1549301742-The_Theory_of_Difference_Schemes__Samarskii

Conservative difference schernes of nonstationary gas dynamics 531

and

(25)

(26)

(27)

(^8) i+l/2
j v ds::::::: vi h,
(^8) i-l/2
(^8) i+l
j T/ ds ::::::: T/i+l/ 2 h, etc.
Si
As a final result we get the difference scheme
T (
Pi+1/2 - Pi-1/2) ( er1 l _

• h - 0,
  j+l j
  T/i+l/2 - T/i+l/2
  T
  (er3) (er,) (er3) (er. 1 )
  Pi+l Vi+l - Pi 1!;
  h
  which falls within the category of conservative schemes for any admissible
  values ofpara1neters er 1 , er 2 , er 3 , er 4. In particular, for er 1 = 0, er 2 = 1, er 3 = 1,
  er 4 = 1 we constitute a syste1n of difference equations, whose solution is
  found successively by the explicit formulas: at the first stage - u;+^1 , at the
  second stage - T/~: 1112 and at the third stage - Pi+l/ 2 , i = 0, 1, ... , N - l by
  the elimination method from the energy equation and the state equation
  p 11 = ( / - 1 )E being supplemented with the suitably chosen boundary
  conditions for i = 0 and i = N - l. For example, we might agree to impose
  conditions (19).
  In this context, it should be noted that conservative difference sche1nes
  may be good enough for the equation of the total energy, but approxirnate
  poorly the equation of the internal energy (14)
  OE ov
  ot = -p os ·
  This can result in improper choices of computational procedures in
  giving the temperature. The lack of energetic balances cannot be avoided
  by refining the grid in a spatial variables. A presence of energetic disbal-
  ances in a scheme can be interpreted as a presence of energetic sources of a
  purely difference nature connected with smne "lack of agreement" between
  separate difference equations of a scheme being inconsistent each to other.