Page 129 - Computational Fluid Dynamics for Engineers

4.5 Finite-Difference Methods for Elliptic Equations 115

4.5.1 Direct Methods

The linear equations given by Eq. (4.5.4) have a block tridiagonal structure and
can be written in vector-matrix form given by Eq. (4.4.29). However, in this
case, due to slightly different notation, we write Eq. (4.4.29) again,

AU = F (4.5.6)
Here A a denotes the coefficient matrix same as that defined in Eq. (4.4.30),
but with different indices,
i Ci
2 A2 C2

Bj Aj c 3 (4.5.7)

Bj-i CJ-I
Bj Aj

and with Aj, Bj and Cj denoting /-dimensional matrices and Ij denoting the
identity matrix of order I

-0 X 1 — 6 X
—6 X 1 — 6 X
(4.5.8a)

1
9 X

Bj = Cj = -9 yIi (4.5.8b)

In Eq. (4.5.6), U and F are 3-dimensional compound vectors (i.e., vectors
whose components are /-dimensional vectors) and are denned by

"1 «1J Fi
U2 U %3 F 2

U = , Uj = , F = (4.5.9)
Uj Ui,j F J

u
UJ u Fj
where
9
-^1 = /1 + ^ i + yVo
F 2 < j < J - 1 (4.5.10)
i = /,• + VxWj
~3
Fj = fj + 0 xwj + 6 yu J+l