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

This equation will be solved subsequently by an explicit time step applied to the
discretized form and a complete solution is now possible. The `correction' given
below is available once the pressure increment is evaluated:
From Eq. (3.1) we have
n ‡ 1
Ui ˆ Ui ÿ U n i ˆ Ui ÿ t @pn ‡ 2
ÿ
@xi t2
2 u @Q
k
n i
@xk ˆ 1
c 2
n
p ˆÿ t @Un ‡ 1
i
@xi ˆÿ t @Un i
@x i
@ Ui ‡ 1
@xi …3:24†
…3:25†
n ‡ 1
Replacing Ui by the known intermediate, auxiliary variable Ui and rearranging
after neglecting higher-order terms we have
ˆ 1
c 2
n
p ˆÿ t @Un i @ Ui @
‡ 1 ÿ t 1
@xi @xi 2 p n
@
‡ 2
@xi@xi 2 " ! #
p
@xi@xi …3:26†
where the U i and pressure terms in the above equation come from Eq. (3.24).
The above equation is fully self-adjoint in the variable p (or ) which is the
unknown. Now a standard Galerkin-type procedure can be optimally used for
spatial approximation. It is clear that the governing equations can be solved after
spatial discretization in the following order:
(a) Eq. (3.23) to obtain U i ;
(b) Eq. (3.26) to obtain p or ;
(c) Eq. (3.24) to obtain U i thus establishing the values at t n ‡ 1 .
After completing the calculation to establish U i and p (or ) the energy
equation is dealt with independently and the value of … E† n ‡ 1 is obtained by the
characteristic±Galerkin process applied to Eq. (3.6).
It is important to remark that this sequence allows us to solve the governing
equations (3.1), (3.4) and (3.6), in an e cient manner and with adequate numerical
damping. Note that these equations are written in conservation form. Therefore,
this algorithm is well suited for dealing with supersonic and hypersonic problems,
in which the conservation form ensures that shocks will be placed at the right position
and a unique solution achieved.
Split B
In this split we also introduce an auxiliary variable U i now retaining the known
values of Q n i ˆ @p n =@x i, i.e.
U i ˆ U i ÿ U n i
ˆ t ÿ @
…u
@x
jUi†‡ j
@ ij
‡
@xj @p
ÿ gi ‡
@xi t
2 uk Characteristic-based split (CBS) algorithm 69
@
@x k
@
@x j
…u jU i†ÿQ ‡ g i
n
…3:27†