Abstract:  T his paper presents a novel lattice walk method to simulate the convection diffusion equation. The convection-diffusion equation is widely used to describe solute transport. However, conventional methods, e. g. the
finite difference method or the finite element method, to solve this equation may result in numerical dispersion and oscillation, especially when convection velocity is large.
The lattice walk method combines the virtues of random walk method and lattice Boltzmann method (LBM), to simulate convection-diffusion process. The positions of solute particles are limited on lattice points and the jumping
probability of particles on regular lattices is determined by Gaussian distribution of random walk. We construct the transition matrix by the equality of the moments ( the zeroth, first and second order) of discrete distribution on
lattices and Gaussian distribution of random walk. By iteration, we can calculate the evolution of the concentration of solute. Then one dimensional instantaneous point source model and constant concentration point source model are used
to test this novel method. The results of one dimensional instantaneous point source model are shown. When convection dominates the convection-diffusion process, the result given by lattice walk method agrees well with the
analytical solution while the solution of finite difference method suffers from serious numerical dispersion. For the diffusion dominated example, the concentration obtained from lattice walk method is consistent with that from finite
difference method. T herefore we can see the most important character of lattice walk method that it can simulate solute transport at convection velocities far beyond the range of conventional methods.
The results of constant concentration point source model are shown. Our results of lattice walk method agree well with analytical solutions and have little dispersion in the front of the concentration curve. However, the
concentration obtained by finite difference method suffers rather serious numerical dispersion even if Pe number is not too large. T herefore from the results of this model we can see that lattice ?walk method can exclude numerical
dispersion, which has puzzled finite difference method and finite element method for a long time.