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

where again M is the standard mass matrix, f are the prescribed `forces' and
…
PC… †ˆ N T @Fi @xi d …2:131a†
represents the convective `forces', while
…
PD… †ˆ
N T @G i
@x i
are the di€usive ones.
If an explicit time integration scheme is used, i.e.
d …2:131b†
M M…~ n ‡ 1 ÿ ~ n †ˆ tw n …~ n † …2:132†
the evaluation of the right-hand side does not require the matrix product representation
and Ai does not have to be computed.
Of course the scheme presented is not accurate for the various reasons previously
discussed, and indeed becomes unconditionally unstable in the absence of di€usion and
external force vectors.
The reader can easily verify that in the case of the linear one-dimensional problem
the right-hand side is equivalent to a central di€erence scheme with ~ n i ÿ 1 and ~ n i ‡ 1
n ‡ 1
only being used to ®nd the value of i , as shown in Fig. 2.22(a).
The scheme can, however, be recast as a two-step, predictor±corrector operation
and conditional stability is regained. Now we proceed as follows:
Step 1. Compute ~ n ‡ 1=2 using an explicit approximation of Eq. (2.132), i.e.
~ n ‡ 1=2 ˆ ~ n ‡ t
2 Mÿ1w n
…2:133†
t
t n + ∆t
t n
(a) Single-step explicit
t
t n + ∆t
i – 1 i i + 1
t
i – 1 i i + 1
n
(b) Standard predictor–corrector
t
t n + ∆t
t
i – 1 i i + 1
n
(c) Local prediction–corrector
(c) (two-step Taylor–Galerkin)
Fig. 2.22 Progression of information in explicit one- and two-step schemes.
Vector-valued variables 55
x
t n + 1/2∆t
x
t n + 1/2∆t
x