Discontinuous Particle Method for Diffusion Advection Problems

The particle method is a numerical method for modeling large systems based on their Lagrangian description. The discontinuous particle method belongs to the “particle–particle” type and consists of two main stages: predictor and corrector. At the predictor stage, particles move. At the corrector stage, a partner for interaction is selected from among the neighbors of the particle, which has the greatest influence on the local dynamics of the system. The “discontinuity” lies in the method of correcting the density of only one of the interacting particles. That is why the restoration of the distribution density occurs in the minimum region determined by only two selected particles, so the front is smeared by only one particle. The novelty of our method is to choose the density as a major characteristic of the particle unless its shape. The criterion for the density reconstruction is the preservation of the projection of the mass on the plane passing through the centers of mass of the interacting particles. A neighbor for density correction is selected using the “impact parameter”. The density is constructed using two selected interacting particles, which allows us to reduce a two-dimensional problem to a one-dimensional one. The effectiveness of the method is presented using the Crowley test as an example. Despite the linearity of this problem the trajectories of the particles are not simple. That’s why we need the Runge–Kutta method at the predictor stage to increase the accuracy. Our Lagrangian approach to constructing the particle method contrasts with another frequently used representative of the particle–particle method—the smoothed particle method (SPH). The article is of a methodological nature, its purpose is to demonstrate the capabilities of the new discontinuous particle method.