Page 63 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

3.2 Mathematical Formulation of the Algorithm 49

(3) b 2 = 0, b 3 = 0

2
−bb ± bb − (4aa)cc
η A = , (3.70)
2aa
c 2 − b 3 η A
ξ A = , (3.71)
b 2
where
a 4 b 3
aa = , (3.72)
b 2

a 2 b 3 − a 4 c 2
bb = − b 3 , (3.73)
b 2
−a 2 c 2
cc = . (3.74)
b 2

3.2.2.4 Case 4: a 4 = 0, b 4 = 0
In this case, Eqs. (3.37) and (3.38) can be written in the following form:

∗
∗
∗
a ξ A + a η A + ξ A η A = c , (3.75)
2 3 1
∗
b ξ A + b η A + ξ A η A = c , (3.76)
∗
∗
2 3 2
where
a i b i
∗ ∗
a = , b = (i = 2, 3), (3.77)
i
i
a 4 b 4
c 1 c 2
∗ ∗
c = , c = . (3.78)
2
1
a 4 b 4
Subtracting Eq. (3.78) from Eq. (3.77) yields the following equation:
∗
∗
∗
(a − b )ξ A + (a − b )η A = c − c . (3.79)
∗
∗
∗
2 2 3 3 1 2
Clearly, if a − b = 0, the solutions to Eqs. (3.77) and (3.78) can be straightfor-
∗
∗
2
2
wardly expressed as
c − c ∗ 2
∗
1
η A = , (3.80)
∗
a − b ∗
3 3
∗ ∗
c − a η A
1
3
ξ A = . (3.81)
∗
a + η A
2