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

54 Convection dominated problems
This can be written in a compact matrix form similar to Eq. (2.93) as
in which, with
M ~ ˆÿ t‰…C ‡ K u ‡ K† ~ ‡ fŠ n
@
Gi ˆÿkij @xj …2:129a†
we have (on omitting the third derivative terms and the e€ect of S) matrices of the
form of Eq. (2.94), i.e.
…
C ˆ N T @N
Ai d
@xi … T
@N
Ku ˆ
@xi AiAj t
2
@N
d
@xj … T
@N
K ˆ
…
f ˆ
…
M ˆ
@x i
@N
d
kij @xj N T ‡ t
2 A @N
i
T
@xi N T N d
Q d ‡ boundary terms
…2:129b†
With ˆ 1 3 it can be shown that the order of approximation increases and for this
scheme a simple iterative solution is possible. 71 We note that with the consistent
mass matrix M the stability limit for ˆ 1
3 is increased to C ˆ 1.
Use of ˆ 1
3 apparently requires an implicit solution. However, similar iteration to
that used in Eq. (2.104) is rapidly convergent and the scheme can be used quite
economically.
2.10.2 Two-step predictor±corrector methods. Two-step
Taylor±Galerkin operation
There are of course various alternative procedures for improving the temporal
approximation other than the Taylor expansion used in the previous section. Such
procedures will be particularly useful if the evaluation of the derivative matrix A
can be avoided. In this section we shall consider two predictor±corrector schemes
(of Runge±Kutta type) that avoid the evaluation of this matrix and are explicit.
The ®rst starts with a standard Galerkin space approximation being applied to the
basic equation (2.1). This results in the form
M d~
dt M _~ ˆ P C ‡ P D ‡ f ˆ w …2:130†