Page 285 - Introduction to Computational Fluid Dynamics

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CONVERGENCE ENHANCEMENT
Similarly, evaluating i−1 from Equation 9.23 and substituting in Equation 9.18,
we have
∗
i = β 1i i+1 + β 2i + β 3i , (9.26)
i
where
ae ∗ i
β 1i = ,
∗
1 − aw A i−1
i
aw B i−1 + as i ∗
∗
i
β 2i = ,
∗
1 − aw A i−1
i
aw C i−1 + b ∗
∗
β 3i = i i . (9.27)
∗
1 − aw A i−1
i
If we now substitute Equation 9.26 in Equation 9.24, then comparison with Equa-
tion 9.22 will show that
α 1i
∗
A = ,
i
1 − α 2i β 2i
α 2i β 1i
∗
B = ,
i
1 − α 2i β 2i
α 2i β 3i + α 3i
∗
C = . (9.28)
i
1 − α 2i β 2i
Similarly, substituting Equation 9.24 in Equation 9.26 and comparison with
Equation 9.23 will show that
β 1i
A i = ,
1 − α 2i β 2i
β 2i α 1i
B i = ,
1 − α 2i β 2i
β 2i α 3i + β 3i
C i = . (9.29)
1 − α 2i β 2i
The overall two-line TDMA procedure is thus as follows:
∗
1. Consider j and j = j + 1 lines.
∗
∗
2. Form as, a s, d, and d according to Equation 9.16 for i = 2, 3,..., IN − 1.
3. Form b i and b from Equations 9.20 and 9.21 for i = 2, 3,..., IN − 1.
∗
i
4. Evaluate αs, βs, As, Bs, and Cs for i = 2, 3,..., IN − 1 by recurrence. Note
that A = B = C = A 1 = B 1 = C 1 = 0.
∗
∗
∗
1 1 1
5. Hence solve Equations 9.22 and 9.23 by back substitution (i.e., i = IN − 1
to 2).
∗
6. Set i, j = and i, j+1 = i .
i