Implicit formulation for advection–diffusion simulation based on particle distribution moments

This paper introduces an implicit method for advection–diffusion equations called Implicit DisPar, based on particle displacement moments applied to uniform grids. The present method tries to solve constraints associated with explicit methods also based on particle displacement methods, in which diffusivity-dominated situations can only be handled by considerably increasing the associated computational costs. In fact, a higher particle destination nodes number allows the use of higher diffusion coefficients for the transport simulation without instabilities. The average was evaluated by an analogy between the Fokker–Planck and the transport equations. The variance is considered to be Fickian. The particle displacement distribution is used to predict deterministic mass transfers between domain nodes. Mass conservation was guaranteed by the distribution concept. In the truncation error analysis, it was shown that the linear Implicit DisPar formulation does not have numerical error up to v − 1 order, if the first v particle moments are forced by the Gaussian moments. It was shown by theoretical tests for linear conditions that the model accuracy level is proportional to the number of particle destination nodes.