Page 61 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

3.2 Mathematical Formulation of the Algorithm 47

Since Eqs. (3.37) and (3.38) are two second-order simultaneous equations, the
solution to the local coordinates, ξ A and η A , can be obtained in the following differ-
ent cases.

3.2.2.1 Case 1: a 4 = 0, b 4 = 0
In this case, Eqs. (3.37) and (3.38) can be written as

a 2 ξ A + a 3 η A = c 1 , (3.47)
b 2 ξ A + b 3 η A = c 2 , (3.48)

where
c 1 = 4x A − a 1 , (3.49)

c 2 = 4y A − b 1 . (3.50)
Clearly, Eqs. (3.47) and (3.48) are a set of standard linear simultaneous equations,
so that the corresponding solutions can be immediately obtained as

b 3 c 1 − a 3 c 2
ξ A = , (3.51)
a 2 b 3 − a 3 b 2
−b 2 c 1 + a 2 c 2
η A = . (3.52)
a 2 b 3 − a 3 b 2

3.2.2.2 Case 2: a 4 = 0, b 4 = 0
The corresponding equations in this case can be rewritten as

a 2 ξ A + a 3 η A = c 1 , (3.53)

b 2 ξ A + b 3 η A + b 4 ξ A η A = c 2 . (3.54)

The solutions to Eqs. (3.53) and (3.54) can be expressed for the following three
sub-cases:

(1) a 2 = 0, a 3 = 0
c 1
η A = , (3.55)
a 3
c 2 − b 3 η A
ξ A = . (3.56)
b 2 + b 4 η A
(2) a 2 = 0, a 3 = 0
c 1
ξ A = , (3.57)
a 2