528 Difference Methods for Solving Nonlinear Equationscorresponds to the case when x ___, oo. The system of equations of gas
dynamics for the isotern1ic flow of the ideal gas comprises the equationsav op 8 1 av
(16) - , at (p) = as , p = c2p,
8t aswhere c = const > 0 is the velocity of sound, or
(17)av op 817 av
- p 17 = c2.
at as
,
at as
,
- p 17 = c2.
which are consistent with the adiabatic case indicated above. In the sequel
our exposition is mostly based on more a detailed exploration of three
equations (9), (10) and (11) capable of describing the gas dynamics.
The complete posing of this necessitates specifying the boundary and
initial conditions in addition to equations (9)-(10). Knowing(18) 1'(a:, O), p(x, 0), p(x, 0),
we may attempt, for example, the boundary conditions in the form(19) p(U,t)=p 0 (t) for s=O, p(JVI, t) = p 1 (t) for s = J\;J,
or( 191 ) u(O, t) = v 0 (t) for s --^0 ' p(lv1,t) =p 1 (t) for s = M.
Summarizing, it is required to construct difference schemes for equa-
tions of gas dynamics (9)-(10) in the complex closed domain {O < s < M,
t > O}.- Equations with psevdoviscosity. In practical in1plen1entatio11s there is
a real need for forming homogeneous and conservative difference equations
of gas dynamics. The nleaning of hon10geneity here is that difference equa-
tions are written at· all the nodes of the grid in just the same way regardless
of the possible continuity or discontinuity of a solution so that subsequent
calculations should be carried out by the same ensuing formulas. Homoge-
neous schen1es or "through execution" schemes of gas dynamics contain the
extra n1en1bers with psevdoviscosity, a key role of which is to spread the
front of shock waves over several intervals of the grid. Fron1 a fonnal point
of view, the viscosity w arises as the additional member to the pressure p,
so that equations (9)-(10) contain for now instead of p the sum
g=p+w,