil c Universidade Federal de Santa Maria, Departamento de Fı´sica, Santa Maria, RS, Brazil Received 23 February 2004; received in revised form 11 June 2004; accepted 22 June 2004 Abstract An alternative numerical method to solve the three-dimensional stochastic Langevin equation applied to the air pollution dispersion is proposed and tested. We obtain a first-order differential equation whose solution is known and determined by an integrating factor. A Langevin model for inhomogeneous turbulence is obtained, considering the Gram–Charlier Probability Density Function (PDF) of turbulent velocity. The calculus process is based on an iterative scheme through the Picard Iterative Method. Numerical simulations and comparisons with measured data from two different tracer experiments are realized, showing a good agreement between predicted and observed values. Furthermore, the results obtained with the new approach are compared with the ones obtained by three different models. Ó 2004 Published by Elsevier Ltd. Keywords: Langevin equation; Lagrangian particle model; Gram–Charlier PDF; Picard iterative method; Model evaluation 1. Introduction Lagrangian stochastic particle models are powerful computational tools for the investigation of the atmo- spheric dispersion process. In these models, the particle displacements are produced by random velocities and the movement evolution of a particle is assumed to be a Markovian process (‘‘past and future are statistically independent when the present is known’’). This method is based on Langevin equation, which is derived from the hypothesis that the velocity is given by the combination between a deterministic term and a sto- chastic term. Each particle moves taking into account the transport due to the mean wind velocity and the turbulent fluctuations of the wind velocity components. From the spatial distribution of the particles it is possible to determine the pollutant concentration. The Langevin equation was the first example of a stochastic differential equation and it is normally integrated according to the rules of the Ito calculus (Rodean, 1996). Some special solutions of the Langevin equation are presented by Gardiner (1985) and Rodean (1996). This last author, for instance, describes the solution for stationary homogeneous turbulence as sug- gested by Lin and Reid (1963) and Legg and Raupach (1982). In this work we present an alternative numerical method to solve the Langevin equation applied to the air pollution dispersion in inhomogeneous turbulence con- ditions. The method leads to a first-order differential equation, solution of which is well known and de- termined by an integrating factor. The present scheme is * Corresponding author. Tel.: C55 51 4779285; fax: C55 51 4771313. E-mail address: jonas@ulbra.tche.br (J.C. Carvalho). 1364-8152/$ - see front matter Ó 2004 Published by Elsevier Ltd. doi:10.1016/j.envsoft.2004.06.002 www.elsevier.com/locate/envsoft Environmental Modelling & Software 20 (2005) 285–289