Page 139 - Computational Fluid Dynamics for Engineers

4.5 Finite-Difference Methods for Elliptic Equations 125

• known iterate
•¥ o •• (old value)
• o • 0 next iterate to
• + O + | f + + + + + + + | be determined
(new value)
• O • y t ° + | > O O O O O O O o | |
• + o+ I , + + + + + + , I • known value at
boundary

(a) (b) i (c)

Fig. 4.8. Finite-difference grid and some iterative methods for solving elliptic problems,
(a) Point iteration, (b) block iteration by column, (c) block iteration by row.

Another alternative to solving Eq. (4.5.4) is the alternating direction implicit
(ADI) method, first introduced for time-dependent problems by Peaceman and
Rachford [4]. There are several versions of this method, but in principle they
all alternatively solve rows and columns implictly in a block-iteration scheme.
To solve Eq. (4.5.1) with the ADI scheme, let (3 = Ax/Ay and write Eq.
(4.5.4a) as
2
2
- [ui-ij - 2uij + Ui+ij] - (3 [uij-i - 2uij + Uij+i] = -(Ax) fij (4.5.34)
On "solving" for each bracketed term, this equation can be written identically
as
—Ui-ij + 2uij - Ui+ij + uuij
2
2
= uuij + j3 (uij-i - 2uij + Uij+i) - (Ax) fij (4.5.35a)
and

2
2
2
-0 Ui-ij + 2/3 Uij - f3 Ui+ij + uuij
2
= cjuij + (ui-ij - 2uij + Ui+ij) - (Ax) fij (4.5.35b)
where u is a relaxation parameter. The Peaceman-Rachford ADI iteration is
then defined by the equations which have a tridiagonal structure,
1 2
+ /2
-<_tf + ^ 2)<; / -< i
+
+
2
= umfj + /3 (u^_ x - 2u?j + v? J+1) - (Axfhj (4.5.36a)
and

2
= uu%V> + «Si 2 " t# 1 / 2 + ^ j ) - {Ax? kj (4.5.36b)
(
2
t
where the n-th iterative values u™ are assumed to be calculated at all grid points
from an arbitrary initial approximation u\ • and known boundary values. A new