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

80 A general algorithm for compressible and incompressible ¯ows
Note that Ui is zero from the third of Eq. (3.76). As in Split A we can write the
following system
K = G T
"
G
# ( )
~U
ˆ
0 ~p
f ( )
1
f2 …3:79†
where f 1 and f 2 arise from the forcing terms as in the Split A form. Clearly here the BB
restrictions are not circumvented.
It is interesting to observe that the lower diagonal term which appeared in Eq. (3.75)
is equivalent to the di€erence between the so-called fourth-order and second-order
approximations of the laplacian. This justi®es the use of similar terms introduced
into the computation by some ®nite di€erence proponents. 53
3.5 A single-step version
If the U i term in Eq. (3.26) is omitted, the intermediate variable U i need not be
determined. Instead we can directly calculate (or p), U i and E. This of course
introduces an additional approximation.
The use of the approximation of Eq. (3.1) is not necessary in any expected fully
explicit scheme as the density increment is directly obtained if we note that
M p ~p ˆ M u ~q …3:80†
With the above simpli®cations and Split A we can return to the original equations
and using the Galerkin approximation. We can therefore write directly
~ ˆÿM ÿ1
…
u t N T @Fi ‡
@xi @Gi d ÿ
@xi 1
…
2 t N T n
D d …3:81†
omitting the source terms for clarity (Fi and Gi are explained in Chapter 1, Eq. (1.25))
and noting that now ~ denotes all the variables. The added stabilizing terms D are
de®ned below and have to be integrated by parts in the usual manner.
@
2 1
2
p
@xi@x i
@ @
ui …uj u
@xi @x
1†‡
j
@p
@x1 @ @
ui …uj u
@xi @x
2†‡
j
@p
@x2 @ @
ui …uj u
@xi @x
3†‡
j
@p
8
9
><
>=
…3:82†
@x3 @ @
>: ui …uj E ‡ ujp† >;
@x i
@x j
The added `di€usions' are simple and are streamline oriented, and thus do not mask
the true e€ects of viscosity as happens in some schemes (e.g. the Taylor±Galerkin
process).