Particle approximation of convection–diffusion equations

We present a particle method for solving initial-value problems for convection–diffusion equations with constant diffusion coefficients. We sample N particles at locations xj(0) from the initial data. We discretize time into intervals of length Δt. We represent the solution at time tn=nΔt by N particles at locations xj(n). 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.