Page 289 - Introduction to Computational Fluid Dynamics

P2: IWV
P1: IWV/KCX
CB908/Date
0 521 85326 5
0521853265c09
268
CONVERGENCE ENHANCEMENT
and the structure of the product matrix will take the form May 11, 2005 15:41
|U||L|| |=|A + D|| |=|Su|. (9.35)
This structure is clearly not the same as that of Equation 9.32 because matrix A
is augmented by D. Stone, however, postulated that Equation 9.34 will be a good
approximation to Equation 9.32 if the following substitutions are made:
N−1+IN = α s ( N−1 + N+IN − N ), (9.36)

N+1−IN = α s ( N+1 + N−IN − N ), (9.37)

where 0 <α s < 1 is an arbitrary constant to be chosen by the analyst. Making the
above substitutions in Equation 9.34 gives

[CP N + α s (CNW N + CSE N )] N

= (CE N + α s CSE N ) N+1 + (CW N + α s CNW N ) N−1
+ (CN N + α s CNW N ) N+IN + (CS N + α s CSE N ) N−IN + Su N .
(9.38)

Equation 9.38 now has the same structure as Equation 9.31. Therefore, replacing
the Cs in Equation 9.38 via Equations 9.33 and comparing the coefficients with
those in Equation 9.31, we can show that

BE N =−AE N /(1 + α s BS N+1 ), (9.39)
BN N =−AN N /(1 + α s BW N+IN ), (9.40)

BP N = AP N + α s (BN N BW N+IN + BE N BS N+IN )
− (BE N BW N+1 + BN N BS N+IN ), (9.41)

BW N =−(AW N + α s BN N BW N+IN )/ BP N , (9.42)
BS N =−(AS N + α s BE N BS N+1 )/ BP N . (9.43)

Now, it is expected that the product matrix will be a close approximation to the A
matrix (i.e., D → 0). In actual solving, therefore, the product matrix equation is
written as

l
l
|A + D|| l+1 |=|A + D|| |+|Su|−|A|| |. (9.44)
We now define

l
|δ|=| l+1 |−| |, (9.45)
l
|R|=−[|A|| |−|Su|], (9.46)