Micro-macro Kolmogorov–Fokker–Planck models for a hard-sphere gas

Abstract

Using a stochastic microscopic model of a rigid-sphere gas in a phase space, which is diffusive in the velocity space and valid at moderate Knudsen numbers, macroscopic equations of gas dynamics are derived, which are different from the system of Navier–Stokes equations or quasi-gasdynamic systems. The main pecularity of our derivation is more accurate velocity averaging due to the analytical solution of stochastic differential equations with respect to the Wiener measure, which describes our original meso model. The problem of a shock-wave front is used as an example showing that such an approach yields a greater and thus more realistic diffusion of the front than the one based on the Navier–Stokes equation. The numerical solution is based on a “discontinuous” particle method well suited for supercomputer applications.