Faculty of Mathematics and Mechanics St Petersburg State University, Russia

Various forms of the kinetic equations being equivalent from
the mathematical point of view are convenient in devising and
grounding of computational methods in rarefied gas dynamics. It
must be noted that this various forms can be derived from definite
phenomenological hypothesis on which direct simulation method of
rarefied gas flow is founded.
The purpose of the work is to derive and construct the system
of kinetic equations which is equivalent to the Boltzmann equation.
Equations of the system are well known models of mathematical
physics and different methods are applicable for their solution. In
deriving of the system of divergence equations the processes of
convectional transfer and particle collisions in gas are separated
and a construction of statistical models become more easy. In the
Bird method this simplification has been achieved due to an
assumption about the decomposition of both processes into two
steps - moving the particles and computing collisions appropriate
to a small time interval.
To derive the system of kinetic equations in divergence form a
force field **F**, which effect is equivalent to a collision
integral in the Boltzmann equation is introduced. This in contrast
to the Vlasov equation. In divergence form of kinetic equations
self - consistent field **F** is introduced according to the
relation between collision integral J(f,f) and divergence in
velocity space **(u)** - div**ᵤ**,
of vector flux (1/m)(**F**f)

1/m div**ᵤ**(**F**(**r**,**u**,t)
f(**r**,**u**,t)) = - J(**r**,**u**,t).

In the report this approach is generalized for
the case of gas mixture. The models of field **F** are
constructed for solving the problems of spatially homogeneous gas
relaxation and plane steady shock.
One of the interest points is transformation of the equation
from the divergence form to integral one. The integration is
realized along the trajectories which are formed by the use of the
field **F**. The integral form of the divergence equations represents
another variant of a description of rarefied gas flow when the
assumptions of Boltzmann equation take place. The iteration scheme
for the divergence form of kinetic equation is not traditional. One
must solve motion equations for particles in the field **F** on every
step of this procedure. Use of the divergence form of kinetic
equations allows to derive net analog of kinetic equation, thus
for the phase sell we get the balance law on which the derivation
of Boltzmann equation is based. The difference equations based on
the net analog will satisfy conservation equations.

* Filippov B.V., Khristinich V.B. Divergence form of the
kinetic equations of rarefied gas dynamics. // Dynamical processes
in gases and solids. Ser. Physical mechanics, issue 4, Leningrad
State University, 1980, p. 7 - 18.
*