Mathematics and Computers in Simulation 55 (2001) 123–130 Particle approximation of convection–diffusion equations Christian Lécot a,* , Wolfgang Ch. Schmid b,1 a Laboratoire de Mathématiques, Université de Savoie, Campus Scientifique, F-73376 Le Bourget-du-Lac Cedex, France b Institut für Mathematik, Universität Salzburg, Hellbrunnerstraße 34, A-5020 Salzburg, Österreich, Austria Abstract We present a particle method for solving initial-value problems for convection–diffusion equations with constant diffusion coefficients. We sample N particles at locations x (0) j from the initial data. We discretize time into intervals of length 1t. We represent the solution at time t n = n 1t by N particles at locations x (n) j . In each time interval the evolution of the system is obtained in three steps. In the first step the particles are transported under the action of the convective field. In the second step the particles are relabeled according to their position. In the third step the diffusive process is modeled by a random walk. We study the convergence of the scheme when quasi-random numbers are used. We compare several constructions of quasi-random point sets based on the theory of (t, s)-sequences. We show that an improvement in both magnitude of error and convergence rate can be achieved when quasi-random numbers are used in place of pseudo-random numbers. © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Random walk; Convection–diffusion equations; Discrepancy 1. Introduction Mathematical models that involve a combination of convective and diffusive processes are among the most widespread in all engineering fields where mathematical modeling is important. Very often the pa- rameter that measures the relative strength of the diffusion is quite small. So difficulties are experienced with standard numerical approximations. The method of splitting consists of reducing the original evo- lutionary problem to two problems describing the convective and the diffusive processes, respectively. In our method, discrete particles are used to track the evolution of the solution. These particles are advected according to the velocity field and, in addition, undergo a random walk motion to simulate the diffusion. The random walk treatment of diffusion is the same as the treatment of vorticity in the random vortex method [1]. This method introduces however a statistical error which can be large for the solution. The quasi-random approach aims at improving the rate of convergence in the random method by number-theoretic techniques. Instead of random samples as in Monte Carlo methods, one employs uni- * Corresponding author. E-mail addresses: christian.lecot@univ-savoie.fr (C. L´ ecot), wolfgang.schmid@sbg.ac.at (W.Ch. Schmid). 1 Research partially supported by the Austrian National Bank Project No. 6788. 0378-4754/01/$20.00 © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. PII:S0378-4754(00)00252-4