Page 310 - Computational Fluid Dynamics for Engineers

300 10. Inviscid Compressible Flow

BC BC

Fig. 10.3. 1-D computational domain
i for the non-linear wave equation.

£ • ~AF [ c 2 ( ^ 3 - < ) + ^ « - ^ )
2Ax (10.3.3)
+ cj(ug - u%) + c £ « - < ) + c £ K - ut)]

n+1 n
where At = t — , Ax = Xi+\ — xi (constant x spacings) and the numeri-
t
cal solution is a function of the interior domain values (02^3, c^u^ etc.). For
conservation to occur, we must have
Flux Flux,, 0 (10.3.4)
out
which expresses that the flux is conserved. Clearly, Eq. (10.3.3) does not sat-
isfy the condition Eq. (10.3.4) since the cross-terms inside the computational
domain show up as source terms. This is inadmissible, and these schemes are
called non-conservative. It can be shown that a non-conservative equation can
be transformed into a conservative equation by the following procedure.
Let us assume that the convective term of Eq. (10.3.1) can be rewritten as

du dwu
(10.3.5)
dx dx
where w is new function of u. The non-linear wave equation becomes

du dwu
(10.3.6)
dt dx
which now can be discretized using the same forward time, central space algo-
rithm used on the non-conservative equation, as

u n+1 - u 71 - 1
{wu)U) (10.3.7)
^r^ = is""*'
A flux balance over the Domain D now yields

-u n - 1
£ "3T 2Ax [(wu)% - (urn)? + (wu)2 - (wu)% + (wu)% - (wu)%
+ (WU)Q — (wu)2 + (wu)j — (wu)2
(10.3.8)
or