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

of course the modi®cation of the basic equations to a self-adjoint form given in
Sec. 2.2.4 produces the full justi®cation of the special weighting. Which of the
procedures is best used in practice is largely a matter of taste, as all can give excellent
results. However, we shall see from the second part of this chapter, in which transient
problems are dealt with, that other methods can be adopted if time-stepping
procedures are used as an iteration to derive steady-state algorithms.
Indeed most of these procedures will again result in the addition of a di€usion term
in which the parameter is now replaced by another one involving the length of the
time step t. We shall show at the end of the next section a comparison between
various procedures for stabilization and will note essentially the same forms in the
steady-state situation.
In the last part of this chapter (Part III) we shall address the case in which the
unknown is a vector variable. Here only a limited number of procedures described
in the ®rst two parts will be available and even so we do not recommend in general the
use of such methods for vector-valued functions.
Before proceeding further it is of interest to consider the original equation with a
source term proportional to the variable , i.e. writing the one-dimensional equation
(2.11) as
U d d
ÿ
dx dx
k d
dx
‡ m ‡ Q ˆ 0 …2:57†
Equations of this type will arise of course from the transient Eq. (2.10) if we assume
the solution to be decomposed into Fourier components, writing for each component
which on substitution gives
Q ˆ Q e i!t
U d d
ÿ
dx dx
k d
dx
ˆ e i!t
Steady state ± concluding remarks 31
…2:58†
‡ i! ‡ Q ˆ 0 …2:59†
in which can be complex.
The use of Petrov±Galerkin or similar procedures on Eq. (2.57) or (2.59) can again
be made. If we pursue the line of approach outlined in Sec. 2.2.4 we note that
(a) the function p required to achieve self-adjointness remains unchanged;
and hence
(b) the weighting applied to achieve optimal results (see Sec. 2.2.3) again remains
unaltered ± providing of course it is applied to all terms.
Although the above result is encouraging and permits the solution in the frequency
domain for transient problems, it does not readily `transplant' to problems in which
time-stepping procedures are required.
Some further points require mentioning at this stage. These are simply that:
1. When pure convection is considered (that is k ˆ 0) only one boundary condition ±
generally that giving the value of at the inlet ± can be speci®ed, and in such a case
the violent oscillations observed in Fig. 2.2 with standard Galerkin methods will
not occur generally.