Page 283 - Introduction to Computational Fluid Dynamics

P2: IWV
P1: IWV/KCX
15:41
May 11, 2005
CB908/Date
0521853265c09
262
and 0 521 85326 5 CONVERGENCE ENHANCEMENT
JN−1
BS i = AE i, j l + AW i, j l + AN i, j l
i+1, j i−1, j i, j+1
j=2
+ AS i, j l i, j−1 + Su i, j − AP i, j l i, j . (9.12)
It will be recognized that the quantity inside the summation in Equation 9.12 is
simply −R (see Equation 9.2) at iteration level l. Further, Equation 9.10 can be
i, j
easily solved by TDMA. In this equation, BE IN−1 = BW 2 = 0 (see Equation 9.7)
and hence IN and 1 are not needed. A similar exercise in the j direction will
result in an equation for j .
The overall procedure is as follows:
1. Solve Equation 9.5 once using ADI to arrive at the l field.
i, j
2. Form the B coefficients in Equation 9.10 and solve this equation by TDMA to
yield i corrections. Reset i, j according to Equation 9.8.
3. Repeat step 2 to yield j corrections and reset i, j again.
4. Return to step 1 if the convergence criterion is not satisfied.

The block-correction procedure generally produces considerably faster con-
vergence than the ADI method but, in certain circumstances, it may produce an
erroneous solution or even divergence. Such a circumstance may arise when is
highly nonuniform and i or j may produce over- or undercorrections. There-
fore, the block-correction procedure may be treated as an optional convergence
enhancement device.

9.3 Method of Two Lines
In the ADI method, two sweeps are alternately executed in i and j directions (see
Chapter 5). Within each sweep, however, the TDMA is executed only along a
single line so that values of that line are updated simultaneously. To enhance
the convergence rate, it is possible to devise a TDMA procedure for two, three, or
multiple lines. By way of illustration, we consider the method of two lines [56, 21]
in which the following definition is introduced:
∗ i, j+1 = i, j . (9.13)
Consider lines j and j + 1 for the sweep in the i direction. The discretised equations
along these lines will read as

AP i, j ∗ i, j+1 = AE i, j ∗ i+1, j+1 + AW i, j ∗ i−1, j+1
+ AN i, j i, j+1 + AS i, j i, j−1 + Su i, j , (9.14)

AP i, j+1 i, j+1 = AE i, j+1 i+1, j+1 + AW i, j+1 i−1, j+1
+ AN i, j+1 i, j+2 + AS i, j+1 ∗ + Su i, j+1 . (9.15)
i, j+1