On the use of random walk models with spatially variable diffusivity

Abstract

The random walk technique is commonly used to model diffusion in the environment. For a constant diffusivity K and model time-step dt. the random step should be chosen from a distribution with variance 2Kdt. However, if K varies spatially, this choice of step leads to the accumulation of particles in regions of low diffusivity. This problem may be overcome either by the incorporation of an apparent advection velocity, or by transforming to a coordinate system in which the diffusivity is constant. The latter technique requires no immediate approximations, is applicable to any reasonable diffusivity field and is therefore the preferred approach. In this case, as with constant K, the random step should be chosen from a normal distribution, for reasons of both theoretical accuracy and computational efficiency. Three important aspects of model design are discussed: the selection of the random number generator, the time step and the total number of particles.