Conservative difference schemes of nonstationary gas dynamics 541- Equations of gas dynamics with heat conductivity. vVe are now inter-
ested in a complex problem in which the gas flow is moving under the heat
conduction condition. In conformity with ( 1 )-(7), the system of differential
equations for the ideal gas in Lagrangian variables acquires the form
ov
otog 017
os ' otov 0€ ov ow
os ' ot = -g os - os 'p11 =RT, c == cvT,
w h ere w = -x ( p, T)-oT ts. the heat flow anc l g = p + w.
OS
It is plain to create for this syste1n of equations a fully conservative
schen1e such asvt = -17~°') L s ' (^17) t = vCO s 5) '
(60)
Et = -g·Cc>) vCO s 5J - wCf3) s ,
w = - k T 8 , g = p + w , w = w ( 17, vs , v) ,
where k = k; = x(0.'5 (1Jiu 5 + 17;+o 5 ), 0.5(T, 0 5+1~+o 5 )). Here CY > 0
and /3 > 0 are arbitrary nun1bers.
Generally speaking, Newton's n1ethod may be employee! for nonlinear
difference equations on every new layer, but the algorithm of the matrix
elimination for a system of two three-point equations (see Chapter 10, Sec-
tion 1) suits us perfectly for this exceptional case. We will say more about
this later.
The tnain idea behind this approach is to accelerate and sin1plify the
algorithms by tneans of the niethod of separate or successive elin1inations.
To that encl, the difference equations ( 60) are divided into the following
groups:
I: "dynamical group"
V t = -g(<>) s )^1 7t -- V(O 8 5) I g=p+w,
w = w(TJ, v 8 , ZJ), p = p( 17, T).