1549301742-The_Theory_of_Difference_Schemes__Samarskii

Conservative difference schemes of nonstationary gas dynan1ics 533

which, in turn, can be obtained from the equations 77t = v~"2) and v~"^4 ) =

·u~"^2 ) - T(0' 4 - 0' 2 ) ust = 1/t + T(0' 4 - 0' 2 )vst· From here it seems clear that

(31) i.s consistent with equation ( 15) only for 0' 4 = 0' 2 ,

At the next stage sche1ne (28)-(29) is obliged to be conservative. This

can always be done by multiplying the equation ·ut = -p~"'^1 by vC^0 5!

0.5(v + v), adding the resulting equation

(32)

to equation (29):

(33)

and rearranging the right-hand side of (33) by means of formula (30) as

x ( uls^0 5 ) + T (O' 4 - 0.5) u st ) + vC^0 5 ) t-lJ~"s^1 I

• 
  
  
  
  • (p(cr1) (-1) v(05)) s +o 2 E ,
  
  
  
  

where

Pc-11 = Pi-l = Pi-l/2,

b 2 E -- T ( O' ::. - O' l ) vl s^0 51 ]J t + T ( O' 4 - (^0) ,i 5) p^1 "^1 1 u st
The outcome of rearranging equation (33) is
(35)
In the preceding the quantity 02 E means the disbalance of the internal
energy. By equating 02 E to 0 for any p and v we find that 0' 3 = 0' 1 and
0' 4 = 0.5 and, hence, 0' 2 = 0.5.
Thus, under such an approach a one-parameter family of fully conser-
vative schemes is given by
( 36) 1/ t = v s Co 5) ' ' E t = -p(cr1) 1 /0 s 5) ,