1549301742-The_Theory_of_Difference_Schemes__Samarskii

Conservative difference schernes of nonstationary gas dynmnics 535

Putting these together with the supplementary boundary conditions for

i = 0 and i = N, say with the values IJii, p~ of pressure, we must write

down the equation of the motion for vf not only at the inner nodes, but

also on the boundary for i = 0 and i = N:

v~+l - v~ = -(P1;2 - Po)(cri),

T 0.5 h.

j+l j ( )

VIV -VIV= -(PIV-PIV-1/2) "'

T 0.5h '

giving v~+l and vj/^1. The remaining quantities 77, E, pare sought only at

the inner half-integer points s 112 , s 312 , ... , sIV-l/ 2 ·

It seen1s worthwhile giving one possible example of nonconservative

schemes. A "cross" scheme was very popular and much applicable in recent

decades. Any scheme of this structure can be written on a "chess" grid by

regarding E, p, 17 to half-integer nodes (s;+ 1 ; 2 , tj+ 1 ; 2 ) and v, x to integer

nodes ( s;, tj) of the grid at hand.

Within the notations p;:::~ = 'Pi = p and 11;: 11 /~ rif +^1 = 1/, etc.,

the "cross" scheme

j+l/2 j+l/2

P;+1/2 - Pi-1/2

T h

j+3/2 j-1/2

77;+1/2 -^1 71+1/2

u^1 ·+1 - ·u^1 ·+1

z+ 1 z

T h

admits an alternative f01·m

(45)^1 - 7t = vs ,

showing the new members to be sensible ones. In this line the values on
every new layer is found by the explicit formulas. No wishing to load the
book down with full details on this point, we cite only a final result after
multiplying vt = -j5 8 by vC^0 5 ) and repeating the preceding manipulations
in such a setting:

where oE = Tj5tv~^0 5 ) + 0.5TPVst + 0.5T^2 j5tvStl that is, sche1ne (45) is not
conservative, thus causing some limitations in p1·actical implementations.