Page 282 - Introduction to Computational Fluid Dynamics

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9.2 BLOCK CORRECTION
9.2 Block Correction May 11, 2005 15:41 261
The block-correction technique is used to enhance the convergence rate of the ADI
method. Thus, we rewrite Equation 9.1 as
AP i, j i, j = AE i, j i+1, j + AW i, j i−1, j
+ AN i, j i, j+1 + AS i, j i, j−1 + Su i, j , (9.5)

where
AP i, j = AE i, j + AW i, j + AN i, j + AS i, j + Sp i, j . (9.6)

Equation 9.5 is written such that the boundary coefficients of the near-boundary
nodes are zero and the boundary conditions are absorbed through Su and Sp,as
explained in Chapter 5. Thus,

AW 2, j = AE IN−1, j = AS i,2 = AN i,JN−1 = 0. (9.7)
The central idea of the block-correction technique is that an unconverged field
l i, j is corrected by adding uniform correction i along lines of constant i. Thus,
let
i, j = l i, j + i . (9.8)

Now, the correction i is chosen such that the integral conservation over all control-
volumes on a constant-i strip is exactly satisfied. The equation governing i is thus
obtained by a two-step procedure. First, Equation 9.8 is substituted in Equation 9.5
so that

l
l
l
AP i, j i, j + i = AE i, j i+1, j + i+1 + AW i, j i−1, j + i−1

l
l
+ AN i, j + i + AS i, j + i
i, j+1 i, j−1
+ Su i, j . (9.9)
Then all such equations for j = 2, 3,...., JN − 1 are added. Thus, one obtains

BP i i = BE i i+1 + BW i i−1 + BS i , i = 2,..., IN − 1, (9.10)

where
JN−1

BP i = (AP i, j − AN i, j − AS i, j ),
j=2
JN−1

BE i = AE i, j ,
j=2
JN−1

BW i = AW i, j , (9.11)
j=2