Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

40 Convection dominated problems
An alternative approximation for U recently recommended is 62
Using the Taylor expansion
U ˆ Un ‡ 1 ‡ U n j …x ÿ †
2
U n j …x ÿ † U n ÿ tU n @U n
…2:85†
@x ‡ O… t2 † …2:86†
from Eqs (2.78)±(2.81) and Eqs (2.85) and (2.86) with equal to 0.5 we have
where
1
t … n ‡ 1 ÿ n †ˆÿU n ‡ 1=2 @ n
t
‡
@x 2 Un @U n
@
@x
n
t
‡
@x 2 Un ‡ 1=2 U n ‡ 1=2 @ 2
@x2 ‡ @
@x
k @
@x
n ‡ 1=2
ÿ Q ‡ t
2 Un ‡ 1=2 @Q
@x
ÿ t
2 Un ‡ 1=2 @
@x
U n ‡ 1=2 ˆ Un ‡ 1 ‡ U n
2
@
@x
k @
@x
n
…2:87†
…2:88†
We can further approximate, as mentioned earlier, n ‡ 1=2 terms using n, to get the
fully explicit version of the scheme. Thus we have
U n ‡ 1=2 ˆ U n ‡ O… t† …2:89†
and similarly the di€usion term is approximated. The ®nal form of the explicit
characteristic±Galerkin method can be written as
ˆ n ‡ 1 ÿ n ˆÿ t U n @ @
ÿ
@x @x
k @
@x
‡ t2
2 Un @
@x Un @ @
ÿ
@x @x
‡ Q
k @
@x
n
‡ Q
n
…2:90†
Generalization to multidimensions is direct and can be written in indicial notation for
equations of the form Eq. (2.5):
ˆÿ t @…Uj †
ÿ
@xj @
@xi ‡ t2
2 Un k
@
@x k
k @
@x i
@…Uj †
ÿ
@xj @
@xi ‡ Q
n
k @
@x i
‡ Q
n
…2:91†
The reader will notice the di€erence in the stabilizing terms obtained by two
di€erent approximations for U. However, as we can see the di€erence between
them is small and when U is constant both approximations give identical stabilizing
terms. In the rest of the book we shall follow the latter approximation and always use
the conservative form of the equations (Eq. 2.91).