Conservative difference schemes of nonstationary gas dynamics 525
8.2 CONSERVATIVE DIFFERENCE SCHEMES
OF NONSTATIONARY GAS DYNAMICS- One-di1nensional equations of nonstationary gas dynainics in Lagrangian
variables. Plenty of problems a.rising in 111echanics and physics such as
the harnessing of nuclear energy, the creation of nuclear reactors, aircrafr
(planes and space-vehicles) design, the dynamics of space flights, plasma
physics (governed thermo-nuclear synthesis) led to equations of gas dy-
namics that are, generally speaking, nonlinear and can be solved by the
universal difference inethods.
In spite of the fact that problen1s in gas dynamics began to spread
to more and more branches of science, engineering and technology as they
gradually took on an important place in real-life situations, for the time
being there are no rigorous mathematical results regarding convergence of
schemes even in the simplest and typical situations. The desirable qual-
ity of schemes are verified with the aid of linear models in the acoustic
approximation by applying numerical tests for specimen problems, whose
analytical solutions are known to the users in explicit form.
As a rule, equations of gas dynan1ics are discontinuous. From a phys-
ical point of view it is fairly common to distinguish weak discontinuities
relating to "cutting waves'' and strong discontinuities relating to "shock
waves". For these reasons successive grid refinement can be made with
caution when the accurate account of accuracy of numerical methods is
performed.
In this section we initiate the design of difference n1ethods for nu-
1nerical solutions of the simplest proble1ns in gas dyna1nics. Of our initial
concern is the problem about one-dimensional nonstationary gas flow in a
plane with the following ingredients: velocity v, density p, temperature T,
pressure p, internal energy E.
In preparation for this, the equations of gas dynamics will reproduce
the conservation lqws of impuls, mass and energy that can be written in
a number of different ways with respect to Eulerian ( x, t) or Lagrangian
(s, t) variables, where :r is the coordinate of a particle and s is the initial
coordinate of a particle or the quantity
.c
8 = J p( ~ ) 0) cl~ )
0that is, the value of inass rn the volume 0 < ~ < i:. The usual practice