Page 142 - Introduction to Computational Fluid Dynamics

P1: IWV
CB908/Date
0521853265c05
5.3 METHOD OF SOLUTION
ADI Method 0 521 85326 5 May 20, 2005 12:28 121
The ADI method is a line-by-line method in which Equation 5.65 is first solved for
all j = constant lines (say). This is called the j-direction sweep. The solution thus
obtained may be called the l+1/2 solution. Now, using this solution, Equation 5.65
is again solved for i = constant lines to generate the l+1 solution. This is called
the i-direction sweep. The implementation details are as follows. For the j sweep,
Equation 5.65 is written as
l+1/2 l+1/2 l+1/2
(AP i, j + Sp i, j ) = AE i, j + AW i, j + SJ i, j , (5.66)
i, j i+1, j i−1, j
where

SJ i, j = AN i, j l + AS i, j l + Su i, j . (5.67)
i, j+1 i, j−1
Now, dividing by coefficient of i, j , Equation 5.66 for fixed j can also be written as

l+1/2 l+1/2 l+1/2
= a i + b i + c i , i = 2,..., IN − 1, (5.68)
i i+1 i−1
where a i = AE i, j /(AP i, j + Sp i, j ), b i = AW i, j /(AP i, j + Sp i, j ), and c i = SJ i, j /
(AP i, j + Sp i, j ).
It is clear that Equation 5.68 can be solved using TDMA for each j = 2to
JN − 1 to complete the j sweep. To execute the i sweep, Equation 5.65 is again
written as
l+1 l+1 l+1
(AP i, j + Sp i, j ) = AN i, j + AS i, j + SI i, j , (5.69)
i, j i, j+1 i, j−1
where
l+1/2 l+1/2
SI i, j = AE i, j + AW i, j + Su i, j . (5.70)
i+1, j i−1, j
Equation 5.69 can again be cast in the form of Equation 5.68 and subsequently
solved for each i = constant line by TDMA. The two sweeps complete one iteration.
Thus, in the ADI method, the domain is swept twice per iteration. In spite of this,
the procedure proves to be much faster than the GS procedure. In Chapter 9, some
additional methods for convergence enhancement are described.

5.3.2 Evaluation of Residuals

The convergence of the iterative procedure is checked by evaluating the imbalance
in Equation 5.12. Thus, for each , we evaluate

⎡ ⎤ 0.5
2

R = ⎣ AP P − A k k − D ⎦ . (5.71)
all nodes k
When the maximum value of R among all s is less than the convergence criterion
−5
(typically 10 ), the iteration is stopped. Often, R is normalized with a reference
quantity specific to a problem having units of AP .