Deterministic Advection-Diffusion Model Based on Markov Processes

A new deterministic numerical formulation named DisPar-k based on particle displacement probability distribution for Markov processes was developed to solve advection-diffusion problems in a one-dimensional discrete spatial grid. DisPar-k is an extension of DisPar, and the major difference is the possibility of establishing a number of consecutive particle destination nodes. This was achieved by solving an algebraic linear system where the particle displacement distribution moments are known parameters taken from the Gaussian distribution. The average was evaluated by an analogy between the Fokker-Planck and the transport equations, being the variance Fickian. The particle displacement distribution is used to predict deterministic mass transfers between domain nodes. Mass conservation was guaranteed by the distribution concept. It was shown that, for linear conditions, the accuracy order is proportional to the number of particle destination nodes. DisPar-k showed to be very sensible to physical discontinuities in the transport parameters (water depth, dispersion, and velocity), showing that this type of problem can only be disguised by introducing numerical dispersion (i.e., changing the Fickian variance).