Пределы применимости модели непрерывной среды и регуляризация уравнений динамики жидкости и газа

The continuous media is a fruitful mathematical notion for the
descriptions of the variety of mechanical processes. In continuous
mechanics basic equations of the balance of mass, momentum, and energy
are derived first for the finite volumes and next transformed to the
differential form. Together with physically reasonable boundary and
initial conditions they are expected to give full description required
processes. However the transition from a finite volume description to a
differential one involves the mathematical assumption of the
infinitesimal volume.
Notice that the governing equations of hydro and gas dynamics are
nonlinear and a variety of crucial mathematical features are to be solved.

But taking into account the molecular structure of matter the
infinitesimal volume necessarily requires dozens of molecules inside.
It means that physically there is the lower limit on its size.
For instance the air of normal density is containing 1019 particles per
cm3. It means that physically infinitesimal volume has the minimum size
of the order d~10–6 cm3. It is much less than the mean free path, which
is in the range of 10–4–10–5 cm, and is much greater than the
characteristic size of gas molecules which are ~10–8 cm. Thus on the d
scale values of gas-dynamic parameters must be constant.
Note that for the computer calculations the boundary-initial value
problem must be discretized. That in turn requires a discrete
description of the continuous media. That is why in the computer
modelling differential equations serves for the transition from one
discrete description to another one. That is why a regularization of
equations of hydro and gas dynamics should be physically reasonable and
additional terms ensuring regularization of solutions at distance d must
be included in the differential equations. The absence of such terms in
the classical equations of gas dynamics can be explained by the fact
that the distance d = 10–6 cm, over which an essential change of the gas
dynamic parameters is ensured, is significantly smaller than the
discretization error. These resolution limits are determined by both the
physical necessity of the detailed problem description, the capabilities
of the numerical method, and the performance characteristics of the
computer facilities employed in the calculations. The main aim of the
lectures is to give an introduction to quasi-gas dynamical systems of
equations and theory of lattice Boltzmann and kinetically consistent
difference schemes.