Bogomolov S.V.

The particle method is a numerical method for modeling large systems based on their Lagrangian description.

The discontinuous particle method is of the “particle–particle” type and consists of two main stages: predictor and corrector. At the predictor stage, a particle shift occurs. At the corrector stage, a partner for interaction is selected among the neighbors of the particle, most influencing the local dynamics of the system. The “discontinuity” of the method lies in the method of density correction only one of the interacting particles, due to which the restoration of the distribution density occurs in a minimal region defined by only two selected particles, which leads to smearing of the front by only one particle.

The novelty of the method presented in this article is that The density of the particles is put in the foreground, not their shape. The criterion for restructuring is the preservation of the projection of the mass onto the plane passing through the centers masses of interacting particles. The neighbor for density correction is selected using the “impact parameter”. The density is constructed using two selected interacting particles, which makes it possible 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. It is shown that the Runge–Kutta method at the predictor stage significantly increases the accuracy of the numerical solution.

Our Lagrangian approach to constructing the particle method contrasts with another frequently used particle-particle method, the smoothed particle method (SPH).

This paper demonstrates the process of deriving macroscopic equations of gas dynamics from a stochastic microscopic model describing the behavior of gases from hard spheres. As the Knudsen number decreases, the resulting equations acquire a form similar to quasi-gasdynamic (QGD) equations with regularizing components that differ from the classical Navier–Stokes system.

A classic example of a simple discrete problem of the motion of a gas consisting of absolutely elastic spheres is considered. This example illustrates the key stages of constructing mathematical models for large systems using stochastic differential equations (SDE) and their computer implementation.

Ludwig Boltzmann derived his fundamental equation based on the balance of the distribution function describing the gas as a continuous medium in the phase space of coordinates and velocities, that is, using the Eulerian approach. We consider a gas as a set of individual particles, the dynamics of which is described by the SDE system, which means the Lagrange formalism.

The proposed model is self-sufficient and does not require setting additional equations of state or configurable parameters.